Business and Economic Forecasting

The Questions are listed in the Word File

9 781292 021270
ISBN 978-1-29202-127-0
Using Econometrics
A Practical Guide
A.H. Studenmund
Sixth Edition
U
sing Econom
etrics Studenm
und Sixth Edition

Using Econometrics
A Practical Guide
A.H. Studenmund
Sixth Edition

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Table of Contents
P E A R S O N C U S T O M L I B R A R Y
I
1. An Overview of Regression Analysis
1
1A. H. Studenmund
2. Ordinary Least Squares
35
35A. H. Studenmund
3. Learning to Use Regression Analysis
71
71A. H. Studenmund
4. The Classical Model
97
97A. H. Studenmund
5. Hypothesis Testing
127
127A. H. Studenmund
6. Specification: Choosing the Independent Variables
177
177A. H. Studenmund
7. Specification: Choosing a Functional Form
219
219A. H. Studenmund
8. Multicollinearity
261
261A. H. Studenmund
9. Serial Correlation
321
321A. H. Studenmund
10. Running Your Own Regression Project
357
357A. H. Studenmund
11. Time-Series Models
389
389A. H. Studenmund
12. Dummy Dependent Variable Techniques
417
417A. H. Studenmund
13. Simultaneous Equations
443
443A. H. Studenmund

II
14. Forecasting
483
483A. H. Studenmund
15. Statistical Principles
507
507A. H. Studenmund
Appendix: Statistical Tables
539
539A. H. Studenmund
555
555Index

1 What Is Econometrics?
2 What Is Regression Analysis?
3 The Estimated Regression Equation
4 A Simple Example of Regression Analysis
5 Using Regression to Explain Housing Prices
6 Summary and Exercises
An Overview of
Regression Analysis
What Is Econometrics?
“Econometrics is too mathematical; it’s the reason my best friend isn’t
majoring in economics.”
“There are two things you are better off not watching in the making:
sausages and econometric estimates.”1
“Econometrics may be defined as the quantitative analysis of actual eco-
nomic phenomena.”2
“It’s my experience that ‘economy-tricks’ is usually nothing more than a
justification of what the author believed before the research was begun.”
Obviously, econometrics means different things to different people. To
beginning students, it may seem as if econometrics is an overly complex
obstacle to an otherwise useful education. To skeptical observers, econo-
metric results should be trusted only when the steps that produced those
1
1. Ed Leamer, “Let’s take the Con out of Econometrics,” American Economic Review, Vol. 73,
No. 1, p. 37.
2. Paul A. Samuelson, T. C. Koopmans, and J. R. Stone, “Report of the Evaluative Committee for
Econometrica,” Econometrica, 1954, p. 141.
From Chapter 1 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
1

results are completely known. To professionals in the field, econometrics is
a fascinating set of techniques that allows the measurement and analysis of
economic phenomena and the prediction of future economic trends.
You’re probably thinking that such diverse points of view sound like the
statements of blind people trying to describe an elephant based on what they
happen to be touching, and you’re partially right. Econometrics has both a
formal definition and a larger context. Although you can easily memorize the
formal definition, you’ll get the complete picture only by understanding the
many uses of and alternative approaches to econometrics.
That said, we need a formal definition. Econometrics—literally,“economic
measurement”—is the quantitative measurement and analysis of actual eco-
nomic and business phenomena. It attempts to quantify economic reality
and bridge the gap between the abstract world of economic theory and the
real world of human activity. To many students, these worlds may seem far
apart. On the one hand, economists theorize equilibrium prices based on
carefully conceived marginal costs and marginal revenues; on the other,
many firms seem to operate as though they have never heard of such con-
cepts. Econometrics allows us to examine data and to quantify the actions of
firms, consumers, and governments. Such measurements have a number of
different uses, and an examination of these uses is the first step to under-
standing econometrics.
Uses of Econometrics
Econometrics has three major uses:
1. describing economic reality
2. testing hypotheses about economic theory
3. forecasting future economic activity
The simplest use of econometrics is description. We can use economet-
rics to quantify economic activity because econometrics allows us to esti-
mate numbers and put them in equations that previously contained only
abstract symbols. For example, consumer demand for a particular com-
modity often can be thought of as a relationship between the quantity
demanded (Q) and the commodity’s price (P), the price of a substitute
good (Ps), and disposable income (Yd). For most goods, the relationship
between consumption and disposable income is expected to be positive,
because an increase in disposable income will be associated with an in-
crease in the consumption of the good. Econometrics actually allows us to
estimate that relationship based upon past consumption, income, and
AN OVERVIEW OF REGRESSION ANALYSIS
2

AN OVERVIEW OF REGRESSION ANALYSIS
prices. In other words, a general and purely theoretical functional relation-
ship like:
(1)
can become explicit:
(2)
This technique gives a much more specific and descriptive picture of the
function.3 Let’s compare Equations 1 and 2. Instead of expecting consump-
tion merely to “increase” if there is an increase in disposable income, Equa-
tion 2 allows us to expect an increase of a specific amount (0.23 units for
each unit of increased disposable income). The number 0.23 is called an esti-
mated regression coefficient, and it is the ability to estimate these coefficients
that makes econometrics valuable.
The second and perhaps most common use of econometrics is hypothesis
testing, the evaluation of alternative theories with quantitative evidence.
Much of economics involves building theoretical models and testing them
against evidence, and hypothesis testing is vital to that scientific approach.
For example, you could test the hypothesis that the product in Equation 1 is
what economists call a normal good (one for which the quantity demanded
increases when disposable income increases). You could do this by applying
various statistical tests to the estimated coefficient (0.23) of disposable in-
come (Yd) in Equation 2. At first glance, the evidence would seem to support
this hypothesis, because the coefficient’s sign is positive, but the “statistical
significance” of that estimate would have to be investigated before such a
conclusion could be justified. Even though the estimated coefficient is posi-
tive, as expected, it may not be sufficiently different from zero to convince us
that the true coefficient is indeed positive.
The third and most difficult use of econometrics is to forecast or predict
what is likely to happen next quarter, next year, or further into the future,
based on what has happened in the past. For example, economists use
econometric models to make forecasts of variables like sales, profits, Gross
Q 5 27.7 2 0.11P 1 0.03Ps 1 0.23Yd
Q 5 f(P, Ps , Yd)
3. The results in Equation 2 are from a model of the demand for chicken. It’s of course naïve to
build a model of the demand for chicken without taking the supply of chicken into considera-
tion. Unfortunately, it’s very difficult to learn how to estimate a system of simultaneous
equations until you’ve learned how to estimate a single equation. You should be aware that we
sometimes will encounter right-hand-side variables that are not truly “independent” from a
theoretical point of view.
3

AN OVERVIEW OF REGRESSION ANALYSIS
Domestic Product (GDP), and the inflation rate. The accuracy of such fore-
casts depends in large measure on the degree to which the past is a good guide
to the future. Business leaders and politicians tend to be especially interested
in this use of econometrics because they need to make decisions about the
future, and the penalty for being wrong (bankruptcy for the entrepreneur and
political defeat for the candidate) is high. To the extent that econometrics can
shed light on the impact of their policies, business and government leaders
will be better equipped to make decisions. For example, if the president of a
company that sold the product modeled in Equation 1 wanted to decide
whether to increase prices, forecasts of sales with and without the price in-
crease could be calculated and compared to help make such a decision.
Alternative Econometric Approaches
There are many different approaches to quantitative work. For example, the
fields of biology, psychology, and physics all face quantitative questions similar
to those faced in economics and business. However, these fields tend to use
somewhat different techniques for analysis because the problems they face
aren’t the same. For example, economics typically is an observational discipline
rather than an experimental one. “We need a special field called econometrics,
and textbooks about it, because it is generally accepted that economic data
possess certain properties that are not considered in standard statistics texts or
are not sufficiently emphasized there for use by economists.”4
Different approaches also make sense within the field of economics. The
kind of econometric tools used depends in part on the uses of that equation.
A model built solely for descriptive purposes might be different from a fore-
casting model, for example.
To get a better picture of these approaches, let’s look at the steps used in
nonexperimental quantitative research:
1. specifying the models or relationships to be studied
2. collecting the data needed to quantify the models
3. quantifying the models with the data
The specifications used in step 1 and the techniques used in step 3 differ
widely between and within disciplines. Choosing the best specification for a
given model is a theory-based skill that is often referred to as the “art” of
4. Clive Granger, “A Review of Some Recent Textbooks of Econometrics,” Journal of Economic Lit-
erature, Vol. 32, No. 1, p. 117.
4

AN OVERVIEW OF REGRESSION ANALYSIS
econometrics. There are many alternative approaches to quantifying the same
equation, and each approach may produce somewhat different results. The
choice of approach is left to the individual econometrician (the researcher
using econometrics), but each researcher should be able to justify that choice.
This text will focus primarily on one particular econometric approach:
single-equation linear regression analysis. The majority of this text will thus con-
centrate on regression analysis, but it is important for every econometrician
to remember that regression is only one of many approaches to econometric
quantification.
The importance of critical evaluation cannot be stressed enough; a good
econometrician can diagnose faults in a particular approach and figure out
how to repair them. The limitations of the regression analysis approach must
be fully perceived and appreciated by anyone attempting to use regression
analysis or its findings. The possibility of missing or inaccurate data, incor-
rectly formulated relationships, poorly chosen estimating techniques, or im-
proper statistical testing procedures implies that the results from regression
analyses always should be viewed with some caution.
What Is Regression Analysis?
Econometricians use regression analysis to make quantitative estimates of eco-
nomic relationships that previously have been completely theoretical in nature.
After all, anybody can claim that the quantity of compact discs demanded will
increase if the price of those discs decreases (holding everything else constant),
but not many people can put specific numbers into an equation and estimate by
how many compact discs the quantity demanded will increase for each dollar
that price decreases. To predict the direction of the change, you need a knowl-
edge of economic theory and the general characteristics of the product in ques-
tion. To predict the amount of the change, though, you need a sample of data,
and you need a way to estimate the relationship. The most frequently used
method to estimate such a relationship in econometrics is regression analysis.
Dependent Variables, Independent Variables, and Causality
Regression analysis is a statistical technique that attempts to “explain” move-
ments in one variable, the dependent variable, as a function of movements in
a set of other variables, called the independent (or explanatory) variables,
through the quantification of a single equation. For example, in Equation 1:
(1)Q 5 f(P, Ps , Yd)
2
5

Don’t be deceived by the words “dependent” and “independent,” how-
ever. Although many economic relationships are causal by their very na-
ture, a regression result, no matter how statistically significant, cannot
prove causality. All regression analysis can do is test whether a significant
quantitative relationship exists. Judgments as to causality must also in-
clude a healthy dose of economic theory and common sense. For exam-
ple, the fact that the bell on the door of a flower shop rings just before a
customer enters and purchases some flowers by no means implies that
the bell causes purchases! If events A and B are related statistically, it may
be that A causes B, that B causes A, that some omitted factor causes both,
or that a chance correlation exists between the two.
AN OVERVIEW OF REGRESSION ANALYSIS
Q is the dependent variable and P, Ps, and Yd are the independent vari-
ables. Regression analysis is a natural tool for economists because most
(though not all) economic propositions can be stated in such single-equation
functional forms. For example, the quantity demanded (dependent vari-
able) is a function of price, the prices of substitutes, and income (indepen-
dent variables).
Much of economics and business is concerned with cause-and-effect
propositions. If the price of a good increases by one unit, then the quantity
demanded decreases on average by a certain amount, depending on the price
elasticity of demand (defined as the percentage change in the quantity de-
manded that is caused by a one percent increase in price). Similarly, if the
quantity of capital employed increases by one unit, then output increases by
a certain amount, called the marginal productivity of capital. Propositions
such as these pose an if-then, or causal, relationship that logically postulates
that a dependent variable’s movements are determined by movements in a
number of specific independent variables.
The cause-and-effect relationship often is so subtle that it fools even the
most prominent economists. For example, in the late nineteenth century,
English economist Stanley Jevons hypothesized that sunspots caused an in-
crease in economic activity. To test this theory, he collected data on national
output (the dependent variable) and sunspot activity (the independent vari-
able) and showed that a significant positive relationship existed. This result
led him, and some others, to jump to the conclusion that sunspots did
indeed cause output to rise. Such a conclusion was unjustified because re-
gression analysis cannot confirm causality; it can only test the strength and
direction of the quantitative relationships involved.
6

AN OVERVIEW OF REGRESSION ANALYSIS
Single-Equation Linear Models
The simplest single-equation linear regression model is:
(3)
Equation 3 states that Y, the dependent variable, is a single-equation linear
function of X, the independent variable. The model is a single-equation
model because it’s the only equation specified. The model is linear be-
cause if you were to plot Equation 3 it would be a straight line rather than
a curve.
The s are the coefficients that determine the coordinates of the straight
line at any point. is the constant or intercept term; it indicates the value
of Y when X equals zero. is the slope coefficient, and it indicates the
amount that Y will change when X increases by one unit. The solid line in
Figure 1 illustrates the relationship between the coefficients and the graphical
meaning of the regression equation. As can be seen from the diagram, Equa-
tion 3 is indeed linear.
�1
�0
�
Y 5 �0 1 �1X
Y
Y2
Y1
0 X1
ΔX
ΔY
X2
Y = �0 + �1X
Y = �0 + �1X
2
X
�0
ΔY
ΔX
Slope = �1 =
Figure 1 Graphical Representation of the Coefficients
of the Regression Line
The graph of the equation is linear with a constant slope equal to
The graph of the equation on the other hand, is nonlin-
ear with an increasing slope (if �1 . 0).
Y 5 �0 1 �1X
2,�1 5 �Y>�X.
Y 5 �0 1 �1X
7

AN OVERVIEW OF REGRESSION ANALYSIS
The slope coefficient, , shows the response of Y to a one-unit increase
in X. Much of the emphasis in regression analysis is on slope coefficients
such as . In Figure 1 for example, if X were to increase by one from X1 to
X2 ( X), the value of Y in Equation 3 would increase from Y1 to Y2 ( Y).
For linear (i.e., straight-line) regression models, the response in the pre-
dicted value of Y due to a change in X is constant and equal to the slope
coefficient :
where is used to denote a change in the variables. Some readers may recog-
nize this as the “rise” ( Y) divided by the “run” ( X). For a linear model, the
slope is constant over the entire function.
If linear regression techniques are going to be applied to an equation, that
equation must be linear. An equation is linear if plotting the function in
terms of X and Y generates a straight line. For example, Equation 3:
(3)
is linear, but Equation 4:
(4)
is not linear, because if you were to plot Equation 4 it would be a quadratic,
not a straight line. This difference5 can be seen in Figure 1.
If regression analysis requires that an equation be linear, how can we deal
with nonlinear equations like Equation 4? The answer is that we can redefine
most nonlinear equations to make them linear. For example, Equation 4 can
be converted into a linear equation if we create a new variable equal to the
square of X:
Z � X2 (5)
and if we substitute Equation 5 into Equation 4:
Y � �0 � �1Z (6)
Y 5 �0 1 �1X
2
Y 5 �0 1 �1X
��
�
(Y
2
2 Y
1
)
(X2 2 X1)
5
�Y
�X
5 �1
�1
��
�1
�1
5. Equations 3 and 4 have the same in Figure 1 for comparison purposes only. If the equa-
tions were applied to the same data, the estimated values would be different. Not surpris-
ingly, the estimated �1 values would be different as well.
�0
�0
8

AN OVERVIEW OF REGRESSION ANALYSIS
This redefined equation is now linear6 and can be estimated by regression
analysis.
The Stochastic Error Term
Besides the variation in the dependent variable (Y) that is caused by the in-
dependent variable (X), there is almost always variation that comes from
other sources as well. This additional variation comes in part from omitted
explanatory variables (e.g., X2 and X3). However, even if these extra vari-
ables are added to the equation, there still is going to be some variation in
Y that simply cannot be explained by the model.7 This variation probably
comes from sources such as omitted influences, measurement error, incor-
rect functional form, or purely random and totally unpredictable occur-
rences. By random we mean something that has its value determined entirely
by chance.
Econometricians admit the existence of such inherent unexplained variation
(“error”) by explicitly including a stochastic (or random) error term in their re-
gression models. A stochastic error term is a term that is added to a regression
equation to introduce all of the variation in Y that cannot be explained by the
included Xs. It is, in effect, a symbol of the econometrician’s ignorance or in-
ability to model all the movements of the dependent variable. The error term
(sometimes called a disturbance term) usually is referred to with the symbol
epsilon ( ), although other symbols (like u or v) sometimes are used.
The addition of a stochastic error term ( ) to Equation 3 results in a typical
regression equation:
(7)Y 5 �0 1 �1X 1 �
�
�
6. Technically, this equation is linear in the coefficients �0 and �1 and linear in the variables Y
and Z, but it is nonlinear in the variables Y and X. The application of regression techniques to
equations that are nonlinear in the coefficients, however, is much more difficult.
7. The exception would be the extremely rare case where the data can be explained by some sort
of physical law and are measured perfectly. Here, continued variation would point to an omit-
ted independent variable. A similar kind of problem is often encountered in astronomy, where
planets can be discovered by noting that the orbits of known planets exhibit variations that can
be caused only by the gravitational pull of another heavenly body. Absent these kinds of physi-
cal laws, researchers in economics and business would be foolhardy to believe that all variation
in Y can be explained by a regression model because there are always elements of error in any
attempt to measure a behavioral relationship.
9

AN OVERVIEW OF REGRESSION ANALYSIS
Equation 7 can be thought of as having two components, the deterministic
component and the stochastic, or random, component. The expression
is called the deterministic component of the regression equation be-
cause it indicates the value of Y that is determined by a given value of X,
which is assumed to be nonstochastic. This deterministic component can
also be thought of as the expected value of Y given X, the mean value of the
Ys associated with a particular value of X. For example, if the average height of
all 13-year-old girls is 5 feet, then 5 feet is the expected value of a girl’s height
given that she is 13. The deterministic part of the equation may be written:
(8)
which states that the expected value of Y given X, denoted as is a linear
function of the independent variable (or variables if there are more than one).8
Unfortunately, the value of Y observed in the real world is unlikely to be
exactly equal to the deterministic expected value . After all, not all 13-
year-old girls are 5 feet tall. As a result, the stochastic element ( ) must be
added to the equation:
(9)Y 5 E(Y k X) 1 � 5 �0 1 �1X 1 �
�
E(Y k X)
E(Y k X),
E(Y k X) 5 �0 1 �1X
�0 1 �1X
8. This property holds as long as (read as “the expected value of epsilon, given X”
equals zero), which is true as long as the Classical Assumptions are met. It’s easiest to think of
as the mean of but the expected value operator E technically is a summation or integra-
tion of all the values that a function can take, weighted by the probability of each value. The ex-
pected value of a constant is that constant, and the expected value of a sum of variables equals
the sum of the expected values of those variables.
�,E(�)
E(� k X) 5 0
The stochastic error term must be present in a regression equation be-
cause there are at least four sources of variation in Y other than the varia-
tion in the included Xs:
1. Many minor influences on Y are omitted from the equation (for
example, because data are unavailable).
2. It is virtually impossible to avoid some sort of measurement error
in the dependent variable.
3. The underlying theoretical equation might have a different functional
form (or shape) than the one chosen for the regression. For example,
the underlying equation might be nonlinear.
4. All attempts to generalize human behavior must contain at least
some amount of unpredictable or purely random variation.
10

AN OVERVIEW OF REGRESSION ANALYSIS
To get a better feeling for these components of the stochastic error term,
let’s think about a consumption function (aggregate consumption as a func-
tion of aggregate disposable income). First, consumption in a particular year
may have been less than it would have been because of uncertainty over the
future course of the economy. Since this uncertainty is hard to measure, there
might be no variable measuring consumer uncertainty in the equation. In
such a case, the impact of the omitted variable (consumer uncertainty)
would likely end up in the stochastic error term. Second, the observed
amount of consumption may have been different from the actual level of
consumption in a particular year due to an error (such as a sampling error) in
the measurement of consumption in the National Income Accounts. Third,
the underlying consumption function may be nonlinear, but a linear con-
sumption function might be estimated. (To see how this incorrect functional
form would cause errors, see Figure 2.) Fourth, the consumption function at-
tempts to portray the behavior of people, and there is always an element of
Y
0
Errors
“True” Relationship
(nonlinear)
Linear Functional Form
X
�2
�1
�3
Figure 2 Errors Caused by Using a Linear Functional Form to Model
a Nonlinear Relationship
One source of stochastic error is the use of an incorrect functional form. For example, if a
linear functional form is used when the underlying relationship is nonlinear, systematic er-
rors will occur. These nonlinearities are just one component of the stochastic error
term. The others are omitted variables, measurement error, and purely random variation.
(the �s)
11

AN OVERVIEW OF REGRESSION ANALYSIS
unpredictability in human behavior. At any given time, some random event
might increase or decrease aggregate consumption in a way that might never
be repeated and couldn’t be anticipated.
These possibilities explain the existence of a difference between the ob-
served values of Y and the values expected from the deterministic component
of the equation, These sources of error will be covered to recognize
that in econometric research there will always be some stochastic or random
element, and, for this reason, an error term must be added to all regression
equations.
Extending the Notation
Our regression notation needs to be extended to allow the possibility of more
than one independent variable and to include reference to the number of obser-
vations. A typical observation (or unit of analysis) is an individual person, year,
or country. For example, a series of annual observations starting in 1985 would
have Y1 = Y for 1985, Y2 for 1986, etc. If we include a specific reference to the
observations, the single-equation linear regression model may be written as:
(10)
where: Yi � the ith observation of the dependent variable
Xi � the ith observation of the independent variable
� the ith observation of the stochastic error term
� the regression coefficients
N � the number of observations
This equation is actually N equations, one for each of the N observations:
That is, the regression model is assumed to hold for each observation. The
coefficients do not change from observation to observation, but the values of
Y, X, and do.
A second notational addition allows for more than one independent vari-
able. Since more than one independent variable is likely to have an effect on
�
YN 5 �0 1 �1XN 1 �N
(
Y3 5 �0 1 �1X3 1 �3
Y2 5 �0 1 �1X2 1 �2
Y1 5 �0 1 �1X1 1 �1
�0, �1
�i
Yi 5 �0 1 �1Xi 1 �i (i 5 1, 2, . . . , N)
E(Y k X).
12

The resulting equation is called a multivariate (more than one indepen-
dent variable) linear regression model:
(11)
The meaning of the regression coefficient in this equation is the im-
pact of a one-unit increase in X1 on the dependent variable Y, holding
constant X2 and X3. Similarly, gives the impact of a one-unit increase
in X2 on Y, holding X1 and X3 constant.
�2
�1
Yi 5 �0 1 �1X1i 1 �2X2i 1 �3X3i 1 �i
AN OVERVIEW OF REGRESSION ANALYSIS
the dependent variable, our notation should allow these additional explana-
tory Xs to be added. If we define:
X1i � the ith observation of the first independent variable
X2i � the ith observation of the second independent variable
X3i � the ith observation of the third independent variable
then all three variables can be expressed as determinants of Y.
These multivariate regression coefficients (which are parallel in nature to
partial derivatives in calculus) serve to isolate the impact on Y of a change in
one variable from the impact on Y of changes in the other variables. This is
possible because multivariate regression takes the movements of X2 and X3
into account when it estimates the coefficient of X1. The result is quite similar
to what we would obtain if we were capable of conducting controlled labora-
tory experiments in which only one variable at a time was changed.
In the real world, though, it is very difficult to run controlled economic ex-
periments,9 because many economic factors change simultaneously, often in
opposite directions. Thus the ability of regression analysis to measure the im-
pact of one variable on the dependent variable, holding constant the influence
of the other variables in the equation, is a tremendous advantage. Note that if a
variable is not included in an equation, then its impact is not held constant in
the estimation of the regression coefficients.
9. Such experiments are difficult but not impossible.
13

AN OVERVIEW OF REGRESSION ANALYSIS
This material is pretty abstract, so let’s look at an example. Suppose we
want to understand how wages are determined in a particular field, perhaps
because we think that there might be discrimination in that field. The wage
of a worker would be the dependent variable (WAGE), but what would be
good independent variables? What variables would influence a person’s wage
in a given field? Well, there are literally dozens of reasonable possibilities,
but three of the most common are the work experience (EXP), education
(EDU), and gender (GEND) of the worker, so let’s use these. To create a re-
gression equation with these variables, we’d redefine the variables in Equa-
tion 11 to meet our definitions:
Y � WAGE � the wage of the worker
X1 � EXP � the years of work experience of the worker
X2 � EDU � the years of education beyond high school of the worker
X3 � GEND � the gender of the worker (1 � male and 0 � female)
The last variable, GEND, is unusual in that it can take on only two values, 0
and 1; this kind of variable is called a dummy variable, and it’s extremely
useful when we want to quantify a concept that is inherently qualitative
(like gender).
If we substitute these definitions into Equation 11, we get:
WAGEi � �0 � �1EXPi � �2EDUi � �3GENDi � i (12)
Equation 12 specifies that a worker’s wage is a function of the experience,
education, and gender of that worker. In such an equation, what would the
meaning of �1 be? Some readers will guess that �1 measures the amount by
which the average wage increases for an additional year of experience, but
such a guess would miss the fact that there are two other independent vari-
ables in the equation that also explain wages. The correct answer is that �1
gives us the impact on wages of a one-year increase in experience, holding con-
stant education and gender. This is a significant difference, because it allows
researchers to control for specific complicating factors without running con-
trolled experiments.
Before we conclude this section, it’s worth noting that the general multi-
variate regression model with K independent variables is written as:
(13)
where i goes from 1 to N and indicates the observation number.
Yi 5 �0 1 �1X1i 1 �2X2i 1
c1 �KXKi 1 �i
�
14

AN OVERVIEW OF REGRESSION ANALYSIS
10. The order of the subscripts doesn’t matter as long as the appropriate definitions are pre-
sented. We prefer to list the variable number first (X1i) because we think it’s easier for a begin-
ning econometrician to understand. However, as the reader moves on to matrix algebra and
computer spreadsheets, it will become common to list the observation number first, as in Xi1.
Often the observational subscript is deleted, and the reader is expected to understand that the
equation holds for each observation in the sample.
11. Our use of the word ”true” throughout the text should be taken with a grain of salt. Many
philosophers argue that the concept of truth is useful only relative to the scientific research pro-
gram in question. Many economists agree, pointing out that what is true for one generation
may well be false for another. To us, the true coefficient is the one that you’d obtain if you could
run a regression on the entire relevant population. Thus, readers who so desire can substitute
the phrase “population coefficient” for “true coefficient” with no loss in meaning.
If the sample consists of a series of years or months (called a time series),
then the subscript i is usually replaced with a t to denote time.10
The Estimated Regression Equation
Once a specific equation has been decided upon, it must be quantified. This
quantified version of the theoretical regression equation is called the
estimated regression equation and is obtained from a sample of data for ac-
tual Xs and Ys. Although the theoretical equation is purely abstract in nature:
(14)
the estimated regression equation has actual numbers in it:
(15)
The observed, real-world values of X and Y are used to calculate the coeffi-
cient estimates 103.40 and 6.38. These estimates are used to determine
(read as “Y-hat”), the estimated or fitted value of Y.
Let’s look at the differences between a theoretical regression equation and
an estimated regression equation. First, the theoretical regression coefficients
in Equation 14 have been replaced with estimates of those coeffi-
cients like 103.40 and 6.38 in Equation 15. We can’t actually observe the val-
ues of the true11 regression coefficients, so instead we calculate estimates of
those coefficients from the data. The estimated regression coefficients,
more generally denoted by (read as “beta-hats”), are empirical best
guesses of the true regression coefficients and are obtained from data from a
sample of the Ys and Xs. The expression
(16)Ŷi 5 �̂0 1 �̂1Xi
�̂0 and �̂1
�0 and �1
Ŷ
Ŷi 5 103.40 1 6.38Xi
Yi 5 �0 1 �1Xi 1 �i
3
15

AN OVERVIEW OF REGRESSION ANALYSIS
is the empirical counterpart of the theoretical regression Equation 14. The
calculated estimates in Equation 15 are examples of the estimated regression
coefficients For each sample we calculate a different set of esti-
mated regression coefficients.
is the estimated value of Yi, and it represents the value of Y calculated
from the estimated regression equation for the ith observation. As such, is
our prediction of from the regression equation. The closer these s
are to the Ys in the sample, the better the fit of the equation. (The word fit is
used here much as it would be used to describe how well clothes fit.)
The difference between the estimated value of the dependent variable
and the actual value of the dependent variable (Yi) is defined as the residual (ei):
(Ŷi)
ŶE(Yi k Xi)
Ŷi
Ŷi
�̂0 and �̂1.
(17)ei 5 Yi 2 Ŷi
Note the distinction between the residual in Equation 17 and the error
term:
(18)
The residual is the difference between the observed Y and the estimated re-
gression line while the error term is the difference between the observed
Y and the true regression equation (the expected value of Y). Note that the
error term is a theoretical concept that can never be observed, but the resid-
ual is a real-world value that is calculated for each observation every time a
regression is run. The residual can be thought of as an estimate of the error
term, and e could have been denoted as Most regression techniques not
only calculate the residuals but also attempt to compute values of
that keep the residuals as low as possible. The smaller the residuals, the better
the fit, and the closer the will be to the Ys.
All these concepts are shown in Figure 3. The (X, Y) pairs are shown as
points on the diagram, and both the true regression equation (which cannot
be seen in real applications) and an estimated regression equation are in-
cluded. Notice that the estimated equation is close to but not equivalent to
the true line. This is a typical result.
In Figure 3, the computed value of Y for the sixth observation, lies on
the estimated (dashed) line, and it differs from Y6, the actual observed value
of Y for the sixth observation. The difference between the observed and esti-
mated values is the residual, denoted by e6. In addition, although we usually
would not be able to see an observation of the error term, we have drawn the
Ŷ6,
Ŷs
�̂0 and �̂1
�̂.
(Ŷ),
�i 5 Yi 2 E(Yi k Xi)
16

AN OVERVIEW OF REGRESSION ANALYSIS
assumed true regression line here (the solid line) to see the sixth observation
of the error term, which is the difference between the true line and the ob-
served value of Y, Y6.
The following table summarizes the notation used in the true and esti-
mated regression equations:
True Regression Equation Estimated Regression Equation
The estimated regression model can be extended to more than one inde-
pendent variable by adding the additional Xs to the right side of the equa-
tion. The multivariate estimated regression counterpart of Equation 13 is:
(19)Ŷi 5 �̂0 1 �̂1X1i 1 �̂2X2i 1
c1 �̂KXKi
ei�i
�̂1�1
�̂0�0
�6,
�6
Y
Y6
0 X6
Yi = �0 + �1Xi
(Estimated Line)
E(Yi|Xi) = �0 + �1Xi
(True Line)
X
�0
Y6
e6
e6
�0
Figure 3 True and Estimated Regression Lines
The true relationship between X and Y (the solid line) typically cannot be observed, but
the estimated regression line (the dashed line) can. The difference between an observed
data point (for example, i = 6) and the true line is the value of the stochastic error term
The difference between the observed Y6 and the estimated value from the regres-
sion line is the value of the residual for this observation, e6.(Ŷ6)
(�6).
17

AN OVERVIEW OF REGRESSION ANALYSIS
Diagrams of such multivariate equations, by the way, are not possible for
more than two independent variables and are quite awkward for exactly two
independent variables.
A Simple Example of Regression Analysis
Let’s look at a fairly simple example of regression analysis. Suppose you’ve
accepted a summer job as a weight guesser at the local amusement park,
Magic Hill. Customers pay two dollars each, which you get to keep if you
guess their weight within 10 pounds. If you miss by more than 10 pounds,
then you have to return the two dollars and give the customer a small prize
that you buy from Magic Hill for three dollars each. Luckily, the friendly
managers of Magic Hill have arranged a number of marks on the wall behind
the customer so that you are capable of measuring the customer’s height accu-
rately. Unfortunately, there is a five-foot wall between you and the customer,
so you can tell little about the person except for height and (usually) gender.
On your first day on the job, you do so poorly that you work all day and
somehow manage to lose two dollars, so on the second day you decide to
collect data to run a regression to estimate the relationship between weight
and height. Since most of the participants are male, you decide to limit your
sample to males. You hypothesize the following theoretical relationship:
(20)
where: Yi � the weight (in pounds) of the ith customer
Xi � the height (in inches above 5 feet) of the ith customer
� the value of the stochastic error term for the ith customer
In this case, the sign of the theoretical relationship between height and
weight is believed to be positive (signified by the positive sign above Xi in the
general theoretical equation), but you must quantify that relationship in
order to estimate weights given heights. To do this, you need to collect a data
set, and you need to apply regression analysis to the data.
The next day you collect the data summarized in Table 1 and run your re-
gression on the Magic Hill computer, obtaining the following estimates:
This means that the equation
(21)Estimated weight 5 103.40 1 6.38 ? Height (inches above five feet)
�̂0 5 103.40 �̂1 5 6.38
�i
Yi 5 f( X
1
i) 1 �i 5 �0 1 �1Xi 1 �i
4
18

Table 1 Data for and Results of the Weight-Guessing Equation
Observation Height Weight Predicted Residual $ Gain or
i Above 5’ Xi Yi Weight ei Loss
(1) (2) (3) (4) (5) (6)
1 5.0 140.0 135.3 4.7 �2.00
2 9.0 157.0 160.8 �3.8 �2.00
3 13.0 205.0 186.3 18.7 �3.00
4 12.0 198.0 179.9 18.1 �3.00
5 10.0 162.0 167.2 �5.2 �2.00
6 11.0 174.0 173.6 0.4 �2.00
7 8.0 150.0 154.4 �4.4 �2.00
8 9.0 165.0 160.8 4.2 �2.00
9 10.0 170.0 167.2 2.8 �2.00
10 12.0 180.0 179.9 0.1 �2.00
11 11.0 170.0 173.6 �3.6 �2.00
12 9.0 162.0 160.8 1.2 �2.00
13 10.0 165.0 167.2 �2.2 �2.00
14 12.0 180.0 179.9 0.1 �2.00
15 8.0 160.0 154.4 5.6 �2.00
16 9.0 155.0 160.8 �5.8 �2.00
17 10.0 165.0 167.2 �2.2 �2.00
18 15.0 190.0 199.1 �9.1 �2.00
19 13.0 185.0 186.3 �1.3 �2.00
20 11.0 155.0 173.6 �18.6 �3.00
TOTAL � $25.00
Note: This data set, and every other data set in the text, is available on the text’s website in four
formats.
Ŷi
AN OVERVIEW OF REGRESSION ANALYSIS
is worth trying as an alternative to just guessing the weights of your cus-
tomers. Such an equation estimates weight with a constant base of 103.40
pounds and adds 6.38 pounds for every inch of height over 5 feet. Note that
the sign of is positive, as you expected.
How well does the equation work? To answer this question, you need to
calculate the residuals (Yi minus ) from Equation 21 to see how many were
greater than ten. As can be seen in the last column in Table 1, if you had ap-
plied the equation to these 20 people, you wouldn’t exactly have gotten rich,
but at least you would have earned $25.00 instead of losing $2.00. Figure 4
shows not only Equation 21 but also the weight and height data for all
20 customers used as the sample.
Equation 21 would probably help a beginning weight guesser, but it could
be improved by adding other variables or by collecting a larger sample.
Ŷi
�̂1
19

AN OVERVIEW OF REGRESSION ANALYSIS
Y
200
190
180
170
160
150
140
130
120
110
0 1 2 3 4 5 6 7 8
Height (over five feet in inches)
Observations
Y-hats
W
e
ig
h
t
9 10 11 12 13 14 15 X
Yi = 103.40 + 6.38Xi
Figure 4 A Weight-Guessing Equation
If we plot the data from the weight-guessing example and include the estimated regres-
sion line, we can see that the estimated come fairly close to the observed Ys for all
but three observations. Find a male friend’s height and weight on the graph; how well
does the regression equation work?
Ŷs
Such an equation is realistic, though, because it’s likely that every successful
weight guesser uses an equation like this without consciously thinking about
that concept.
Our goal with this equation was to quantify the theoretical weight/height
equation, Equation 20, by collecting data (Table 1) and calculating an esti-
mated regression, Equation 21. Although the true equation, like observations
of the stochastic error term, can never be known, we were able to come up
with an estimated equation that had the sign we expected for and that
helped us in our job. Before you decide to quit school or your job and try to
make your living guessing weights at Magic Hill, there is quite a bit more to
learn about regression analysis, so we’d better move on.
Using Regression to Explain Housing Prices
As much fun as guessing weights at an amusement park might be, it’s hardly
a typical example of the use of regression analysis. For every regression run
on such an off-the-wall topic, there are literally hundreds run to describe the
5
�̂1
20

AN OVERVIEW OF REGRESSION ANALYSIS
reaction of GDP to an increase in the money supply, to test an economic
theory with new data, or to forecast the effect of a price change on a firm’s
sales.
As a more realistic example, let’s look at a model of housing prices. The
purchase of a house is probably the most important financial decision in an
individual’s life, and one of the key elements in that decision is an appraisal
of the house’s value. If you overvalue the house, you can lose thousands of
dollars by paying too much; if you undervalue the house, someone might
outbid you.
All this wouldn’t be much of a problem if houses were homogeneous
products, like corn or gold, that have generally known market prices with
which to compare a particular asking price. Such is hardly the case in the real
estate market. Consequently, an important element of every housing pur-
chase is an appraisal of the market value of the house, and many real estate
appraisers use regression analysis to help them in their work.
Suppose your family is about to buy a house in Southern California, but
you’re convinced that the owner is asking too much money. The owner says
that the asking price of $230,000 is fair because a larger house next door sold
for $230,000 about a year ago. You’re not sure it’s reasonable to compare the
prices of different-sized houses that were purchased at different times. What
can you do to help decide whether to pay the $230,000?
Since you’re taking an econometrics class, you decide to collect data on
all local houses that were sold within the last few weeks and to build a re-
gression model of the sales prices of the houses as a function of their
sizes.12 Such a data set is called cross-sectional because all of the observa-
tions are from the same point in time and represent different individual
economic entities (like countries or, in this case, houses) from that same
point in time.
To measure the impact of size on price, you include the size of the house
as an independent variable in a regression equation that has the price of that
house as the dependent variable. You expect a positive sign for the coefficient
of size, since big houses cost more to build and tend to be more desirable
than small ones. Thus the theoretical model is:
(22)PRICEi 5 f(SIZE
1
i) 1 �i 5 �0 1 �1SIZEi 1 �i
12. It’s unusual for an economist to build a model of price without including some measure of
quantity on the right-hand side. Such models of the price of a good as a function of the attributes
of that good are called hedonic models.
21

AN OVERVIEW OF REGRESSION ANALYSIS
PRICEi
0
Size of the house (square feet)
Slope = .138
Intercept = 40.0
PRICE
(thousands of $)
PRICEi = 40.0 + 0.138SIZEi
SIZEi
Figure 5 A Cross-Sectional Model of Housing Prices
A regression equation that has the price of a house in Southern California as a function of
the size of that house has an intercept of 40.0 and a slope of 0.138, using Equation 23.
where: PRICEi � the price (in thousands of $) of the ith house
SIZEi � the size (in square feet) of that house
� the value of the stochastic error term for that house
You collect the records of all recent real estate transactions, find that 43
local houses were sold within the last 4 weeks, and estimate the following re-
gression of those 43 observations:
(23)
What do these estimated coefficients mean? The most important coefficient
is since the reason for the regression is to find out the impact of
size on price. This coefficient means that if size increases by 1 square foot,
price will increase by 0.138 thousand dollars ($138). thus measures the
change in PRICEi associated with a one-unit increase in SIZEi. It’s the slope of
the regression line in a graph like Figure 5.
What does mean? is the estimate of the constant or intercept
term. In our equation, it means that price equals 40.0 when size equals zero.
As can be seen in Figure 5, the estimated regression line intersects the price
�̂0�̂0 5 40.0
�̂1
�̂1 5 0.138,
PRICEi 5 40.0 1 0.138SIZEi
�i
22

AN OVERVIEW OF REGRESSION ANALYSIS
axis at 40.0. While it might be tempting to say that the average price of a
vacant lot is $40,000, such a conclusion would be unjustified for a num-
ber of reasons. It’s much safer either to interpret as nothing
more than the value of the estimated regression when Si � 0, or to not in-
terpret at all.
What does mean? is the estimate of the coefficient of SIZE
in Equation 22, and as such it’s also an estimate of the slope of the line in
Figure 5. It implies that an increase in the size of a house by one square foot
will cause the estimated price of the house to go up by 0.138 thousand dol-
lars or $138. It’s a good habit to analyze estimated slope coefficients to see
whether they make sense. The positive sign of certainly is what we
expected, but what about the magnitude of the coefficient? Whenever you
interpret a coefficient, be sure to take the units of measurement into consider-
ation. In this case, is $138 per square foot a plausible number? Well, it’s hard
to know for sure, but it certainly is a lot more reasonable than $1.38 per
square foot or $13,800 per square foot!
How can you use this estimated regression to help decide whether to pay
$230,000 for the house? If you calculate a (predicted price) for a house that
is the same size (1,600 square feet) as the one you’re thinking of buying, you
can then compare this with the asking price of $230,000. To do this, substi-
tute 1600 for SIZEi in Equation 23, obtaining:
The house seems to be a good deal. The owner is asking “only” $230,000
for a house when the size implies a price of $260,800! Perhaps your original
feeling that the price was too high was a reaction to the steep housing prices
in Southern California in general and not a reflection of this specific price.
On the other hand, perhaps the price of a house is influenced by more than
just the size of the house. (After all, what good’s a house in Southern California
unless it has a pool or air-conditioning?) Such multivariate models are the
heart of econometrics.
Summary
1. Econometrics—literally, “economic measurement”—is a branch of
economics that attempts to quantify theoretical relationships. Regres-
sion analysis is only one of the techniques used in econometrics, but
it is by far the most frequently used.
6
PRICEi 5 40.0 1 0.138(1600) 5 40.0 1 220.8 5 260.8
Ŷ
Ŷ
�̂1
�̂1�̂1 5 0.138
�̂0
�̂0 5 40.0
23

AN OVERVIEW OF REGRESSION ANALYSIS
2. The major uses of econometrics are description, hypothesis testing,
and forecasting. The specific econometric techniques employed may
vary depending on the use of the research.
3. While regression analysis specifies that a dependent variable is a func-
tion of one or more independent variables, regression analysis alone
cannot prove or even imply causality.
4. A stochastic error term must be added to all regression equations to
account for variations in the dependent variable that are not ex-
plained completely by the independent variables. The components of
this error term include:
a. omitted or left-out variables
b. measurement errors in the data
c. an underlying theoretical equation that has a different functional
form (shape) than the regression equation
d. purely random and unpredictable events
5. An estimated regression equation is an approximation of the true
equation that is obtained by using data from a sample of actual Ys
and Xs. Since we can never know the true equation, econometric
analysis focuses on this estimated regression equation and the esti-
mates of the regression coefficients. The difference between a particu-
lar observation of the dependent variable and the value estimated
from the regression equation is called the residual.
EXERCISES
(The answer to Exercise 2 is at the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or your notes), and compare your definition with the ver-
sion in the text for each:
a. stochastic error term
b. regression analysis
c. linear
d. slope coefficient
e. multivariate regression model
f. expected value
g. residual
h. time series
i. cross-sectional data set
24

AN OVERVIEW OF REGRESSION ANALYSIS
2. Use your own computer’s regression software and the weight (Y) and
height (X) data from Table 1 to see if you can reproduce the estimates
in Equation 21. There are three different ways to load the data: You
can type in the data yourself, you can open datafile HTWT1 on the
EViews CD-ROM, or you can download datafile HTWT1
(in Excel, Stata or ASCII formats) from the text’s website: www
.pearsonhighered.com/studenmund. Once the datafile is loaded,
run Y � f(X), and your results should match Equation 21. Different
programs require different commands to run a regression. For help
in how to do this with EViews and Stata, see the answer to this
question at the end of the chapter.
3. Decide whether you would expect relationships between the follow-
ing pairs of dependent and independent variables (respectively) to be
positive, negative, or ambiguous. Explain your reasoning.
a. Aggregate net investment in the United States in a given year and
GDP in that year.
b. The amount of hair on the head of a male professor and the age of
that professor.
c. The number of acres of wheat planted in a season and the price of
wheat at the beginning of that season.
d. Aggregate net investment and the real rate of interest in the same
year and country.
e. The growth rate of GDP in a year and the average hair length in that
year.
f. The quantity of canned tuna demanded and the price of a can of
tuna.
4. Let’s return to the height/weight example in Section 4:
a. Go back to the data set and identify the three customers who seem
to be quite a distance from the estimated regression line. Would we
have a better regression equation if we dropped these customers
from the sample?
b. Measure the height of a male friend and plug it into Equation 21.
Does the equation come within 10 pounds? If not, do you think
you see why? Why does the estimated equation predict the same
weight for all males of the same height when it is obvious that all
males of the same height don’t weigh the same?
c. Look over the sample with the thought that it might not be ran-
domly drawn. Does the sample look abnormal in any way? (Hint:
Are the customers who choose to play such a game a random sam-
ple?) If the sample isn’t random, would this have an effect on the
regression results and the estimated weights?
25

AN OVERVIEW OF REGRESSION ANALYSIS
d. Think of at least one other factor besides height that might be a
good choice as a variable in the weight/height equation. How
would you go about obtaining the data for this variable? What
would the expected sign of your variable’s coefficient be if the vari-
able were added to the equation?
5. Continuing with the height/weight example, suppose you collected
data on the heights and weights of 29 different male customers and
estimated the following equation:
(24)
where: Yi � the weight (in pounds) of the ith person
Xi � the height (in inches over five feet) of the ith person
a. Why aren’t the coefficients in Equation 24 the same as those we es-
timated previously (Equation 21)?
b. Compare the estimated coefficients of Equation 24 with those in
Equation 21. Which equation has the steeper estimated relation-
ship between height and weight? Which equation has the higher
intercept? At what point do the two intersect?
c. Use Equation 24 to “predict” the 20 original weights given the
heights in Table 1. How many weights does Equation 24 miss by
more than 10 pounds? Does Equation 24 do better or worse than
Equation 21? Could you have predicted this result beforehand?
d. Suppose you had one last day on the weight-guessing job. What
equation would you use to guess weights? (Hint: There is more
than one possible answer.)
6. Not all regression coefficients have positive expected signs. For exam-
ple, a Sports Illustrated article by Jaime Diaz reported on a study of
golfing putts of various lengths on the Professional Golfers’ Associa-
tion (PGA) Tour.13 The article included data on the percentage of
putts made (Pi) as a function of the length of the putt in feet (Li).
Since the longer the putt, the less likely even a professional is to make
it, we’d expect Li to have a negative coefficient in an equation explain-
ing Pi. Sure enough, if you estimate an equation on the data in the ar-
ticle, you obtain:
(25)P̂i 5 f(Li) 5 83.6 2 4.1Li
Ŷi 5 125.1 1 4.03Xi
13. Jaime Diaz, “Perils of Putting,” Sports Illustrated, April 3, 1989, pp. 76–79.
26

AN OVERVIEW OF REGRESSION ANALYSIS
a. Carefully write out the exact meaning of the coefficient of Li.
b. Suppose someone else took the data from the article and estimated:
Is this the same result as that of Equation 25? If so, what definition
do you need to use to convert this equation back to Equation 25?
c. Use Equation 25 to determine the percent of the time you’d expect a
PGA golfer to make a 10-foot putt. Does this seem realistic? How
about a 1-foot putt or a 25-foot putt? Do these seem as realistic?
d. Your answer to part c should suggest that there’s a problem in apply-
ing a linear regression to these data. What is that problem? (Hint: If
you’re stuck, first draw the theoretical diagram you’d expect for Pi as
a function of Li, then plot Equation 25 onto the same diagram.)
7. Return to the housing price model of Section 5 and consider the fol-
lowing equation:
(26)
where: SIZEi � the size (in square feet) of the ith house
PRICEi � the price (in thousands of $) of that house
a. Carefully explain the meaning of each of the estimated regression
coefficients.
b. Suppose you’re told that this equation explains a significant por-
tion (more than 80 percent) of the variation in the size of a house.
Have we shown that high housing prices cause houses to be large?
If not, what have we shown?
c. What do you think would happen to the estimated coefficients of
this equation if we had measured the price variable in dollars in-
stead of in thousands of dollars? Be specific.
8. If an equation has more than one independent variable, we have to be
careful when we interpret the regression coefficients of that equation.
Think, for example, about how you might build an equation to ex-
plain the amount of money that different states spend per pupil on
public education. The more income a state has, the more they proba-
bly spend on public schools, but the faster enrollment is growing, the
less there would be to spend on each pupil. Thus, a reasonable equa-
tion for per pupil spending would include at least two variables: in-
come and enrollment growth:
(27)Si 5 �0 1 �1Yi 1 �2Gi 1 �i
SIZEi 5 2290 1 3.62 PRICEi
Pi 5 83.6 2 4.1Li 1 ei
27

where: Si � educational dollars spent per public school student in
the ith state
Yi � per capita income in the ith state
Gi � the percent growth of public school enrollment in the
ith state
a. State the economic meaning of the coefficients of Y and G. (Hint:
Remember to hold the impact of the other variable constant.)
b. If we were to estimate Equation 27, what signs would you expect
the coefficients of Y and G to have? Why?
c. Silva and Sonstelie estimated a cross-sectional model of per stu-
dent spending by state that is very similar to Equation 27:14
(28)
N � 49
Do these estimated coefficients correspond to your expectations?
Explain Equation 28 in common sense terms.
d. The authors measured G as a decimal, so if a state had a 10 percent
growth in enrollment, then G equaled .10. What would
Equation 28 have looked like if the authors had measured G in per-
centage points, so that if a state had 10 percent growth, then G
would have equaled 10? (Hint: Write out the actual numbers for
the estimated coefficients.)
9. Your friend has an on-campus job making telephone calls to alumni
asking for donations to your college’s annual fund, and she wonders
whether her calling is making any difference. In an attempt to mea-
sure the impact of student calls on fund raising, she collects data from
50 alums and estimates the following equation:
(29)
where: GIFTi � the 2008 annual fund donation (in dollars)
from the ith alum
INCOMEi � the 2008 estimated income (in dollars) of the
ith alum
CALLSi � the # of calls to the ith alum asking for a do-
nation in 2008
GIFTi 5 2.29 1 0.001INCOMEi 1 4.62CALLSi
Ŝi 5 2183 1 0.1422Yi 2 5926Gi
14. Fabio Silva and Jon Sonstelie, “Did Serrano Cause a Decline in School Spending?” National
Tax Review, Vol. 48, No. 2, pp. 199–215. The authors also included the tax price for spending
per pupil in the ith state as a variable.
AN OVERVIEW OF REGRESSION ANALYSIS
28

a. Carefully explain the meaning of each estimated coefficient. Are
the estimated signs what you expected?
b. Why is the left-hand variable in your friend’s equation GIFTi and
not GIFTi?
c. Your friend didn’t include the stochastic error term in the estimated
equation. Was this a mistake? Why or why not?
d. Suppose that your friend decides to change the units of INCOME
from “dollars” to “thousands of dollars.” What will happen to the
estimated coefficients of the equation? Be specific.
e. If you could add one more variable to this equation, what would it
be? Explain.
10. Housing price models can be estimated with time-series as well as
cross-sectional data. If you study aggregate time-series housing prices
(see Table 2 for data and sources), you have:
N � 38 (annual 1970–2007)
where: Pt � the nominal median price of new single-family houses
in the United States in year t
Yt � the U.S. GDP in year t (billions of current $)
a. Carefully interpret the economic meaning of the estimated coefficients.
b. What is Yt doing on the right side of the equation? Isn’t Y always
supposed to be on the left side?
c. Both the price and GDP variables are measured in nominal (or cur-
rent, as opposed to real, or inflation-adjusted) dollars. Thus a
major portion of the excellent explanatory power of this equation
(almost 99 percent of the variation in Pt can be explained by Yt
alone) comes from capturing the huge amount of inflation that
took place between 1970 and 2007. What could you do to elimi-
nate the impact of inflation in this equation?
d. GDP is included in the equation to measure more than just infla-
tion. What factors in housing prices other than inflation does the
GDP variable help capture? Can you think of a variable that might
do a better job?
e. To be sure that you understand the difference between a cross-
sectional data set and a time-series data set, compare the variable
you thought of in part d with a variable that you could add to
Equation 22. The dependent variable in both equations is the price
of a house. Could you add the same independent variable to both
equations? Explain.
P̂t 5 f(GDP
1
) 5 12,928 1 17.08Yt
AN OVERVIEW OF REGRESSION ANALYSIS
29

Table 2 Data for the Time-Series Model of Housing Prices
t Year Price (Pt) GDP (Yt)
1 1970 23,400 1,038.5
2 1971 25,200 1,127.1
3 1972 27,600 1,238.3
4 1973 32,500 1,382.7
5 1974 35,900 1,500.0
6 1975 39,300 1,638.3
7 1976 44,200 1,825.3
8 1977 48,800 2,030.9
9 1978 55,700 2,294.7
10 1979 62,900 2,563.3
11 1980 64,600 2,789.5
12 1981 68,900 3,128.4
13 1982 69,300 3,255.0
14 1983 75,300 3,536.7
15 1984 79,900 3,933.2
16 1985 84,300 4,220.3
17 1986 92,000 4,462.8
18 1987 104,500 4,739.5
19 1988 112,500 5,103.8
20 1989 120,000 5,484.4
21 1990 122,900 5,803.1
22 1991 120,000 5,995.9
23 1992 121,500 6,337.7
24 1993 126,500 6,657.4
25 1994 130,000 7,072.2
26 1995 133,900 7,397.7
27 1996 140,000 7,816.9
28 1997 146,000 8,304.3
29 1998 152,500 8,747.0
30 1999 161,000 9,268.4
31 2000 169,000 9,817.0
32 2001 175,200 10,128.0
33 2002 187,600 10,469.6
34 2003 195,000 10,960.8
35 2004 221,000 11,685.9
36 2005 240,900 12,421.9
37 2006 246,500 13,178.4
38 2007 247,900 13,807.5
Pt � the nominal median price of new single-family houses in the United States in year t.
(Source: The Statistical Abstract of the U.S.)
Yt � the U.S. GDP in year t (billions of current dollars). (Source: The Economic Report of the
President )
Datafile � HOUSE1
30

AN OVERVIEW OF REGRESSION ANALYSIS
11. The distinction between the stochastic error term and the residual is
one of the most difficult concepts to master in this chapter.
a. List at least three differences between the error term and the residual.
b. Usually, we can never observe the error term, but we can get around
this difficulty if we assume values for the true coefficients. Calculate
values of the error term and residual for each of the following six
observations given that the true equals 0.0, the true equals
1.5, and the estimated regression equation is :
Yi 2 6 3 8 5 4
Xi 1 4 2 5 3 4
(Hint: To answer this question, you’ll have to solve Equation 14 for
.) Note: Datafile � EX1.
12. Let’s return to the wage determination example of Section 2. In that
example, we built a model of the wage of the ith worker in a particular
field as a function of the work experience, education, and gender of
that worker:
WAGEi � �0 � �1EXPi � �2EDUi � �3GENDi � i (12)
where: Yi � WAGEi � the wage of the ith worker
X1i � EXPi � the years of work experience of the ith worker
X2i � EDUi � the years of education beyond high school
of the ith worker
X3i � GENDi � the gender of the ith worker (1 � male and
0 � female)
a. What is the real-world meaning of �2? (Hint: If you’re unsure
where to start, review Section 2.)
b. What is the real-world meaning of �3? (Hint: Remember that
GEND is a dummy variable.)
c. Suppose that you wanted to add a variable to this equation to mea-
sure whether there might be discrimination against people of color.
How would you define such a variable? Be specific.
d. Suppose that you had the opportunity to add another variable to
the equation. Which of the following possibilities would seem
best? Explain your answer.
i. the age of the ith worker
ii. the number of jobs in this field
iii. the average wage in this field
�
�
Ŷi 5 0.48 1 1.32Xi
�1�0
31

AN OVERVIEW OF REGRESSION ANALYSIS
iv. the number of “employee of the month” awards won by the ith
worker
v. the number of children of the ith worker
13. Have you heard of “RateMyProfessors.com”? On this website, students
evaluate a professor’s overall teaching ability and a variety of other attrib-
utes. The website then summarizes these student-submitted ratings for
the benefit of any student considering taking a class from the professor.
Two of the most interesting attributes that the website tracks are how
“easy” the professor is (in terms of workload and grading), and how
“hot” the professor is (presumably in terms of physical attractiveness).
A recently published article15 indicates that being “hot” improves a
professor’s rating more than being “easy.” To investigate these ideas
ourselves, we created the following equation for RateMyProfessors.com:
RATINGi � �0 � �1EASEi � �2HOTi � �i (30)
where: RATINGi � the overall rating (5 � best) of the ith professor
EASEi � the easiness rating (5 � easiest) of the ith
professor
HOTi � 1 if the ith professor is considered “hot,” 0
otherwise
To estimate Equation 30, we need data, and Table 3 contains
data for these variables from 25 randomly chosen professors on
RateMyProfessors.com. If we estimate Equation 30 with the data in
Table 3, we obtain:
RATINGi � 3.23 � 0.01EASEi � 0.59HOTi (31)
a. Take a look at Equation 31. Do the estimated coefficients support
our expectations? Explain.
b. See if you can reproduce the results in Equation 31 on your own. To
do this, take the data in Table 3 and use EViews, Stata, or your own re-
gression program to estimate the coefficients from these data. If you
do everything correctly, you should be able to verify the estimates in
Equation 31. (If you’re not sure how to get started on this question,
take a look at the answer to Exercise 2 at the end of the chapter.)
c. This model includes two independent variables. Does it make
sense to think that the teaching rating of a professor depends on
15. James Otto, Douglas Sanford, and Douglas Ross, “Does RateMyProfessors.com Really Rate
My Professor?” Assessment and Evaluation in Higher Education, August 2008, pp. 355–368.
32

AN OVERVIEW OF REGRESSION ANALYSIS
just these two variables? What other variable(s) do you think might
be important?
d. Suppose that you were able to add your suggested variable(s) to
Equation 31. What do you think would happen to the coefficients
of EASE and HOT when you added the variable(s)? Would you ex-
pect them to change? Would you expect them to remain the same?
Explain.
e. (optional) Go to the RateMyProfessors.com website, choose 25 obser-
vations at random, and estimate your own version of Equation 30.
Now compare your regression results to those in Equation 31.
Do your estimated coefficients have the same signs as those in
Equation 31? Are your estimated coefficients exactly the same as
those in Equation 31? Why or why not?
Table 3 RateMyProfessors.com Ratings
Observation RATING EASE HOT
1 2.8 3.7 0
2 4.3 4.1 1
3 4.0 2.8 1
4 3.0 3.0 0
5 4.3 2.4 0
6 2.7 2.7 0
7 3.0 3.3 0
8 3.7 2.7 0
9 3.9 3.0 1
10 2.7 3.2 0
11 4.2 1.9 1
12 1.9 4.8 0
13 3.5 2.4 1
14 2.1 2.5 0
15 2.0 2.7 1
16 3.8 1.6 0
17 4.1 2.4 0
18 5.0 3.1 1
19 1.2 1.6 0
20 3.7 3.1 0
21 3.6 3.0 0
22 3.3 2.1 0
23 3.2 2.5 0
24 4.8 3.3 0
25 4.6 3.0 0
Datafile � RATE1
33

Answers
Exercise 2
Using EViews:
a. Install and launch the software.
b. Open the datafile. All datafiles can be found in EViews format at
www.pearsonhighered.com/studenmund. Alternatively, on your
EViews disc, you can click through File � Open � Workfile. Then
browse to the CD-ROM, select the folder “Studenmund,” and
double-click on “HTW T1” followed by “OK.”
c. Run the regression. Type “LS Y C X” on the top line, making sure
to leave spaces between the variable names. (LS stands for Least
Squares and C stands for constant.) Press Enter, and the regres-
sion results will appear on your screen.
Using Stata:
a. Install and launch the regression software.
b. Open the datafile. All datafiles can be found in Stata format at
www.pearsonhighered.com/studenmund. This particular datafile
is “HTW T1.”
c. Run the regression. Click through Statistics � Linear Models and
Related � Linear Regression. Select Y as your dependent variable
and X as your independent variable. Then click “OK,” and the
regression results will appear on your screen.
AN OVERVIEW OF REGRESSION ANALYSIS
34

From Chapter 2 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
Ordinary Least Squares
35

The bread and butter of regression analysis is the estimation of the coeffi-
cients of econometric models with a technique called Ordinary Least Squares
(OLS). The first two sections of this chapter summarize the reasoning behind
and the mechanics of OLS. Regression users rely on computers to do the ac-
tual OLS calculations, so the emphasis here is on understanding what OLS
attempts to do and how it goes about doing it.
How can you tell a good equation from a bad one once it has been esti-
mated? There are a number of useful criteria, including the extent to which the
estimated equation fits the actual data. A focus on fit is not without perils, how-
ever, so the chapter concludes with an example of the misuse of this criterion.
Estimating Single-Independent-Variable
Models with OLS
The purpose of regression analysis is to take a purely theoretical equation like:
(1)
and use a set of data to create an estimated equation like:
(2)
where each “hat” indicates a sample estimate of the true population value.
(In the case of Y, the “true population value” is The purpose of theEfY k Xg.)
Ŷi 5 �̂0 1 �̂1Xi
Yi 5 �0 1 �1Xi 1 �i
1
1 Estimating Single-Independent-Variable Models with OLS
2 Estimating Multivariate Regression Models with OLS
3 Evaluating the Quality of a Regression Equation
4 Describing the Overall Fit of the Estimated Model
5 An Example of the Misuse of
6 Summary and Exercises
R2
Ordinary Least Squares
36

estimation technique is to obtain numerical values for the coefficients of an
otherwise completely theoretical regression equation.
The most widely used method of obtaining these estimates is Ordinary
Least Squares (OLS), which has become so standard that its estimates are pre-
sented as a point of reference even when results from other estimation tech-
niques are used. Ordinary Least Squares (OLS) is a regression estimation
technique that calculates the so as to minimize the sum of the squared
residuals, thus:1
(3)
Since these residuals (eis) are the differences between the actual Ys and the es-
timated Ys produced by the regression (the in Equation 2), Equation 3 is
equivalent to saying that OLS minimizes
Why Use Ordinary Least Squares?
Although OLS is the most-used regression estimation technique, it’s not the
only one. Indeed, econometricians have developed what seem like zillions of
different estimation techniques.
There are at least three important reasons for using OLS to estimate regres-
sion models:
1. OLS is relatively easy to use.
2. The goal of minimizing is quite appropriate from a theoretical
point of view.
3. OLS estimates have a number of useful characteristics.
ge 2i
g (Yi 2 Ŷi)
2.
Ŷs
OLS minimizes g
N
i 5 1
e˛2i (i 5 1, 2, . . . , N)
�̂s
ORDINARY LEAST SQUARES
1. The summation symbol, , means that all terms to its right should be added (or summed)
over the range of the i values attached to the bottom and top of the symbol. In Equation 3, for
example, this would mean adding up for all integer values between 1 and N:
Often the notation is simply written as , and it is assumed that the summation is over all
observations from i � 1 to i � N. Sometimes, the i is omitted entirely and the same assumption
is made implicitly. For more practice in the basics of summation algebra, see Exercise 3.
g

i
g
g
N
i 5 1
e2i 5 e
2
1 1 e
2
2 1
c1 e2N
e2i
g
37

The first reason for using OLS is that it’s the simplest of all econometric
estimation techniques. Most other techniques involve complicated non-
linear formulas or iterative procedures, many of which are extensions of
OLS itself. In contrast, OLS estimates are simple enough that, if you had
to, you could compute them without using a computer or a calculator
(for a single-independent-variable model). Indeed, in the “dark ages” be-
fore computers and calculators, econometricians calculated OLS estimates
by hand!
The second reason for using OLS is that minimizing the summed, squared
residuals is a reasonable goal for an estimation technique. To see this, recall
that the residual measures how close the estimated regression equation
comes to the actual observed data:
(17)
Since it’s reasonable to want our estimated regression equation to be as
close as possible to the observed data, you might think that you’d want to
minimize these residuals. The main problem with simply totaling the resid-
uals is that ei can be negative as well as positive. Thus, negative and positive
residuals might cancel each other out, allowing a wildly inaccurate equa-
tion to have a very low For example, if Y � 100,000 for two consecu-
tive observations and if your equation predicts 1.1 million and �900,000,
respectively, your residuals will be �1 million and �1 million, which add
up to zero!
We could get around this problem by minimizing the sum of the absolute
values of the residuals, but absolute values are difficult to work with mathe-
matically. Luckily, minimizing the summed squared residuals does the job.
Squared functions pose no unusual mathematical difficulties in terms of ma-
nipulations, and the technique avoids canceling positive and negative residu-
als because squared terms are always positive.
The final reason for using OLS is that its estimates have at least two useful
characteristics:
1. The sum of the residuals is exactly zero.
2. OLS can be shown to be the “best” estimator possible under a set of
specific assumptions.
An estimator is a mathematical technique that is applied to a sample of
data to produce real-world numerical estimates of the true population re-
gression coefficients (or other parameters). Thus, OLS is an estimator, and a
produced by OLS is an estimate.�̂
gei.
ei 5 Yi 2 Ŷi (i 5 1, 2, . . ., N)
ORDINARY LEAST SQUARES
38

How Does OLS Work?
How would OLS estimate a single-independent-variable regression model
like Equation 1?
(1)
OLS selects those estimates of that minimize the squared residuals,
summed over all the sample data points.
For an equation with just one independent variable, these coefficients
are:2
�0 and �1
Yi 5 �0 1 �1Xi 1 �i
2. Since
and OLS actually minimizes
by choosing the that do so. For those with a moderate grasp of calculus and algebra, the
derivation of these equations is informative. See Exercise 12.
�̂s
g
i
e2i 5 g
i
(Yi 2 �̂0 2 �̂1Xi)
2
Ŷi 5 �̂0 1 �̂X1i,
g
N
i 5 1
e2i 5 g
N
i 5 1
(Yi 2 Ŷi)
2
and, given this estimate of �1,
where the mean of X, or and the mean of Y, or
Note that for each different data set, we’ll get different estimates of
depending on the sample.
�1 and �0,
gYi>N.Y 5gXi>N,X 5
(4)�̂1 5
g
N
i 5 1
f (X
i
2 X) (Y
i
2 Y) g
g
N
i 5 1
(Xi 2 X)
2
(5)�̂0 5 Y 2 �̂1X
ORDINARY LEAST SQUARES
39

ORDINARY LEAST SQUARES
An Illustration of OLS Estimation
The equations for calculating regression coefficients might seem a little for-
bidding, but it’s not hard to apply them yourself to data sets that have only a
few observations and independent variables. Although you’ll usually want to
use regression software packages to do your estimation, you’ll understand
OLS better if you work through the following illustration.
To keep things simple, let’s attempt to estimate the regression coef-
ficients of the height and weight data given in Table 1. The formulas for
OLS estimation for a regression equation with one independent variable
are Equations 4 and 5:
(4)
(5)
If we undertake the calculations outlined in Table 1 and substitute them into
Equations 4 and 5, we obtain these values:
or
(6)
As can be seen in Table 1, the sum of the (column 8) equals the sum of the
Ys (column 2), so the sum of the residuals (column 9) does indeed equal
zero (except for rounding errors).
Ŷs
Ŷi 5 103.4 1 6.38Xi
�̂0 5 169.4 2 (6.38 ? 10.35) 5 103.4
�̂1 5
590.20
92.50
5 6.38
�̂0 5 Y 2 �̂1X
�̂1 5
g
N
i 5 1
f (X
i
2 X) (Y
i
2 Y) g
g
N
i 5 1
(Xi 2 X)
2
40

ORDINARY LEAST SQUARES
Table 1 The Calculation of Estimated Regression Coefficients
for the Weight/Height Example
Raw Data Required Intermediate Calculations
i Yi Xi
(1) (2) (3) (4) (5) (6) (7) (8) (9)
1 140 5 �29.40 �5.35 28.62 157.29 135.3 4.7
2 157 9 �12.40 �1.35 1.82 16.74 160.8 �3.8
3 205 13 35.60 2.65 7.02 94.34 186.3 18.7
4 198 12 28.60 1.65 2.72 47.19 179.9 18.1
5 162 10 �7.40 �0.35 0.12 2.59 167.2 �5.2
6 174 11 4.60 0.65 0.42 2.99 173.6 0.4
7 150 8 �19.40 �2.35 5.52 45.59 154.4 �4.4
8 165 9 �4.40 �1.35 1.82 5.94 160.8 4.2
9 170 10 0.60 �0.35 0.12 �0.21 167.2 2.8
10 180 12 10.60 1.65 2.72 17.49 179.9 0.1
11 170 11 0.60 0.65 0.42 0.39 173.6 �3.6
12 162 9 �7.40 �1.35 1.82 9.99 160.8 1.2
13 165 10 �4.40 �0.35 0.12 1.54 167.2 2.2
14 180 12 10.60 1.65 2.72 17.49 179.9 0.1
15 160 8 �9.40 �2.35 5.52 22.09 154.4 5.6
16 155 9 �14.40 �1.35 1.82 19.44 160.8 �5.8
17 165 10 �4.40 �0.35 0.12 1.54 167.2 �2.2
18 190 15 20.60 4.65 21.62 95.79 199.1 �9.1
19 185 13 15.60 2.65 7.02 41.34 186.3 �1.3
20 155 11 �14.40 0.65 0.42 �9.36 173.6 �18.6
Sum 3388 207 0.0 0.0 92.50 590.20 3388.3 �0.3
Mean 169.4 10.35 0.0 0.0 169.4 0.0
ei 5 Yi 2 ŶiŶi(Xi 2 X) (Yi 2 Y)(Xi 2 X)2(Xi 2 X)(Yi 2 Y)
Estimating Multivariate Regression
Models with OLS
Let’s face it: only a few dependent variables can be explained fully by a single
independent variable. A person’s weight, for example, is influenced by more
than just that person’s height. What about bone structure, percent body fat,
exercise habits, or diet?
As important as additional explanatory variables might seem to the
height/weight example, there’s even more reason to include a variety of in-
dependent variables in economic and business applications. Although the
quantity demanded of a product is certainly affected by price, that’s not the
2
41

ORDINARY LEAST SQUARES
3. The term “partial regression coefficient” will seem especially appropriate to those readers
who have taken calculus, since multivariate regression coefficients correspond to partial
derivatives.
whole story. Advertising, aggregate income, the prices of substitutes, the influ-
ence of foreign markets, the quality of customer service, possible fads, and
changing tastes all are important in real-world models. As a result, it’s vital
to move from single-independent-variable regressions to multivariate regres-
sion models, or equations with more than one independent variable.
The Meaning of Multivariate Regression Coefficients
The general multivariate regression model with K independent variables can
be represented by Equation 13:
(13)
where i, as before, goes from 1 to N and indicates the observation number.
Thus, X1i indicates the ith observation of independent variable X1, while X2i
indicates the ith observation of another independent variable, X2.
The biggest difference between a single-independent-variable regression
model and a multivariate regression model is in the interpretation of the lat-
ter’s slope coefficients. These coefficients, often called partial regression coef-
ficients,3 are defined to allow a researcher to distinguish the impact of one
variable from that of other independent variables.
Yi 5 �0 1 �1X1i 1 �2X2i 1
c1 �KXKi 1 �i
This last italicized phrase is a key to understanding multiple regression (as
multivariate regression is often called). The coefficient measures the im-
pact on Y of a one-unit increase in X1, holding constant X2, X3, . . . and XK
but not holding constant any relevant variables that might have been omitted
�1
Specifically, a multivariate regression coefficient indicates the change
in the dependent variable associated with a one-unit increase in the in-
dependent variable in question holding constant the other independent
variables in the equation.
42

ORDINARY LEAST SQUARES
from the equation (e.g., XK�1). The coefficient is the value of Y when all
the Xs and the error term equal zero. You should always include a constant
term in a regression equation, but you should not rely on estimates of for
inference.
As an example, let’s consider the following annual model of the per capita
demand for beef in the United States:
(7)
where: CBt � the per capita consumption of beef in year t (in pounds per
person)
Pt � the price of beef in year t (in cents per pound)
Ydt � the per capita disposable income in year t (in thousands of
dollars)
The estimated coefficient of income, 9, tells us that beef consumption will in-
crease by 9 pounds per person if per capita disposable income goes up by
$1,000, holding constant the price of beef. The ability to hold price constant
is crucial because we’d expect such a large increase in per capita income to
stimulate demand, therefore pushing up prices and making it hard to distin-
guish the effect of the income increase from the effect of the price increase.
The multivariate regression estimate allows us to focus on the impact of the
income variable by holding the price variable constant.
Note, however, that the equation does not hold constant other possible
variables (like the price of a substitute) because these variables are not in-
cluded in Equation 7. Before you move on to the next section, take the time
to think through the meaning of the estimated coefficient of P in Equation 7;
do you agree that the sign and relative size fit with economic theory?
OLS Estimation of Multivariate Regression Models
The application of OLS to an equation with more than one independent vari-
able is quite similar to its application to a single-independent-variable
model. To see this, consider the estimation of the simplest possible multi-
variate model, one with just two independent variables:
(8)
The goal of OLS is to choose those that minimize the summed square resid-
uals. These residuals are now from a multivariate model, but they can be mini-
mized using the same mathematical approach used in Section 1. Thus the
�̂s
Yi 5 �0 1 �1X1i 1 �2X2i 1 �i
CBt 5 37.54 2 0.88Pt 1 11.9Ydt
�0
�0
43

ORDINARY LEAST SQUARES
4. For Equation 8, the estimated coefficients are:
where lowercase variables indicate deviations from the mean, as in
and x2 5 X2i 2 X2.
y 5 Yi 2 Y; x1 5 X1i 2 X1;
�̂0 5 Y 2 �̂1X1 2 �̂2X2
�̂2 5
(gyx2) (gx
2
1) 2 (gyx1) (gx1x2)
(gx21) (gx
2
2) 2 (gx1x2)
2
�̂1 5
(gyx1) (gx
2
2) 2 (gyx2) (gx1x2)
(gx21) (gx
2
2) 2 (gx1x2)
2
OLS estimation of multivariate models is identical in general approach to the
OLS estimation of models with just one independent variable. The equations
themselves are more cumbersome,4 but the underlying principle of estimating
that minimize the summed squared residuals remains the same.
Luckily, user-friendly computer packages can calculate estimates with
these cumbersome equations in less than a second of computer time. Indeed,
only someone lost in time or stranded on a desert island would bother esti-
mating a multivariate regression model without a computer. The rest of us
will use EViews, Stata, SPSS, SAS, or any of the other commercially available
regression packages.
An Example of a Multivariate Regression Model
As an example of multivariate regression, let’s take a look at a model of financial
aid awards at a liberal arts college. The dependent variable in such a study
would be the amount, in dollars, awarded to a particular financial aid applicant:
FINAIDi � the financial aid (measured in dollars of grant per year)
awarded to the ith applicant
What kinds of independent variables might influence the amount of finan-
cial aid received by a given student? Well, most aid is either need-based or
merit-based, so it makes sense to consider a model that includes at least these
two attributes:
� �
FINAIDi � f(PARENTi, HSRANKi) (9)
and
FINAIDi � �0 � �1PARENTi, � �2HSRANKi � i (10)�
�̂s
44

ORDINARY LEAST SQUARES
5. These data are from an unpublished analysis of financial aid awards at Occidental College.
The fourth variable in Table 2 is MALEi, which equals 1 if the ith student is male and 0 otherwise.
where: PARENTi � the amount (in dollars per year) that the parents of
the ith student are judged able to contribute to col-
lege expenses
HSRANKi � the ith student’s GPA rank in high school, measured
as a percentage (ranging from a low of 0 to a high
of 100)
Note from the signs over the independent variables in Equation 9 that we
anticipate that the more parents can contribute to their child’s education, the
less the financial aid award will be. Similarly, we expect that the higher the
student’s rank in high school, the higher the financial aid award will be. Do
you agree with these expectations?
If we estimate Equation 10 using OLS and the data5 in Table 2, we get:
FINAIDi � 8927� 0.36PARENTi � 87.4HSRANKi (11)
What do these coefficients mean? Well, the –0.36 means that the model
implies that the ith student’s financial aid grant will fall by $0.36 for every
dollar increase in his or her parents’ ability to pay, holding constant high
school rank. Does the sign of the estimated coefficient meet our expecta-
tions? Yes. Does the size of the coefficient make sense? Yes.
To be sure that you understand this concept, take the time to write down
the meaning of the coefficient of HSRANK in Equation 11. Do you agree
that the model implies that the ith student’s financial aid grant will in-
crease by $87.40 for each percentage point increase in high school rank,
holding constant parents’ ability to pay? Does this estimated coefficient
seem reasonable?
To illustrate, take a look at Figures 1 and 2. These figures contain two dif-
ferent views of Equation 11. Figure 1 is a diagram of the effect of PARENT
on FINAID, holding HSRANK constant, and Figure 2 shows the effect of
HSRANK on FINAID, holding PARENT constant. These two figures are graph-
ical representations of multivariate regression coefficients, since they mea-
sure the impact on the dependent variable of a given independent variable,
holding constant the other variables in the equation.
45

ORDINARY LEAST SQUARES
0
Slope = 87.40 = �2 (holding PARENTi constant)
HSRANK i
FINAID i
Figure 2 Financial Aid as a Function of High School Rank
In Equation 11, an increase of one percentage point in high school rank increases the
financial aid award by $87.40, holding constant parents’ ability to pay.
0
Slope = 20.36 = �1 (holding HSRANK i constant)
PARENTi
FINAID i
Figure 1 Financial Aid as a Function of Parents’ Ability to Pay
In Equation 11, an increase of one dollar in the parents’ ability to pay decreases the
financial aid award by $0.36, holding constant high school rank.
46

ORDINARY LEAST SQUARES
Table 2 Data for the Financial Aid Example
i FINAID PARENT HSRANK MALE
1 19,640 0 92 0
2 8,325 9,147 44 1
3 12,950 7,063 89 0
4 700 33,344 97 1
5 7,000 20,497 95 1
6 11,325 10,487 96 0
7 19,165 519 98 1
8 7,000 31,758 70 0
9 7,925 16,358 49 0
10 11,475 10,495 80 0
11 18,790 0 90 0
12 8,890 18,304 75 1
13 17,590 2,059 91 1
14 17,765 0 81 0
15 14,100 15,602 98 0
16 18,965 0 80 0
17 4,500 22,259 90 1
18 7,950 5,014 82 1
19 7,000 34,266 98 1
20 7,275 11,569 50 0
21 8,000 30,260 98 1
22 4,290 19,617 40 1
23 8,175 12,934 49 1
24 11,350 8,349 91 0
25 15,325 5,392 82 1
26 22,148 0 98 0
27 17,420 3,207 99 0
28 18,990 0 90 0
29 11,175 10,894 97 0
30 14,100 5,010 59 0
31 7,000 24,718 97 1
32 7,850 9,715 84 1
33 0 64,305 84 0
34 7,000 31,947 98 1
35 16,100 8,683 95 1
36 8,000 24,817 99 0
37 8,500 8,720 20 1
38 7,575 12,750 89 1
39 13,750 2,417 41 1
40 7,000 26,846 92 1
41 11,200 7,013 86 1
42 14,450 6,300 87 0
(continued)
47

ORDINARY LEAST SQUARES
Table 2 (continued)
i FINAID PARENT HSRANK MALE
43 15,265 3,909 84 0
44 20,470 2,027 99 1
45 9,550 12,592 89 0
46 15,970 0 57 0
47 12,190 6,249 84 0
48 11,800 6,237 81 0
49 21,640 0 99 0
50 9,200 10,535 68 0
Datafile � FINAID2
Total, Explained, and Residual Sums of Squares
Before going on, let’s pause to develop some measures of how much of the
variation of the dependent variable is explained by the estimated regression
equation. Such comparison of the estimated values with the actual values can
help a researcher judge the adequacy of an estimated regression.
Econometricians use the squared variations of Y around its mean as a
measure of the amount of variation to be explained by the regression. This
computed quantity is usually called the total sum of squares, or TSS, and is
written as:
(12)
For Ordinary Least Squares, the total sum of squares has two components,
variation that can be explained by the regression and variation that cannot:
(13)
Total Sum � Explained � Residual
of Sum of Sum of
Squares Squares Squares
(TSS) (ESS) (RSS)
This is usually called the decomposition of variance.
Figure 3 illustrates the decomposition of variance for a simple regression
model. The estimated values of Yi lie on the estimated regression line
g
i
(Yi 2 Y)
2 5 g
i
(Ŷi 2 Y)
2 1 g
i
e2i
TSS 5 g
N
i 5 1
(Yi 2 Y)
2
48

ORDINARY LEAST SQUARES
Y
0 XiX
= Yi – Yi
Yi – Y
X
�0
ei
Y
(Xi, Yi)
(Xi, Yi)
Yi – Y
Yi = + �iXi
Figure 3 Decomposition of the Variance in Y
The variation of Y around its mean can be decomposed into two parts:
(1) the difference between the estimated value of and the mean value of
and (2) the difference between the actual value of Y and the estimated
value of Y.
(Yi 2 Ŷi),Y (Y);
Y(Ŷ)(Ŷi 2 Y),
(Y 2 Y)
6. Note that some authors reverse the definitions of RSS and ESS (defining ESS as ), and
other authors reverse the order of the letters, as in SSR.
ge˛2i
. The variation of Y around its mean can be decom-
posed into two parts: (1) the difference between the estimated
value of and the mean value of and (2) the difference
between the actual value of Y and the estimated value of Y.
The first component of Equation 13 measures the amount of the squared
deviation of Yi from its mean that is explained by the regression line. This
component of the total sum of the squared deviations, called the explained
sum of squares, or ESS, is attributable to the fitted regression line. The un-
explained portion of TSS (that is, unexplained in an empirical sense by the
estimated regression equation), is called the residual sum of squares, or
RSS.6
(Yi 2 Ŷi),Y (Y);Y (Ŷ)
(Ŷi 2 Y),
(Yi 2 Y)Ŷi 5 �̂0 1 �̂1Xi
49

ORDINARY LEAST SQUARES
We can see from Equation 13 that the smaller the RSS is relative to the TSS,
the better the estimated regression line fits the data. OLS is the estimating
technique that minimizes the RSS and therefore maximizes the ESS for a
given TSS.
Evaluating the Quality of a Regression Equation
If the bread and butter of regression analysis is OLS estimation, then the heart
and soul of econometrics is figuring out how good these OLS estimates are.
Many beginning econometricians have a tendency to accept regression es-
timates as they come out of a computer, or as they are published in an article,
without thinking about the meaning or validity of those estimates. Such
blind faith makes as much sense as buying an entire wardrobe of clothes
without trying them on. Some of the clothes will fit just fine, but many oth-
ers will turn out to be big (or small) mistakes.
Instead, the job of an econometrician is to carefully think about and eval-
uate every aspect of the equation, from the underlying theory to the quality
of the data, before accepting a regression result as valid. In fact, most good
econometricians spend quite a bit of time thinking about what to expect
from an equation before they estimate that equation.
Once the computer estimates have been produced, however, it’s time to
evaluate the regression results. The list of questions that should be asked dur-
ing such an evaluation is long. For example:
1. Is the equation supported by sound theory?
2. How well does the estimated regression fit the data?
3. Is the data set reasonably large and accurate?
4. Is OLS the best estimator to be used for this equation?
5. How well do the estimated coefficients correspond to the expectations
developed by the researcher before the data were collected?
6. Are all the obviously important variables included in the equation?
7. Has the most theoretically logical functional form been used?
8. Does the regression appear to be free of major econometric problems?
The goal of this text is to help you develop the ability to ask and appropri-
ately answer these kinds of questions. The rest of the chapter will be devoted
to the second of these topics—the overall fit of the estimated model.
3
50

ORDINARY LEAST SQUARES
Describing the Overall Fit of the Estimated Model
Let’s face it: we expect that a good estimated regression equation will explain
the variation of the dependent variable in the sample fairly accurately. If it
does, we say that the estimated model fits the data well.
Looking at the overall fit of an estimated model is useful not only for eval-
uating the quality of the regression, but also for comparing models that have
different data sets or combinations of independent variables. We can never
be sure that one estimated model represents the truth any more than another,
but evaluating the quality of the fit of the equation is one ingredient in a
choice between different formulations of a regression model. Be careful, how-
ever! The quality of the fit is a minor ingredient in this choice, and many be-
ginning researchers allow themselves to be overly influenced by it.
R2
The simplest commonly used measure of fit is R2 or the coefficient of deter-
mination. R2 is the ratio of the explained sum of squares to the total sum of
squares:
(14)
The higher R2 is, the closer the estimated regression equation fits the sam-
ple data. Measures of this type are called “goodness of fit” measures. R2
measures the percentage of the variation of Y around that is explained
by the regression equation. Since OLS selects the coefficient estimates that
minimize RSS, OLS provides the largest possible R2, given a linear model.
Since TSS, RSS, and ESS are all nonnegative (being squared deviations),
and since R2 must lie in the interval , a value of R2
close to one shows an excellent overall fit, whereas a value near zero
shows a failure of the estimated regression equation to explain the values
of Yi better than could be explained by the sample mean Y.
0 # R2 # 1ESS # TSS,
Y
R2 5
ESS
TSS
5 1 2
RSS
TSS
5 1 2
ge˛2i
g (Yi 2 Y)
2
4
51

ORDINARY LEAST SQUARES
Y
Y
0 X
Regression Line
R2 = 0
Figure 4
X and Y are not related; in such a case, R2 would be 0.
Figures 4 through 6 demonstrate some extremes. Figure 4 shows an X and
Y that are unrelated. The fitted regression line might as well be the
same value it would have if X were omitted. As a result, the estimated linear
regression is no better than the sample mean as an estimate of Yi. The ex-
plained portion, ESS, � 0, and the unexplained portion, RSS, equals the total
squared deviations TSS; thus, R2 � 0.
Figure 5 shows a relationship between X and Y that can be “explained”
quite well by a linear regression equation: the value of R2 is .95. This kind of
result is typical of a time-series regression with a good fit. Most of the varia-
tion has been explained, but there still remains a portion of the variation that
is essentially random or unexplained by the model.
Goodness of fit is relative to the topic being studied. In time series data,
we often get a very high R2 because there can be significant time trends on
both sides of the equation. In cross-sectional data, we often get low R2s
because the observations (say, countries) differ in ways that are not easily
quantified. In such a situation, an R2 of .50 might be considered a good
fit, and researchers would tend to focus on identifying the variables that
have a substantive impact on the dependent variable, not on R2. In other
words, there is no simple method of determining how high R2 must be for
the fit to be considered satisfactory. Instead, knowing when R2 is relatively
large or small is a matter of experience. It should be noted that a high
R2 does not imply that changes in X lead to changes in Y, as there may be
an underlying variable whose changes lead to changes in both X and Y
simultaneously.
Ŷ 5 Y,
52

ORDINARY LEAST SQUARES
Figure 6 shows a perfect fit of R2 � 1. Such a fit implies that no estima-
tion is required. The relationship is completely deterministic, and the
slope and intercept can be calculated from the coordinates of any two
points. In fact, reported equations with R2s equal to (or very near) one
should be viewed with suspicion; they very likely do not explain the move-
ments of the dependent variable Y in terms of the causal proposition ad-
vanced, even though they explain them empirically. This caution applies to
economic applications, but not necessarily to those in fields like physics or
chemistry.
Y
0 X
R2 = .95
Figure 5
A set of data for X and Y that can be “explained” quite well with a regression line
(R2 � .95).
Y
0 X
R2 = 1
Figure 6
A perfect fit: all the data points are on the regression line, and the resulting R2 is 1.
53

ORDINARY LEAST SQUARES
The Simple Correlation Coefficient, r
A related measure that will prove useful in future chapters is “r,” the simple
correlation coefficient. The simple correlation coefficient, r, is a measure of
the strength and direction of the linear relationship between two variables.7
The range of r is from �1 to �1, and the sign of r indicates the direction of
the correlation between the two variables. The closer the absolute value of r is
to 1, the stronger the correlation between the two variables. Thus:
7. The equation for r12, the simple correlation coefficient between X1 and X2, is:
r12 5
g f (X1i 2 X1) (X2i 2 X2)g
“g (X1i 2 X1)
2 g (X2i 2 X2)
2
If two variables are perfectly positively correlated, then r � �1
If two variables are perfectly negatively correlated, then r � �1
If two variables are totally uncorrelated, then r � 0
We’ll use the simple correlation coefficient to describe the correlation be-
tween two variables. Interestingly, it turns out that r and R2 are related if the
estimated equation has exactly one independent variable. The square of r
equals R2 for a regression where one of the two variables is the dependent
variable and the other is the only independent variable.
, The Adjusted R2
A major problem with R2 is that adding another independent variable to a
particular equation can never decrease R2. That is, if you compare two equa-
tions that are identical (same dependent variable and independent variables),
except that one has an additional independent variable, the equation with the
greater number of independent variables will always have a better (or equal)
fit as measured by R2.
To see this, recall the equation for R2, Equation 14.
(14)R2 5
ESS
TSS
5 1 2
RSS
TSS
5 1 2
ge˛2i
g (Yi 2 Y)
2
R2
54

ORDINARY LEAST SQUARES
What will happen to R2 if we add a variable to the equation? Adding a vari-
able can’t change TSS (can you figure out why?), but in most cases the added
variable will reduce RSS, so R2 will rise. You know that RSS will never increase
because the OLS program could always set the coefficient of the added vari-
able equal to zero, thus giving the same fit as the previous equation. The coef-
ficient of the newly added variable being zero is the only circumstance in
which R2 will stay the same when a variable is added. Otherwise, R2 will
always increase when a variable is added to an equation.
Perhaps an example will make this clear. Let’s return to our weight guess-
ing regression:
The R2 for this equation is .74. If we now add a completely nonsensical
variable to the equation (say, the campus post office box number of each in-
dividual in question), then it turns out that the results become:
but the R2 for this equation is .75! Thus, an individual using R2 alone as the
measure of the quality of the fit of the regression would choose the second
version as better fitting.
The inclusion of the campus post office box variable not only adds a non-
sensical variable to the equation, but it also requires the estimation of another
coefficient. This lessens the degrees of freedom, or the excess of the number of
observations (N) over the number of coefficients (including the intercept) esti-
mated (K � 1). For instance, when the campus box number variable is added
to the weight/height example, the number of observations stays constant at 20,
but the number of estimated coefficients increases from 2 to 3, so the number
of degrees of freedom falls from 18 to 17. This decrease has a cost, since the
lower the degrees of freedom, the less reliable the estimates are likely to be.
Thus, the increase in the quality of the fit caused by the addition of a variable
needs to be compared to the decrease in the degrees of freedom before a deci-
sion can be made with respect to the statistical impact of the added variable.
To sum, R2 is of little help if we’re trying to decide whether adding a variable
to an equation improves our ability to meaningfully explain the dependent
variable. Because of this problem, econometricians have developed another
measure of the quality of the fit of an equation. That measure is (pro-
nounced R-bar-squared), which is R2 adjusted for degrees of freedom:
(15)R2 5 1 2
ge˛2i >(N 2 K 2 1)
g (Yi 2 Y)
2
>(N 2 1)
R2
Estimated weight 5 102.35 1 6.36 (Height . five feet) 1 0.02 (Box#)
Estimated weight 5 103.40 1 6.38 ? Height (over five feet)
55

ORDINARY LEAST SQUARES
will increase, decrease, or stay the same when a variable is added to an
equation, depending on whether the improvement in fit caused by the addi-
tion of the new variable outweighs the loss of the degree of freedom. Indeed,
the for the weight-guessing equation decreases to .72 when the mail box
variable is added. The mail box variable, since it has no theoretical relation to
weight, should never have been included in the equation, and the measure
supports this conclusion.
The highest possible is 1.00, the same as for R2. The lowest possible
however, is not .00; if R2 is extremely low, can be slightly negative.R2
R2,R2
R2
R2
R2
Finally, a warning is in order. Always remember that the quality of fit of an
estimated equation is only one measure of the overall quality of that regres-
sion. As mentioned previously, the degree to which the estimated coefficients
conform to economic theory and the researcher’s previous expectations
about those coefficients are just as important as the fit itself. For instance, an
estimated equation with a good fit but with an implausible sign for an esti-
mated coefficient might give implausible predictions and thus not be a very
useful equation. Other factors, such as theoretical relevance and usefulness,
also come into play. Let’s look at an example of these factors.
An Example of the Misuse of
Section 4 implies that the higher the overall fit of a given equation, the
better. Unfortunately, many beginning researchers assume that if a high is
good, then maximizing is the best way to maximize the quality of an
equation. Such an assumption is dangerous because a good overall fit is only
one measure of the quality of an equation.
R2
R2
R25
measures the percentage of the variation of Y around its mean that is
explained by the regression equation, adjusted for degrees of freedom.
R2
can be used to compare the fits of equations with the same dependent
variable and different numbers of independent variables. Because of this
property, most researchers automatically use instead of R2 when evaluat-
ing the fit of their estimated regression equations.
R2
R2
56

ORDINARY LEAST SQUARES
8. The principle involved in this section is the same one that was discussed during the actual
research, but these coefficients are hypothetical because the complexities of the real equation
are irrelevant to our points.
Perhaps the best way to visualize the dangers inherent in maximizing
without regard to the economic meaning or statistical significance of an
equation is to look at an example of such misuse. This is important because it
is one thing for a researcher to agree in theory that “ maximizing” is bad,
and it is another thing entirely for that researcher to avoid subconsciously
maximizing on projects. It is easy to agree that the goal of regression is not
to maximize but many researchers find it hard to resist that temptation.
As an example, assume that you’ve been hired by the State of California to
help the legislature evaluate a bill to provide more water to Southern Califor-
nia.8 This issue is important because a decision must be made whether to
ruin, through a system of dams, one of the state’s best trout fishing areas. On
one side of the issue are Southern Californians who claim that their desert-
like environment requires more water; on the other side are nature lovers and
environmentalists who want to retain the natural beauty for which California
is famous. Your job is to forecast the amount of water demanded in Los An-
geles County, the biggest user of water in the state.
Because the bill is about to come before the state legislature, you’re forced to
choose between two regressions that already have been run for you, one by the
state econometrician and the other by an independent consultant. You will base
your forecast on one of these two equations. The state econometrician’s equation:
(16)
or the independent consultant’s equation:
(17)
where: � the total amount of water consumed in Los Angeles County
in a given year (measured in millions of gallons)
PR � the price of a gallon of water that year (measured in real
dollars)
P � the population in Los Angeles County that year
RF � the amount of rainfall that year (measured in inches)
DF � degrees of freedom, which equal the number of observations
(N � 29) minus the number of coefficients estimated
W
R2 5 .847 DF 5 26
Ŵ 5 30,000 1 0.62P 2 400RF
R2 5 .859 DF 5 25
Ŵ 5 24,000 1 48,000PR 1 0.40P 2 370RF
R2,
R2
R2
R2
57

9. A couple of caveats to this example are in order. First, we normally wouldn’t leave price out
of a demand equation, but it’s appropriate to do so here because the unexpected sign for the
coefficient of price would otherwise cause forecast errors. Second, average rainfall would be
used in forecasts, because future rainfall would not be known. Finally, income does indeed
belong in the equation, but it turns out to have a relatively small coefficient, because water
expenditure is minor in relation to the overall budget.
ORDINARY LEAST SQUARES
Review these two equations carefully before going on with the rest of the
section. What do you think the arguments of the state econometrician were
for using his equation? What case did the independent econometrician make
for her work?
The question is whether the increased is worth the unexpected sign in
the price of water coefficient in Equation 16. The state econometrician ar-
gued that given the better fit of his equation, it would do a better job of fore-
casting water demand. The independent consultant argued that it did not
make sense to expect that an increase in price in the future would, holding
the other variables in the equation constant, increase the quantity of water
demanded in Los Angeles. Furthermore, given the unexpected sign of the co-
efficient, it seemed much more likely that the demand for water was unre-
lated to price during the sample period or that some important variable
(such as real per capita income) had been left out of both equations. Since
the amount of money spent on water was fairly low compared with other
expenditures during the sample years, the consultant pointed out, it was pos-
sible that the demand for water was fairly price-inelastic. The economic argu-
ment for the positive sign observed by the state econometrician is difficult to
justify; it implies that as the price of water goes up, so does the quantity of
water demanded.
Was this argument simply academic? The answer, unfortunately, is no. If a
forecast is made with Equation 16, it will tend to overforecast water demand
in scenarios that foresee rising prices and underforecast water demand with
lower price scenarios. In essence, the equation with the better fit would do a
worse job of forecasting.9
Thus, a researcher who uses as the sole measure of the quality of an
equation (at the expense of economic theory or statistical significance) in-
creases the chances of having unrepresentative or misleading results. This
practice should be avoided at all costs. No simple rule of econometric esti-
mation is likely to work in all cases. Instead, a combination of technical
competence, theoretical judgment, and common sense makes for a good
econometrician.
R2
R2
58

ORDINARY LEAST SQUARES
To help avoid the natural urge to maximize without regard to the
rest of the equation, you might find it useful to imagine the following
conversation:
You: Sometimes, it seems like the best way to choose between two models
is to pick the one that gives the highest
Your Conscience: But that would be wrong.
You: I know that the goal of regression analysis is to obtain the best possi-
ble estimates of the true population coefficients and not to get a high but
my results “look better” if my fit is good.
Your Conscience: Look better to whom? It’s not at all unusual to get a high
but find that some of the regression coefficients have signs that are con-
trary to theoretical expectations.
You: Well, I guess I should be more concerned with the logical relevance of
the explanatory variables than with the fit, huh?
Your Conscience: Right! If in this process we obtain a high , well and
good, but if is high, it doesn’t mean that the model is good.
Summary
1. Ordinary Least Squares (OLS) is the most frequently used method of
obtaining estimates of the regression coefficients from a set of data.
OLS chooses those that minimize the summed squared residuals
for a particular sample.
2. R-bar-squared measures the percentage of the variation of Y
around its mean that has been explained by a particular regression
equation, adjusted for degrees of freedom. increases when a vari-
able is added to an equation only if the improvement in fit caused
by the addition of the new variable more than offsets the loss of the
degree of freedom that is used up in estimating the coefficient of the
new variable. As a result, most researchers will automatically use
when evaluating the fit of their estimated regression equations.
3. Always remember that the fit of an estimated equation is only one of
the measures of the overall quality of that regression. A number of
other criteria, including the degree to which the estimated coefficients
conform to economic theory and expectations (developed by the re-
searcher before the data were collected) are more important than the
size of .R2
R2
R2
(R2)
(ge2i )
�̂s
6
R2
R2
R2
R2,
R2.
R2
59

ORDINARY LEAST SQUARES
EXERCISES
(The answer to Exercise 2 is at the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or your notes), and compare your definition with the ver-
sion in the text for each:
a. Ordinary Least Squares
b. the meaning of a multivariate regression coefficient
c. total, explained, and residual sums of squares
d. simple correlation coefficient
e. degrees of freedom
f.
2. Just as you are about to estimate a regression (due tomorrow), mas-
sive sunspots cause magnetic interference that ruins all electrically
powered machines (e.g., computers). Instead of giving up and flunk-
ing, you decide to calculate estimates from your data (on per capita
income in thousands of U.S. dollars as a function of the percent of
the labor force in agriculture in 10 developed countries) using meth-
ods like those used in Section 1 without a computer. Your data are:
Country A B C D E F G H I J
Per Capita Income 6 8 8 7 7 12 9 8 9 10
% in Agriculture 9 10 8 7 10 4 5 5 6 7
a. Calculate and
b. Calculate R2 and
c. If the percent of the labor force in agriculture in another developed
country was 8 percent, what level of per capita income (in thou-
sands of U.S. dollars) would you guess that country had?
3. To get more practice in the use of summation notation, use the data
in Exercise 2 on per capita income (Y) and percent of the labor force
in agriculture (X) to answer the following questions. (Hint: Before
starting this exercise, reread footnote 1 in this chapter which defines
a. Calculate (Hint: Note that N � 10.)
b. Calculate
c. Calculate Does it equal 3 ?
d. Calculate . Does it equal ?gX 1 gYg (X 1 Y)
gXg3X.
gY.
gX.
gX 5 X1 1 X2 1
c1 XN.)
R2.
�̂1.�̂0
R2
60

ORDINARY LEAST SQUARES
4. Consider the following two least-squares estimates of the relationship
between interest rates and the federal budget deficit in the United States:
Model A:
where: Y1 � the interest rate on Aaa corporate bonds
X1 � the federal budget deficit as a percentage of GNP
(quarterly model: N � 56)
Model T:
where: Y2 � the interest rate on 3-month Treasury bills
X2 � the federal budget deficit in billions of dollars
X3 � the rate of inflation (in percent)
(quarterly model: N � 38)
a. What does “least-squares estimates” mean? What is being estimated?
What is being squared? In what sense are the squares “least”?
b. What does it mean to have an R2 of .00? Is it possible for an R2 to
be negative?
c. Based on economic theory, what signs would you have expected for
the estimated slope coefficients of the two models?
d. Compare the two equations. Which model has estimated signs that
correspond to your prior expectations? Is Model T automatically
better because it has a higher R2? If not, which model do you prefer
and why?
5. Let’s return to the height-weight example presented earlier and recall
what happened when we added a nonsensical variable that measured
the student’s campus post office box number (MAIL) to the equation.
The estimated equation changed from:
WEIGHT � 103.40 � 6.38HEIGHT
to:
WEIGHT � 102.35 � 6.36HEIGHT � 0.02MAIL
a. The estimated coefficient of HEIGHT changed when we added
MAIL to the equation. Does that make sense? Why?
b. In theory, someone’s weight has nothing to do with their campus
mail box number, yet R2 went up from .74 to .75 when MAIL was
added to the equation! How is it possible that adding a nonsensi-
cal variable to an equation can increase R2?
Ŷ2 5 0.089 1 0.369X2 1 0.887X3 R2 5 .40
Ŷ1 5 0.103 2 0.079X1 R2 5 .00
61

ORDINARY LEAST SQUARES
10. David Romer, “Do Students Go to Class? Should They?” Journal of Economic Perspectives,
Vol. 7, No. 3, pp. 167–174.
c. Adding the nonsensical variable to the equation decreased from
.73 to .72. Explain how it’s possible that can go down at the
same time that R2 goes up.
d. If a person’s campus mail box number truly is unrelated to their
weight, shouldn’t the estimated coefficient of that variable equal
exactly 0.00? How is it possible for a nonsensical variable to get a
nonzero estimated coefficient?
6. In an effort to determine whether going to class improved student aca-
demic performance, David Romer10 developed the following equation:
where: Gi � the grade of the ith student in Romer’s class (A � 4,
B � 3, etc.)
ATTi � the percent of class lectures that the ith student
attended
PSi � the percent of the problem sets that the ith student
completed
a. What signs do you expect for the coefficients of the independent
variables in this equation? Explain your reasoning.
b. Romer then estimated the equation:
Do the estimated results agree with your expectations?
c. It’s usually easier to develop expectations about the signs of coeffi-
cients than about the size of those coefficients. To get an insight
into the size of the coefficients, let’s assume that there are 25 hours
of lectures in a semester and that it takes the average student
approximately 50 hours to complete all the problem sets in a se-
mester. If a student in one of Romer’s classes had only one more
hour to devote to class and wanted to maximize the impact on his
or her grade, should the student go to class for an extra hour or
work on problem sets for an extra hour? (Hint: Convert the extra
hour to percentage terms and then multiply those percentages by
the estimated coefficients.)
d. From the given information, it’d be easy to draw the conclusion
that the bigger a variable’s coefficient, the greater its impact on the
N 5 195 R2 5 .33
Ĝi 5 1.07 1 1.74ATTi 1 0.60PSi
Gi 5 f(ATTi, PSi) 1 �i
R2
R2
62

ORDINARY LEAST SQUARES
dependent variable. To test this conclusion, what would your an-
swer to part c have been if there had been 50 hours of lecture in a
semester and if it had taken 10 hours for the average student to
complete the problem sets? Were we right to conclude that the
larger the estimated coefficient, the more important the variable?
e. What’s the real-world meaning of having R2 � .33? For this specific
equation, does .33 seem high, low, or just about right?
f. Is it reasonable to think that only class attendance and problem-set
completion affect your grade in a class? If you could add just one more
variable to the equation, what would it be? Explain your reasoning.
What should adding your variable to the equation do to R2? To ?
7. Suppose that you have been asked to estimate a regression model to
explain the number of people jogging a mile or more on the school
track to help decide whether to build a second track to handle all the
joggers. You collect data by living in a press box for the spring semes-
ter, and you run two possible explanatory equations:
where: Y � the number of joggers on a given day
X1 � inches of rain that day
X2 � hours of sunshine that day
X3 � the high temperature for that day (in degrees F)
X4 � the number of classes with term papers due the
next day
a. Which of the two (admittedly hypothetical) equations do you pre-
fer? Why?
b. How is it possible to get different estimated signs for the coefficient
of the same variable using the same data?
8. David Katz11 studied faculty salaries as a function of their “productivity”
and estimated a regression equation with the following coefficients:
Ŝi 5 22,310 1 460Bi 1 36Ai 1 204Ei 1 978Di 1 378Yi 1
c
B: Ŷ 5 123.0 2 14.0X1 1 5.5X2 2 3.7X4 R
2 5 .73
A: Ŷ 5 125.0 2 15.0X1 2 1.0X2 1 1.5X3 R2 5 .75
R2
11. David A. Katz, “Faculty Salaries, Promotions, and Productivity at a Large University,”
American Economic Review, Vol. 63, No. 3, pp. 469–477. Katz’s equation included other variables
as well, as indicated by the “ ” at the end of the equation. Estimated coefficients have been
adjusted for inflation.
1 c
63

12. Charles A. Lave, “Speeding, Coordination, and the 55 MPH Limit,” American Economic
Review, Vol. 75, No. 5, pp. 1159–1164.
ORDINARY LEAST SQUARES
where: Si � the salary of the ith professor in dollars per year
Bi � the number of books published, lifetime
Ai � the number of articles published, lifetime
Ei � the number of “excellent” articles published, lifetime
Di � the number of dissertations supervised
Yi � the number of years teaching experience
a. Do the signs of the coefficients match your prior expectations?
b. Do the relative sizes of the coefficients seem reasonable? (Hint:
Most professors think that it’s much more important to write an ex-
cellent article than to supervise a dissertation.)
c. Suppose a professor had just enough time (after teaching, etc.) to
write a book, write two excellent articles, or supervise three disser-
tations. Which would you recommend? Why?
d. Would you like to reconsider your answer to part b? Which coeffi-
cient seems out of line? What explanation can you give for that re-
sult? Is the equation in some sense invalid? Why or why not?
9. What’s wrong with the following kind of thinking: “I understand that
R2 is not a perfect measure of the quality of a regression equation be-
cause it always increases when a variable is added to the equation.
Once we adjust for degrees of freedom by using , though, it seems
to me that the higher the , the better the equation.”
10. Charles Lave12 published a study of driver fatality rates. His overall con-
clusion was that the variance of driving speed (the extent to which vehi-
cles sharing the same highway drive at dramatically different speeds) is
important in determining fatality rates. As part of his analysis, he esti-
mated an equation with cross-state data from two different years:
where: Fi � the fatalities on rural interstate highways (per 100
million vehicle miles traveled) in the ith state
� an unspecified estimated intercept�̂0
R2 5 .532 N 5 44
Year 2: F̂i 5 �̂0 1 0.190 Vi 1 0.0071Ci 2 5.29Hi
R2 5 .624 N 5 41
Year 1: F̂i 5 �̂0 1 0.176 Vi 1 0.0136Ci 2 7.75Hi
R2
R2
64

ORDINARY LEAST SQUARES
Vi � the driving speed variance in the ith state
Ci � driving citations per driver in the ith state
Hi � hospitals per square mile (adjusted) in the ith state
a. Think through the theory behind each variable, and develop ex-
pected signs for each coefficient. (Hint: Be careful with C.) Do
Lave’s estimates support your expectations?
b. Should we attach much meaning to the differences between the esti-
mated coefficients from the two years? Why or why not? Under what
circumstances might you be concerned about such differences?
c. The equation for the first year has the higher but which equa-
tion has the higher R2? (Hint: You can calculate the R2s with the in-
formation given, but such a calculation isn’t required.)
11. In Exercise 5 in Chapter 1, we estimated a height/weight equation on
a new data set of 29 male customers, Equation 1.24:
where: Yi � the weight (in pounds) of the ith person
Xi � the height (in inches above five feet) of the ith person
Suppose that a friend now suggests adding Fi, the percent body fat
of the ith person, to the equation.
a. What is the theory behind adding Fi to the equation? How does the
meaning of the coefficient of X change when you add F?
b. Assume you now collect data on the percent body fat of the 29
males and estimate:
(18)
Do you prefer Equation 18 or the first equation listed above? Why?
c. Suppose you learn that the of Equation the first equation is .75
and the of Equation 18 is .72. Which equation do you prefer
now? Explain your answer.
d. Suppose that you learn that the mean of F for your sample is 12.0.
Which equation do you prefer now? Explain your answer.
12. For students with a background in calculus, the derivation of Equa-
tions 4 and 5 is useful. Derive these two equations by carrying out the
following steps. (Hint: Be sure to write out each step of the proof.)
a. Differentiate the second equation in footnote 2 with respect to
and then with respect to
b. Set these two derivatives equal to zero, thus creating what are called
the “normal equations.”
�̂1.
�̂0
R2
R2
Ŷi 5 120.8 1 4.11Xi 1 0.28Fi
Ŷi 5 125.1 1 4.03Xi
R2,
65

ORDINARY LEAST SQUARES
c. Solve the normal equations for obtaining Equation 4.
d. Solve the normal equations for obtaining Equation 5.
13. Suppose that you work in the admissions office of a college that
doesn’t allow prospective students to apply by using the Common
Application.13 How might you go about estimating the number of
extra applications that your college would generate if it allowed the
use of the Common Application? An econometric approach to this
question would be to build the best possible model of the number of
college applications and then to examine the estimated coefficient of
a dummy variable that equaled one if the college in question allowed
the use of the “common app” (and zero otherwise).
For example, if we estimate an equation using the data in Table 3
for high-quality coed national liberal arts colleges, we get:
APPLICATIONi � 523.3 � 2.15SIZEi � 32.1RANKi
� 1222COMMONAPPi
(19)
N � 49 R2 � .724 � .705
where: APPLICATIONi � the number of applications received by
the ith college in 2007
SIZEi � the total number of undergraduate stu-
dents at the ith college in 2006
RANKi � the U.S. News
14 rank of the ith college
(1 � best) in 2006
COMMONAPPi � a dummy variable equal to 1 if the ith
college allowed the use of the Common
Application in 2007 and 0 otherwise.
a. Take a look at the signs of each of the three estimated regression
coefficients. Are they what you would have expected? Explain.
b. Carefully state the real-world meaning of the coefficients of SIZE
and RANK. Does the fact that the coefficient of RANK is 15 times
bigger (in absolute value) than the coefficient of SIZE mean that
the ranking of a college is 15 times more important than the size
R2
�̂0,
�̂1,
13. The Common Application is a computerized application form that allows high school stu-
dents to apply to a number of different colleges and universities using the same basic data. For
more information, go to www.commonap.org.
14. U.S. News and World Report Staff, U.S. News Ultimate College Guide. Naperville, Illinois:
Sourcebooks, Inc., 2006–2008.
66

ORDINARY LEAST SQUARES
Table 3 Data for the College Application Example
COLLEGE APPLICATION COMMONAPP RANK SIZE
Amherst College 6680 1 2 1648
Bard College 4980 1 36 1641
Bates College 4434 1 23 1744
Bowdoin College 5961 1 7 1726
Bucknell University 8934 1 29 3529
Carleton College 4840 1 6 1966
Centre College 2159 1 44 1144
Claremont McKenna
College 4140 1 12 1152
Colby College 4679 1 20 1865
Colgate University 8759 1 16 2754
College of the Holy Cross 7066 1 32 2790
Colorado College 4826 1 26 1939
Connecticut College 4742 1 39 1802
Davidson College 3992 1 10 1667
Denison University 5196 1 48 2234
DePauw University 3624 1 48 2294
Dickinson College 5844 1 41 2372
Franklin and Marshall
College 5018 1 41 1984
Furman University 3879 1 41 2648
Gettysburg College 6126 1 45 2511
Grinnell College 3077 1 14 1556
Hamilton College 4962 1 17 1802
Harvey Mudd College 2493 1 14 729
Haverford College 3492 1 9 1168
Kenyon College 4626 1 32 1630
Lafayette College 6364 1 30 2322
Lawrence University 2599 1 53 1409
Macalester College 4967 1 24 1884
Middlebury College 7180 1 5 2363
Oberlin College 7014 1 22 2744
Occidental College 5275 1 36 1783
Pitzer College 3748 1 51 918
Pomona College 5907 1 7 1545
Reed College 3365 1 53 1365
Rhodes College 3709 1 45 1662
Sewanee-University of
the South 2424 0 34 1498
Skidmore College 6768 1 48 2537
St. Lawrence University 4645 0 57 2148
St. Olaf College 4058 0 55 2984
(continued)
67

of that college in terms of explaining the number of applications to
that college? Why or why not?
c. Now carefully state the real-world meaning of the coefficient of
COMMONAPP. Does this prove that 1,222 more students would
apply if your college decided to allow the Common Application?
Explain. (Hint: There are at least two good answers to this question.
Can you get them both?)
d. To get some experience with your computer’s regression software,
use the data in Table 3 to estimate Equation 19. Do you get the
same results?
e. Now use the same data and estimate Equation 19 again without the
COMMONAPP variable. What is the new ? Does go up or down
when you drop the variable? What, if anything, does this change tell
you about whether COMMONAPP belongs in the equation?
R2R2
ORDINARY LEAST SQUARES
Table 3 (continued)
COLLEGE APPLICATION COMMONAPP RANK SIZE
Swarthmore College 5242 1 3 1477
Trinity College 5950 1 30 2183
Union College 4837 1 39 2178
University of Richmond 6649 1 34 2804
Vassar College 6393 1 12 2382
Washington and Lee
University 3719 1 17 1749
Wesleyan University 7750 1 10 2798
Wheaton College 2160 1 55 1548
Whitman College 2892 1 36 1406
Williams College 6478 1 1 2820
Sources: U.S. News & World Report Staff, U.S. News Ultimate College Guide, Naperville,
IL: Sourcebooks, Inc. 2006–2008.
Datafile � COLLEGE2
68

Answers
Exercise 2
a. 1 � �0.5477, 0 � 12.289
b. R2 � .465, � .398
c. Income � 12.289 � 0.5477 (8) � 7.907
R2
�̂�̂
ORDINARY LEAST SQUARES
69

Learning to Use
Regression Analysis
1 Steps in Applied Regression Analysis
2 Using Regression Analysis to Pick Restaurant Locations
3 Summary and Exercises
It’d be easy to conclude that regression analysis is little more than the me-
chanical application of a set of equations to a sample of data. Such a notion
would be similar to deciding that all that matters in golf is hitting the ball
well. Golfers will tell you that it does little good to hit the ball well if you
have used the wrong club or have hit the ball toward a trap, tree, or pond.
Similarly, experienced econometricians spend much less time thinking about
the OLS estimation of an equation than they do about a number of other
factors. Our goal in this chapter is to introduce some of these “real-world”
concerns.
The first section, an overview of the six steps typically taken in applied re-
gression analysis, is the most important in the chapter. We believe that the
ability to learn and understand a specific topic, like OLS estimation, is en-
hanced if the reader has a clear vision of the role that the specific topic plays
in the overall framework of regression analysis. In addition, the six steps
make it hard to miss the crucial function of theory in the development of
sound econometric research.
This is followed by a complete example of how to use the six steps in ap-
plied regression: a location analysis for the “Woody’s” restaurant chain that is
based on actual company data and to which we will return in future chapters
to apply new ideas and tests.
Steps in Applied Regression Analysis
Although there are no hard and fast rules for conducting econometric research,
most investigators commonly follow a standard method for applied regression
analysis. The relative emphasis and effort expended on each step will vary,
1
From Chapter 3 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
71

but normally all the steps are necessary for successful research. Note that we
don’t discuss the selection of the dependent variable; this choice is deter-
mined by the purpose of the research. Once a dependent variable is chosen,
however, it’s logical to follow this sequence:
LEARNING TO USE REGRESSION ANALYSIS
1. Review the literature and develop the theoretical model.
2. Specify the model: Select the independent variables and the
functional form.
3. Hypothesize the expected signs of the coefficients.
4. Collect the data. Inspect and clean the data.
5. Estimate and evaluate the equation.
6. Document the results.
The purpose of suggesting these steps is not to discourage the use of inno-
vative or unusual approaches but rather to develop in the reader a sense of
how regression ordinarily is done by professional economists and busi-
ness analysts.
Step 1: Review the Literature and Develop the Theoretical Model
The first step in any applied research is to get a good theoretical grasp of the
topic to be studied. That’s right: the best data analysts don’t start with data,
but with theory! This is because many econometric decisions, ranging from
which variables to include to which functional form to employ, are deter-
mined by the underlying theoretical model. It’s virtually impossible to build
a good econometric model without a solid understanding of the topic you’re
studying.
For most topics, this means that it’s smart to review the scholarly literature
before doing anything else. If a professor has investigated the theory behind
your topic, you want to know about it. If other researchers have estimated
equations for your dependent variable, you might want to apply one of their
models to your data set. On the other hand, if you disagree with the
approach of previous authors, you might want to head off in a new direction.
In either case, you shouldn’t have to “reinvent the wheel.” You should start
your investigation where earlier researchers left off. Any academic paper on
72

an empirical topic should begin with a summary of the extent and quality of
previous research.
The most convenient approaches to reviewing the literature are to obtain
several recent issues of the Journal of Economic Literature or a business-
oriented publication of abstracts, or to run an Internet search or an EconLit
search1 on your topic. Using these resources, find and read several recent arti-
cles on your topic. Pay attention to the bibliographies of these articles. If an
older article is cited by a number of current authors, or if its title hits your
topic on the head, trace back through the literature and find this article
as well.
In some cases, a topic will be so new or so obscure that you won’t be able
to find any articles on it. What then? We recommend two possible strategies.
First, try to transfer theory from a similar topic to yours. For example, if
you’re trying to build a model of the demand for a new product, read articles
that analyze the demand for similar, existing products. Second, if all else
fails, pick up the telephone and call someone who works in the field you’re
investigating. For example, if you’re building a model of housing in an unfa-
miliar city, call a real estate agent who works there.
Step 2: Specify the Model: Select the Independent
Variables and the Functional Form
The most important step in applied regression analysis is the specification of
the theoretical regression model. After selecting the dependent variable, the
specification of a model involves choosing the following components:
1. the independent variables and how they should be measured,
2. the functional (mathematical) form of the variables, and
3. the properties of the stochastic error term.
A regression equation is specified when each of these elements has been
treated appropriately.
Each of the elements of specification is determined primarily on the basis
of economic theory. A mistake in any of the three elements results in a
LEARNING TO USE REGRESSION ANALYSIS
1. EconLit is an electronic bibliography of economics literature. EconLit contains abstracts, reviews,
indexing, and links to full-text articles in economics journals. In addition, it abstracts books
and indexes articles in books, working papers series, and dissertations. EconLit is available at
libraries and on university websites throughout the world. For more, go to www.EconLit.org.
73

specification error. Of all the kinds of mistakes that can be made in applied
regression analysis, specification error is usually the most disastrous to the
validity of the estimated equation. Thus, the more attention paid to economic
theory at the beginning of a project, the more satisfying the regression results
are likely to be.
The emphasis in this text is on estimating behavioral equations, those that
describe the behavior of economic entities. We focus on selecting independent
variables based on the economic theory concerning that behavior. An explana-
tory variable is chosen because it is a theoretical determinant of the dependent
variable; it is expected to explain at least part of the variation in the dependent
variable. Recall that regression gives evidence but does not prove economic
causality. Just as an example does not prove the rule, a regression result does
not prove the theory.
There are dangers in specifying the wrong independent variables. Our goal
should be to specify only relevant explanatory variables, those expected theo-
retically to assert a substantive influence on the dependent variable. Variables
suspected of having little effect should be excluded unless their possible im-
pact on the dependent variable is of some particular (e.g., policy) interest.
For example, an equation that explains the quantity demanded of a con-
sumption good might use the price of the product and consumer income or
wealth as likely variables. Theory also indicates that complementary and sub-
stitute goods are important. Therefore, you might decide to include the prices
of complements and substitutes, but which complements and substitutes? Of
course, selection of the closest complements and/or substitutes is appropri-
ate, but how far should you go? The choice must be based on theoretical
judgment, and such judgments are often quite subjective.
When researchers decide, for example, that the prices of only two other
goods need to be included, they are said to impose their priors (i.e., previous
theoretical belief) or their working hypotheses on the regression equation.
Imposition of such priors is a common practice that determines the number
and kind of hypotheses that the regression equation has to test. The danger is
that a prior may be wrong and could diminish the usefulness of the esti-
mated regression equation. Each of the priors therefore should be explained
and justified in detail.
Some concepts (for example, gender) might seem impossible to include in
an equation because they’re inherently qualitative in nature and can’t be
quantified. Such concepts can be quantified by using dummy (or binary)
variables. A dummy variable takes on the values of one or zero depending on
whether a specified condition holds.
As an illustration of a dummy variable, suppose that Yi represents
the salary of the ith high school teacher, and that the salary level depends
LEARNING TO USE REGRESSION ANALYSIS
74

primarily on the experience of the teacher and the type of degree earned. All
teachers have a B.A., but some also have a graduate degree, like an M.A. An
equation representing the relationship between earnings and the type of de-
gree might be:
(1)
where:
the number of years of teaching experience of the ith
teacher
The variable X1 takes on only values of zero or one, so X1 is called a dummy
variable, or just a “dummy.” Needless to say, the term has generated many a
pun. In this case, the dummy variable represents the condition of having a
master’s degree. The coefficient indicates the additional salary that can be
attributed to having a graduate degree, holding teaching experience constant.
Step 3: Hypothesize the Expected Signs of the Coefficients
Once the variables are selected, it’s important to hypothesize the expected
signs of the regression coefficients. For example, in the demand equation for
a final consumption good, the quantity demanded (Qd) is expected to be in-
versely related to its price (P) and the price of a complementary good (Pc),
and positively related to consumer income (Y) and the price of a substitute
good (Ps). The first step in the written development of a regression model
usually is to express the equation as a general function:
(2)
The signs above the variables indicate the hypothesized sign of the respective
regression coefficient in a linear model.
In many cases, the basic theory is general knowledge, so the reasons for
each sign need not be discussed. However, if any doubt surrounds the selec-
tion of an expected sign, you should document the opposing forces at work
and the reasons for hypothesizing a positive or negative coefficient.
Qd 5 f( P
2
, Y
1
, P
2
c, P
1
s) 1 �
�1
X2i 5
X1i 5 e
1 if the ith teacher has a graduate degree
0 otherwise
Yi 5 �0 1 �1X1i 1 �2X2i 1 �i
LEARNING TO USE REGRESSION ANALYSIS
75

Step 4: Collect the Data. Inspect and Clean the Data
Obtaining an original data set and properly preparing it for regression is a
surprisingly difficult task. This step entails more than a mechanical recording
of data, because the type and size of the sample also must be chosen.
A general rule regarding sample size is “the more observations the better,”
as long as the observations are from the same general population. Ordinarily,
researchers take all the roughly comparable observations that are readily
available. In regression analysis, all the variables must have the same number
of observations. They also should have the same frequency (monthly, quar-
terly, annual, etc.) and time period. Often, the frequency selected is deter-
mined by the availability of data.
The reason there should be as many observations as possible concerns the
statistical concept of degrees of freedom. Consider fitting a straight line to two
points on an X, Y coordinate system as in Figure 1. Such an exercise can be done
mathematically without error. Both points lie on the line, so there is no estima-
tion of the coefficients involved. The two points determine the two parameters,
the intercept and the slope, precisely. Estimation takes place only when a straight
line is fitted to three or more points that were generated by some process that is
not exact. The excess of the number of observations (three) over the number of
coefficients to be estimated (in this case two, the intercept and slope) is the
LEARNING TO USE REGRESSION ANALYSIS
Y
0 X
Figure 1 Mathematical Fit of a Line to Two Points
If there are only two points in a data set, as in Figure 1, a straight line can be fitted to
those points mathematically without error, because two points completely determine a
straight line.
76

LEARNING TO USE REGRESSION ANALYSIS
Y
0 X
Figure 2 Statistical Fit of a Line to Three Points
If there are three (or more) points in a data set, as in Figure 2, then the line must
almost always be fitted to the points statistically, using the estimation procedures of
Ordinary Least Squares (OLS).
2. We will calculate the number of degrees of freedom (d.f.) in a regression equation as
where K is the number of independent variables in the equation. Equiva-
lently, some authors will set and define . Since equals the num-
ber of independent variables plus 1 (for the constant), it equals the number of coefficients to
be estimated in the regression.
Krd.f. 5 (N 2 Kr)Kr 5 K 1 1
d.f. 5 (N 2 K 2 1),
degrees of freedom.2 All that is necessary for estimation is a single degree of
freedom, as in Figure 2, but the more degrees of freedom there are, the bet-
ter. This is because when the number of degrees of freedom is large, every
positive error is likely to be balanced by a negative error. When degrees of
freedom are low, the random element is likely to fail to provide such
offsetting observations. For example, the more a coin is flipped, the more
likely it is that the observed proportion of heads will reflect the true proba-
bility of 0.5.
Another area of concern has to do with the units of measurement of the
variables. Does it matter if a variable is measured in dollars or thousands
of dollars? Does it matter if the measured variable differs consistently from
the true variable by 10 units? Interestingly, such changes don’t matter in
terms of regression analysis except in interpreting the scale of the coeffi-
cients. All conclusions about signs, significance, and economic theory are
independent of units of measurement. For example, it makes little difference
77

whether an independent variable is measured in dollars or thousands of
dollars. The constant term and measures of overall fit remain unchanged.
Such a multiplicative factor does change the slope coefficient, but only by
the exact amount necessary to compensate for the change in the units of
measurement of the independent variable. Similarly, a constant factor
added to a variable alters only the intercept term without changing the
slope coefficient itself.
The final step before estimating your equation is to inspect and clean the
data. You should make it a point always to look over your data set to see if
you can find any errors. The reason is obvious: why bother using sophisti-
cated regression analysis if your data are incorrect?
To inspect the data, obtain a printout and a plot (graph) of the data and
look for outliers. An outlier is an observation that lies outside the range of
the rest of the observations, and looking for outliers is an easy way to find
data entry errors. In addition, it’s a good habit to look at the mean, maxi-
mum, and minimum of each variable and then think about possible incon-
sistencies in the data. Are any observations impossible or unrealistic? Did
GDP double in one year? Does a student have a 7.0 GPA on a 4.0 scale? Is
consumption negative?
Typically, the data can be cleaned of these errors by replacing an incor-
rect number with the correct one. In extremely rare circumstances, an obser-
vation can be dropped from the sample, but only if the correct number
can’t be found or if that particular observation clearly isn’t from the same
population as the rest of the sample. Be careful! The mere existence of an
outlier is not a justification for dropping that observation from the sample.
A regression needs to be able to explain all the observations in a sample,
not just the well-behaved ones.
Step 5: Estimate and Evaluate the Equation
Believe it or not, it can take months to complete steps 1–4 for a regression
equation, but a computer program like EViews or Stata can estimate that
equation in less than a second! Typically, estimation is done using OLS, but
if another estimation technique is used, the reasons for that alternative tech-
nique should be carefully explained and evaluated.
You might think that once your equation has been estimated, your work
is finished, but that’s hardly the case. Instead, you need to evaluate your
LEARNING TO USE REGRESSION ANALYSIS
78

results in a variety of ways. How well did the equation fit the data? Were
the signs and magnitudes of the estimated coefficients what you expected?
Most of the rest of this text is concerned with the evaluation of estimated
econometric equations, and beginning researchers should be prepared to
spend a considerable amount of time doing this evaluation.
Once this evaluation is complete, don’t automatically go to step 6. Regres-
sion results are rarely what one expects, and additional model development
often is required. For example, an evaluation of your results might indicate
that your equation is missing an important variable. In such a case, you’d go
back to step 1 to review the literature and add the appropriate variable to
your equation. You’d then go through each of the steps in order until you
had estimated your new specification in step 5. You’d move on to step 6 only
if you were satisfied with your estimated equation. Don’t be too quick to
make such adjustments, however, because we don’t want to adjust the theory
merely to fit the data. A researcher has to walk a fine line between making
appropriate changes and avoiding inappropriate ones, and making these
choices is one of the artistic elements of applied econometrics.
Finally, it’s often worthwhile to estimate additional specifications of an
equation in order to see how stable your observed results are. This approach,
called sensitivity analysis.
Step 6: Document the Results
A standard format usually is used to present estimated regression results:
(3)
The number in parentheses is the estimated standard error of the esti-
mated coefficient, and the t-value is the one used to test the hypothesis
that the true value of the coefficient is different from zero. What is
N 5 20 R2 5 .73
t 5 7.22
(0.88)
Ŷi 5 103.40 1 6.38Xi
LEARNING TO USE REGRESSION ANALYSIS
79

important to note is that the documentation of regression results using an
easily understood format is considered part of the analysis itself. For time-
series data sets, the documentation also includes the frequency (e.g., quarterly
or annual) and the time period of the data.
Most computer programs present statistics to eight or more digits, but
it is important to recognize the difference between the number of digits
computed and the number of meaningful digits, which may be as low as two
or three.
One of the important parts of the documentation is the explanation of the
model, the assumptions, and the procedures and data used. The written doc-
umentation must contain enough information so that the entire study could
be replicated by others.3 Unless the variables have been defined in a glossary
or table, short definitions should be presented along with the equations. If
there is a series of estimated regression equations, then tables should provide
the relevant information for each equation. All data manipulations as well as
data sources should be documented fully. When there is much to explain,
this documentation usually is relegated to a data appendix. If the data are not
available generally or are available only after computation, the data set itself
might be included in this appendix.
Using Regression Analysis to Pick
Restaurant Locations
To solidify your understanding of the six basic steps of applied regression
analysis, let’s work through a complete regression example. Suppose that
you’ve been hired to determine the best location for the next Woody’s
restaurant, where Woody’s is a moderately priced, 24-hour, family restau-
rant chain.4 You decide to build a regression model to explain the gross
sales volume at each of the restaurants in the chain as a function of various
descriptors of the location of that branch. If you can come up with a
sound equation to explain gross sales as a function of location, then you
2
LEARNING TO USE REGRESSION ANALYSIS
3. For example, the Journal of Money, Credit, and Banking has requested authors to submit their
actual data sets so that regression results can be verified. See W. G. Dewald et al., “Replication in
Empirical Economics,” American Economic Review, Vol. 76, No. 4, pp. 587–603 and Daniel S.
Hamermesh, “Replication in Economics,” NBER Working Paper 13026, April 2007.
4. The data in this example are real (they’re from a sample of 33 Denny’s restaurants in South-
ern California), but the number of independent variables considered is much smaller than was
used in the actual research. Datafile � WOODY3
80

can use this equation to help Woody’s decide where to build their newest
eatery. Given data on land costs, building costs, and local building and
restaurant municipal codes, the owners of Woody’s will be able to make an
informed decision.
1. Review the literature and develop the theoretical model. You do some read-
ing about the restaurant industry, but your review of the literature con-
sists mainly of talking to various experts within the firm. They give
you some good ideas about the attributes of a successful Woody’s loca-
tion. The experts tell you that all of the chain’s restaurants are identical
(indeed, this is sometimes a criticism of the chain) and that all the
locations are in what might be called “suburban, retail, or residential”
environments (as distinguished from central cities or rural areas, for
example). Because of this, you realize that many of the reasons that might
help explain differences in sales volume in other chains do not apply in
this case because all the Woody’s locations are similar. (If you were com-
paring Woody’s to another chain, such variables might be appropriate.)
In addition, discussions with the people in the Woody’s strategic
planning department convince you that price differentials and con-
sumption differences between locations are not as important as the
number of customers a particular location attracts. This causes you
concern for a while because the variable you had planned to study orig-
inally, gross sales volume, would vary as prices changed between loca-
tions. Since your company controls these prices, you feel that you
would rather have an estimate of the “potential” for such sales. As a
result, you decide to specify your dependent variable as the number of
customers served (measured by the number of checks or bills that the
waiters and waitresses handed out) in a given location in the most
recent year for which complete data are available.
2. Specify the model: Select the independent variables and the functional form.
Your discussions lead to a number of suggested variables. After a while,
you realize that there are three major determinants of sales (customers)
on which virtually everyone agrees. These are the number of people
who live near the location, the general income level of the location,
and the number of direct competitors close to the location. In addi-
tion, there are two other good suggestions for potential explanatory
variables. These are the number of cars passing the location per day
and the number of months that the particular restaurant has been
open. After some serious consideration of your alternatives, you decide
not to include the last possibilities. All the locations have been open
LEARNING TO USE REGRESSION ANALYSIS
81

long enough to have achieved a stable clientele. In addition, it would
be very expensive to collect data on the number of passing cars for all
the locations. Should population prove to be a poor measure of the
available customers in a location, you’ll have to decide whether to ask
your boss for the money to collect complete traffic data.
The exact definitions of the independent variables you decide to
include are:
: the number of direct market competitors within a
two-mile radius of the Woody’s location
: the number of people living within a three-mile
radius of the Woody’s location
: the average household income of the population
measured in variable P
Since you have no reason to suspect anything other than a linear func-
tional form and a typical stochastic error term, that’s what you decide
to use.
3. Hypothesize the expected signs of the coefficients. After thinking about
which variables to include, you expect hypothesizing signs will be easy.
For two of the variables, you’re right. Everyone expects that the more
competition, the fewer customers (holding constant the population
and income of an area), and also that the more people who live near a
particular restaurant, the more customers (holding constant the com-
petition and income). You expect that the greater the income in a par-
ticular area, the more people will choose to eat in a family restaurant.
However, people in especially high-income areas might want to eat in a
restaurant that has more “atmosphere” than a family restaurant like
Woody’s. As a result, you worry that the income variable might be only
weakly positive in its impact. To sum, you expect:
(4)
where the signs above the variables indicate the expected impact of that
particular independent variable on the dependent variable, holding
constant the other two explanatory variables, and is a typical stochastic
error term.
�i
5 �0 1 �NNi 1 �PPi 1 �IIi 1 �iYi 5 f(N
2
i,
P
1
i, I
1?
i) 1 �i
I 5 Income
P 5 Population
N 5 Competition
LEARNING TO USE REGRESSION ANALYSIS
82

4. Collect the data. Inspect and clean the data. You want to include every
local restaurant in the Woody’s chain in your study, and, after some
effort, you come up with data for your dependent variable and your
independent variables for all 33 locations. You inspect the data, and
you’re confident that the quality of your data is excellent for three rea-
sons: each manager measured each variable identically, you’ve included
each restaurant in the sample, and all the information is from the same
year. [The data set is included in this section, along with a sample com-
puter output for the regression estimated by EViews (Tables 1 and 2)
and Stata (Tables 3 and 4).]
5. Estimate and evaluate the equation. You take the data set and enter it into
the computer. You then run an OLS regression on the data, but you do
so only after thinking through your model once again to see if there are
hints that you’ve made theoretical mistakes. You end up admitting that
although you cannot be sure you are right, you’ve done the best you
can, so you estimate the equation, obtaining:
(5)
This equation satisfies your needs in the short run. In particular, the
estimated coefficients in the equation have the signs you expected. The
overall fit, although not outstanding, seems reasonable for such a
diverse group of locations. To predict Y, you obtain the values of N, P,
and I for each potential new location and then plug them into Equa-
tion 5. Other things being equal, the higher the predicted Y, the better
the location from Woody’s point of view.
6. Document the results. The results summarized in Equation 5 meet our
documentation requirements. (Note that we include the standard
errors of the estimated coefficients and t-values5 for completeness,
N 5 33 R2 5 .579
t 5 24.42 4.88 2.37
(2053) (0.073) (0.543)
Ŷi 5 102,192 2 9075Ni 1 0.355Pi 1 1.288Ii
LEARNING TO USE REGRESSION ANALYSIS
5. The number in parentheses below a coefficient estimate will be the standard error of that
estimated coefficient. Some authors put the t-value in parentheses, though, so be alert when
reading journal articles or other books.
83

LEARNING TO USE REGRESSION ANALYSIS
Table 1 Data for the Woody’s Restaurants Example (Using the
EViews Program)
84

Table 2 Actual Computer Output (Using the EViews Program)
85

LEARNING TO USE REGRESSION ANALYSIS
Table 3 Data for the Woody’s Restaurant Example (Using the Stata Program)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
11.
12.
13.
14.
15.
IPN
3
5
7
2
3
13240
22554
16916
20967
19576
15039
21857
26435
24024
14987
21384
18800
15289
16702
19093
26502
18760
33242
14988
18505
16839
28915
19033
19200
22833
14409
20307
20111
30902
31573
19001
20058
16194
65044
101376
124989
55249
73775
5
8
2
6
2
48484
138809
50244
104300
37852
5
6
6
3
6
139900
171740
149894
57386
185105
114520
52933
203500
39334
95120
3
3
5
4
3
3
4
9
7
7
49200
113566
194125
233844
83416
6
3
2
183953
60457
65065
107919
118866
98579
122015
152827
Y
91259
123550
160931
98496
108052
127030
166755
125343
121886
134594
152937
109622
149884
98388
140791
101260
139517
115236
136749
105067
136872
117146
163538
144788
164571
105564
102568
103342
3
4
3
5
2
66921
166332
61951
100441
39462
1.0000
–0.1442
0.3926
0.5370
Y
N
P
I
1.0000
0.7263
–0.0315
1.0000
0.2452 1.0000
P IPNY
(obs=33)
86

LEARNING TO USE REGRESSION ANALYSIS
Table 4 Actual Computer Output (Using the Stata Program)
Source
Model
Residual
Total
Coef. [ 95% Conf. Interval ]
–4876.485
.5033172
2.399084
128371
–13272.86
.2060195
.1767628
76013.84
P> | t |
0.000
0.000
0.025
0.000
t
–4.42
4.88
2.37
7.98
Std. Err.
–9074.674
.3546684
1.287923
102192.4
2052.674
.0726808
.5432938
12799.83
Y
N
P
I
_cons
32 5019432461.6062e+10
9.9289e+09
6.1333e+09
3.3096e+09
211492485
3
29
SS df MS Number of obs =
F( 3, 29) =
Prob > F =
R – squared =
Adj R – squared =
Root MSE =
33
15.65
0.0000
0.6182
0.5787
14543
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
11.
12.
13.
14.
15.
residu~s
115089.6
121821.7
104785.9
130642
126346.5
93383.88
106976.3
135909.3
115677.4
116770.1
133978.1
132868.1
120598.1
116832.3
137985.6
149717.6
117903.5
171807.2
99147.65
132537.5
114105.4
143412.3
113883.4
146334.9
97661.88
131544.4
122564.5
133021
138502.6
165550
121412.3
118275.5
118895.6
107919
118866
98579
122015
152827
Y
91259
123550
160931
98496
108052
127030
166755
125343
121886
134594
152937
109622
149884
98388
140791
101260
139517
115236
136749
105067
136872
117146
163538
144788
164571
105564
102568
103342
Yhat
115089.6
121821.7
104785.9
130642
126346.5
93383.88
106976.3
135909.3
115677.4
116770.1
133978.1
132868.1
120598.1
116832.3
137985.6
149717.6
117903.5
171807.2
99147.65
132537.5
114105.4
143412.3
113883.4
146334.9
97661.88
131544.4
122564.5
133021
138502.6
165550
121412.3
118275.5
118895.6
87

even though we won’t make use of them.) However, it’s not easy
for a beginning researcher to wade through a computer’s
regression output to find all the numbers required for documentation.
You’ll probably have an easier time reading your own computer sys-
tem’s printout if you take the time to “walk through” the sample com-
puter output for the Woody’s model in Tables 1–4. This sample output
was produced by the EViews and Stata computer programs, but it’s sim-
ilar to those produced by SAS, SHAZAM, TSP, and others.
The first items listed are the actual data. These are followed by the
simple correlation coefficients between all pairs of variables in the
data set. Next comes a listing of the estimated coefficients, their esti-
mated standard errors, and the associated t-values, and follows with
the standard error of the regression, RSS, the F-ratio, and other
items. Finally, we have a listing of the observed Ys, the predicted Ys, the
residuals for each observation and a graph of these residuals. Numbers
followed by “E�06” or “E–01” are expressed in a scientific notation in-
dicating that the printed decimal point should be moved six places to
the right or one place to the left, respectively.
We’ll return to this example in order to apply various tests and ideas
as we learn them.
Summary
1. Six steps typically taken in applied regression analysis for a given de-
pendent variable are:
a. Review the literature and develop the theoretical model.
b. Specify the model: Select the independent variables and the func-
tional form.
c. Hypothesize the expected signs of the coefficients.
d. Collect the data. Inspect and clean the data.
e. Estimate and evaluate the equation.
f. Document the results.
2. A dummy variable takes on only the values of 1 or 0, depending on
whether some condition is met. An example of a dummy variable
would be X equals 1 if a particular individual is female and 0 if the
person is male.
3
R2, R2,
LEARNING TO USE REGRESSION ANALYSIS
88

LEARNING TO USE REGRESSION ANALYSIS
EXERCISES
(The answer to Exercise 2 is at the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or your notes), and compare your definition with the ver-
sion in the text for each:
a. the six steps in applied regression analysis
b. dummy variable
c. cross-sectional data set
d. specification error
e. degrees of freedom
2. Contrary to their name, dummy variables are not easy to understand
without a little bit of practice:
a. Specify a dummy variable that would allow you to distinguish
between undergraduate students and graduate students in your
econometrics class.
b. Specify a regression equation to explain the grade (measured on
a scale of 4.0) each student in your class received on his or her
first econometrics test (Y) as a function of the student’s grade in
a previous course in statistics (G), the number of hours the stu-
dent studied for the test (H), and the dummy variable you cre-
ated above (D). Are there other variables you would want to add?
Explain.
c. What is the hypothesized sign of the coefficient of D? Does the sign
depend on the exact way in which you defined D? (Hint: In particu-
lar, suppose that you had reversed the definitions of 1 and 0 in
your answer to part a.) How?
d. Suppose that you collected the data and ran the regression
and found an estimated coefficient for D that had the expected
sign and an absolute value of 0.5. What would this mean in
real-world terms? By the way, what would have happened if
you had only undergraduates or only graduate students in
your class?
3. Do liberal arts colleges pay economists more than they pay other
professors? To find out, we looked at a sample of 2,929 small-college
89

faculty members and built a model of their salaries that included
a number of variables, four of which were:
(6)
where: Si � the salary of the ith college professor
Mi � a dummy variable equal to 1 if the ith professor is a
male and 0 otherwise
Ai � a dummy variable equal to 1 if the ith professor is
African American and 0 otherwise
Ri � the years in rank of the ith professor
Ti � a dummy variable equal to 1 if the ith professor
teaches economics and 0 otherwise
a. Carefully explain the meaning of the estimated coefficient of M.
b. The equation indicates that African Americans earn $426 more
than members of other ethnic groups, holding constant the other
variables in the equation. Does this coefficient have the sign you
expected? Why or why not?
c. Is R a dummy variable? If not, what is it? Carefully explain the
meaning of the coefficient of R. (Hint: A professor’s salary typically
increases each year based on rank.)
d. What’s your conclusion? Do economists earn more than other pro-
fessors at liberal arts colleges? Explain.
e. The fact that the equation ends with the notation “+ . . .” indi-
cates that there were more than four independent variables in the
equation. If you could add a variable to the equation, what would
it be? Explain.
4. Return to the Woody’s regression example of Section 2.
a. In any applied regression project, there is the distinct possibility
that an important explanatory variable has been omitted. Reread
the discussion of the selection of independent variables and come
up with a suggestion for an independent variable that has not been
included in the model (other than the variables already men-
tioned). Why do you think this variable was not included?
b. What other kinds of criticisms would you have of the sample or
independent variables chosen in this model?
R2 5 .77 N 5 2929
(259) (456) (24) (458)
Ŝi 5 36,721 1 817Mi 1 426Ai 1 406Ri 1 3539Ti 1
c
LEARNING TO USE REGRESSION ANALYSIS
90

5. Suppose you were told that although data on traffic for Equation 5
are still too expensive to obtain, a variable on traffic, called Ti, is
available that is defined as 1 if traffic is “heavy” in front of the restau-
rant and 0 otherwise. Further suppose that when the new variable
(Ti) is added to the equation, the results are:
(7)
a. What is the expected sign of the coefficient of the new variable?
b. Would you prefer this equation to the original one? Why?
c. Does the fact that is higher in Equation 7 mean that it is
necessarily better than Equation 5?
6. Suppose that the population variable in Section 2 had been defined
in different units, as in:
P � Population: thousands of people living within a three-mile
radius of the Woody’s location
a. Given this definition of P, what would the estimated slope coeffi-
cients in Equation 5 have been?
b. Given this definition of P, what would the estimated slope coeffi-
cients in Equation 7 above have been?
c. Is the estimated constant affected by this change?
7. Use EViews, Stata, or your own computer regression software to esti-
mate Equation 5 using the data in Table 1. Can you get the same
results?
8. The Graduate Record Examination (GRE) subject test in economics
was a multiple-choice measure of knowledge and analytical ability
in economics that was used mainly as an entrance criterion for
students applying to Ph.D. programs in the “dismal science.” For
years, critics claimed that the GRE, like the Scholastic Aptitude
Test (SAT), was biased against women and some ethnic groups.
To test the possibility that the GRE subject test in economics
was biased against women, Mary Hirschfeld, Robert Moore, and
R2
N 5 33 R2 5 .617
t 5 23.39 4.24 2.47 1.97
(2153) (0.073) (0.51) (5577)
Ŷi 5 95,236 2 7307Ni 1 0.320Pi 1 1.28Ii 1 10,994Ti
LEARNING TO USE REGRESSION ANALYSIS
91

Eleanor Brown estimated the following equation (standard errors
in parentheses):6
(8)
where: GREi � the score of the ith student in the Graduate
Record Examination subject test in economics
Gi � a dummy variable equal to 1 if the ith student
was a male, 0 otherwise
GPAi � the GPA in economics classes of the ith student
(4 � A, 3 � B, etc.)
SATMi � the score of the ith student on the mathematics
portion of the Scholastic Aptitude Test
SATVi � the score of the ith student on the verbal portion
of the Scholastic Aptitude Test
a. Carefully explain the meaning of the coefficient of G in this equa-
tion. (Hint: Be sure to specify what 39.7 stands for.)
b. Does this result prove that the GRE is biased against women? Why
or why not?
c. If you were going to add one variable to Equation 8, what would it
be? Explain your reasoning.
d. Suppose that the authors had defined their gender variables as Gi
� a dummy variable equal to 1 if the ith student was a female, 0
otherwise. What would the estimated Equation 8 have been in that
case? (Hint: Only the intercept and the coefficient of the dummy
variable change.)
9. Michael Lovell estimated the following model of the gasoline mileage
of various models of cars (standard errors in parentheses):7
R2 5 .82
(0.001) (0.71) (1.41) (0.097)
Ĝi 5 22.008 2 0.002Wi 2 2.76Ai 1 3.28Di 1 0.415Ei
N 5 149 R2 5 .46
(10.9) (10.4) (0.071) (0.058)
GREi 5 172.4 1 39.7Gi 1 78.9GPAi 1 0.203SATMi 1 0.110SATVi
LEARNING TO USE REGRESSION ANALYSIS
6. Mary Hirschfeld, Robert L. Moore, and Eleanor Brown, “Exploring the Gender Gap on the
GRE Subject Test in Economics,” Journal of Economic Education, Vol. 26, No. 1, pp. 3–15.
7. Michael C. Lovell, “ Tests of the Rational Expectations Hypothesis,” American Economic Review,
Vol. 76, No. 1, pp. 110–124.
92

where: Gi � miles per gallon of the ith model as reported by Con-
sumers’ Union based on actual road tests
Wi � the gross weight (in pounds) of the ith model
Ai � a dummy variable equal to 1 if the ith model has an
automatic transmission and 0 otherwise
Di � a dummy variable equal to 1 if the ith model has a
diesel engine and 0 otherwise
Ei � the U.S. Environmental Protection Agency’s estimate
of the miles per gallon of the ith model
a. Hypothesize signs for the slope coefficients of W and E. Which, if
any, of the signs of the estimated coefficients are different from
your expectations?
b. Carefully interpret the meanings of the estimated coefficients of Ai
and Di. (Hint: Remember that E is in the equation.)
c. Lovell included one of the variables in the model to test a specific
hypothesis, but that variable wouldn’t necessarily be in another re-
searcher’s gas mileage model. What variable do you think Lovell
added? What hypothesis do you think Lovell wanted to test?
10. Your boss is about to start production of her newest box-office
smash-to-be, Invasion of the Economists, Part II, when she calls you in
and asks you to build a model of the gross receipts of all the movies
produced in the last five years. Your regression is (standard errors in
parentheses):8
where: Gi � the final gross receipts of the ith motion picture (in
thousands of dollars)
Ti � the number of screens (theaters) on which the ith
film was shown in its first week
Fi � a dummy variable equal to 1 if the star of the ith film
is a female and 0 otherwise
R2 5 .485 N 5 254
(5.9) (674) (800) (1006) (2381)
Ĝi 5 781 1 15.4Ti 2 992Fi 1 1770Ji 1 3027Si 2 3160Bi 1
c
LEARNING TO USE REGRESSION ANALYSIS
8. This estimated equation (but not the question) comes from a final exam in managerial eco-
nomics given at the Harvard Business School.
93

Ji � a dummy variable equal to 1 if the ith movie was re-
leased in June or July and 0 otherwise
Si � a dummy variable equal to 1 if the star of the ith film
is a superstar (like Tom Cruise or Milton) and 0
otherwise
Bi � a dummy variable equal to 1 if at least one member
of the supporting cast of the ith film is a superstar
and 0 otherwise
a. Hypothesize signs for each of the slope coefficients in the equa-
tion. Which, if any, of the signs of the estimated coefficients are dif-
ferent from your expectations?
b. Milton, the star of the original Invasion of the Economists, is demand-
ing $4 million from your boss to appear in the sequel. If your esti-
mates are trustworthy, should she say “yes” or hire Fred (a nobody)
for $500,000?
c. Your boss wants to keep costs low, and it would cost $1.2 million
to release the movie on an additional 200 screens. Assuming
your estimates are trustworthy, should she spring for the extra
screens?
d. The movie is scheduled for release in September, and it would cost
$1 million to speed up production enough to allow a July release
without hurting quality. Assuming your estimates are trustworthy,
is it worth the rush?
e. You’ve been assuming that your estimates are trustworthy. Do
you have any evidence that this is not the case? Explain your
answer. (Hint: Assume that the equation contains no specifica-
tion errors.)
11. Let’s get some more experience with the six steps in applied regres-
sion. Suppose that you’re interested in buying an Apple iPod (either
new or used) on eBay (the auction website) but you want to avoid
overbidding. One way to get an insight into how much to bid would
be to run a regression on the prices9 for which iPods have sold in
previous auctions.
LEARNING TO USE REGRESSION ANALYSIS
9. This is another example of a hedonic model, in which the price of an item is the dependent
variable and the independent variables are the attributes of that item rather than the quantity
demanded/supplied of that item.
94

The first step would be to review the literature, and luckily you find
some good material—particularly a 2008 article by Leonardo Rezende10
that analyzes eBay Internet auctions and even estimates a model of the
price of iPods.
The second step would be to specify the independent variables
and functional form for your equation, but you run into a problem.
The problem is that you want to include a variable that measures the
condition of the iPod in your equation, but some iPods are new, some
are used and unblemished, and some are used and have a scratch or
other defect.
a. Carefully specify a variable (or variables) that will allow you to
quantify the three different conditions of the iPods. Please answer
this question before moving on.
b. The third step is to hypothesize the signs of the coefficients of your
equation. Assume that you choose the following specification.
What signs do you expect for the coefficients of NEW, SCRATCH,
and BIDRS? Explain.
PRICEi � �0 � �1NEWi � �2SCRATCHi � �3BIDRSi � �i
where: PRICEi � the price at which the ith iPod sold on eBay
NEWi � a dummy variable equal to 1 if the ith iPod
was new, 0 otherwise
SCRATCHi � a dummy variable equal to 1 if the ith iPod
had a minor cosmetic defect, 0 otherwise
BIDRSi � the number of bidders on the ith iPod
c. The fourth step is to collect your data. Luckily, Rezende has data for
215 silver-colored, 4 GB Apple iPod minis available on a website,
so you download the data and are eager to run your first regression.
Before you do, however, one of your friends points out that the
iPod auctions were spread over a three-week period and worries
that there’s a chance that the observations are not comparable be-
cause they come from different time periods. Is this a valid con-
cern? Why or why not?
LEARNING TO USE REGRESSION ANALYSIS
10. Leonardo Rezende, “Econometrics of Auctions by Least Squares,” Journal of Applied Econo-
metrics, November/December 2008, pp. 925–948.
95

d. The fifth step is to estimate your specification using Rezende’s data,
producing:
PRICEi � 109.24 � 54.99NEWi � 20.44SCRATCHi � 0.73BIDRSi
(5.34) (5.11) (0.59)
t � 10.28 –4.00 1.23
N � 215
Do the estimated coefficients correspond to your expectations?
Explain.
e. The sixth step is to document your results. Look over the regression
results in part d. What, if anything, is missing that should be in-
cluded in our normal documentation format?
f. (optional) Estimate the equation yourself (Datafile � IPOD3), and
determine the value of the item that you reported missing in your
answer to part e.
LEARNING TO USE REGRESSION ANALYSIS
Answers
Exercise 2
a. D � 1 if graduate student and D � 0 if undergraduate (or D � 1
if undergraduate and D � 0 if graduate).
b. Yes; for example, E � how many exercises the student did.
c. If D is defined as in answer a, then its coefficient’s sign would
be expected to be positive. If D is defined as 0 if graduate
student, 1 if undergraduate, then the expected sign would be
negative.
d. A coefficient with value of 0.5 indicates that holding constant
the other independent variables in the equation, a graduate
student would be expected to earn half a grade point higher
than an undergraduate. If there were only graduate students or
only undergraduates in class, the coefficient of D could not be
estimated.
96

The classical model of econometrics has nothing to do with ancient Greece
or even the classical economic thinking of Adam Smith. Instead, the term
classical refers to a set of fairly basic assumptions required to hold in order for
OLS to be considered the “best” estimator available for regression models.
When one or more of these assumptions do not hold, other estimation tech-
niques (such as Generalized Least Squares) sometimes may be better than
OLS.
As a result, one of the most important jobs in regression analysis is to decide
whether the classical assumptions hold for a particular equation. If so, the OLS
estimation technique is the best available. Otherwise, the pros and cons of al-
ternative estimation techniques must be weighed. These alternatives usually
are adjustments to OLS that take account of the particular assumption that has
been violated. In a sense, most of the rest of this text deals in one way or an-
other with the question of what to do when one of the classical assumptions is
not met. Since econometricians spend so much time analyzing violations of
them, it is crucial that they know and understand these assumptions.
The Classical Assumptions
The Classical Assumptions must be met in order for OLS estimators to be
the best available. Because of their importance in regression analysis, the as-
sumptions are presented here in tabular form as well as in words. Subsequent
1
1 The Classical Assumptions
2 The Sampling Distribution of
3 The Gauss–Markov Theorem and the Properties
of OLS Estimators
4 Standard Econometric Notation
5 Summary and Exercises
�̂
The Classical Model
From Chapter 4 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
97

chapters will investigate major violations of the assumptions and introduce
estimation techniques that may provide better estimates in such cases.
An error term satisfying Assumptions I through V is called a classical error
term, and if Assumption VII is added, the error term is called a classical normal
error term.
I. The regression model is linear, is correctly specified, and has an additive
error term. The regression model is assumed to be linear:
(1)
The assumption that the regression model is linear1 does not require the
underlying theory to be linear. For example, an exponential function:
(2)
where e is the base of the natural log, can be transformed by taking the nat-
ural log of both sides of the equation:
(3)ln(Yi) 5 �0 1 �1 ln(Xi) 1 �i
Yi 5 e
�0Xi
�1e�i
Yi 5 �0 1 �1X1i 1 �2X2i 1
c1 �KXKi 1 �i
THE CLASSICAL MODEL
1. The Classical Assumption that the regression model be “linear” technically requires the
model to be “linear in the coefficients.” We’ll cover the application of regression analysis to
equations that are nonlinear in the variables in that same section, but the application of regres-
sion analysis to equations that are nonlinear in the coefficients is beyond the scope of this
textbook.
The Classical Assumptions
I. The regression model is linear, is correctly specified, and has an
additive error term.
II. The error term has a zero population mean.
III. All explanatory variables are uncorrelated with the error term.
IV. Observations of the error term are uncorrelated with each other
(no serial correlation).
V. The error term has a constant variance (no heteroskedasticity).
VI. No explanatory variable is a perfect linear function of any other
explanatory variable(s) (no perfect multicollinearity).
VII. The error term is normally distributed (this assumption is optional
but usually is invoked).
98

If the variables are relabeled as , then the form
of the equation becomes linear:
(4)
In Equation 4, the properties of the OLS estimator of the still hold because
the equation is linear.
Two additional properties also must hold. First, we assume that the equa-
tion is correctly specified. If an equation has an omitted variable or an in-
correct functional form, the odds are against that equation working well.
Second, we assume that a stochastic error term has been added to the equa-
tion. This error term must be an additive one and cannot be multiplied by or
divided into any of the variables in the equation.
II. The error term has a zero population mean. Econometricians add a sto-
chastic (random) error term to regression equations to account for variation
in the dependent variable that is not explained by the model. The specific
value of the error term for each observation is determined purely by chance.
Probably the best way to picture this concept is to think of each observation
of the error term as being drawn from a random variable distribution such as
the one illustrated in Figure 1.
�s
Y*i 5 �0 1 �1X*i 1 �i
Y*i 5 ln(Yi) and X*i 5 ln(Xi)
THE CLASSICAL MODEL
0
Probability
2 1 �
Figure 1 An Error Term Distribution with a Mean of Zero
Observations of stochastic error terms are assumed to be drawn from a random variable
distribution with a mean of zero. If Classical Assumption II is met, the expected value
(the mean) of the error term is zero.
99

Classical Assumption II says that the mean of this distribution is zero. That
is, when the entire population of possible values for the stochastic error term
is considered, the average value of that population is zero. For a small sam-
ple, it is not likely that the mean is exactly zero, but as the size of the sample
approaches infinity, the mean of the sample approaches zero.
To compensate for the chance that the mean of might not equal zero,
the mean of for any regression is forced to be zero by the existence of the
constant term in the equation. In essence, the constant term equals the
fixed portion of Y that cannot be explained by the independent variables,
whereas the error term equals the stochastic portion of the unexplained
value of Y.
Although it’s true that the error term can never be observed, it’s instructive
to pretend that we can do so to see how the existence of a constant term
forces the mean of the error term to be zero in a sample. Consider a typical
regression equation:
(5)
Suppose that the mean of is 3 instead of 0, then2 If we add
3 to the constant term and subtract it from the error term, we obtain:
(6)
Since Equations 5 and 6 are equivalent (do you see why?), and since
then Equation 6 can be written in a form that has a zero
mean for the error term:
(7)
where As can be seen, Equation 7 conforms
to Assumption II. This form is always assumed to apply for the true model.
Therefore, the second classical assumption is assured as long as a constant
term is included in the equation and all other classical assumptions are met.
�*0 5 �0 1 3 and �*i 5 �i 2 3.
Yi 5 �*0 1 �1Xi 1 �*i
E(�i 2 3) 5 0,
Yi 5 (�0 1 3) 1 �1Xi 1 (�i 2 3)
E(�i 2 3) 5 0.�i
Yi 5 �0 1 �1Xi 1 �i
�i
�
THE CLASSICAL MODEL
2. Here, the “E” refers to the expected value (mean) of the item in parentheses after it. Thus
equals the expected value of the stochastic error term epsilon minus 3. In this specific
example, since we’ve defined One way to think about
expected value is as our best guess of the long-run average value a specific random variable will
have.
E(�i) 5 3, we know that E(�i 2 3) 5 0.
E(�i 2 3)
100

III. All explanatory variables are uncorrelated with the error term. It is as-
sumed that the observed values of the explanatory variables are independent
of the values of the error term.
If an explanatory variable and the error term were instead correlated with
each other, the OLS estimates would be likely to attribute to the X some of
the variation in Y that actually came from the error term. If the error term
and X were positively correlated, for example, then the estimated coefficient
would probably be higher than it would otherwise have been (biased up-
ward), because the OLS program would mistakenly attribute the variation
in Y caused by to X instead. As a result, it’s important to ensure that the ex-
planatory variables are uncorrelated with the error term.
Classical Assumption III is violated most frequently when a researcher
omits an important independent variable from an equation. One of the
major components of the stochastic error term is omitted variables, so if a
variable has been omitted, then the error term will change when the omitted
variable changes. If this omitted variable is correlated with an included inde-
pendent variable (as often happens in economics), then the error term is cor-
related with that independent variable as well. We have violated Assumption
III! Because of this violation, OLS will attribute the impact of the omitted
variable to the included variable, to the extent that the two variables are cor-
related.
An important economic application that violates this assumption is any
model that is simultaneous in nature. In most economic applications, there
are several related propositions that, when taken as a group, suggest a system
of regression equations. In most situations, interrelated equations should be
considered simultaneously instead of separately. Unfortunately, such simul-
taneous systems violate Classical Assumption III.
IV. Observations of the error term are uncorrelated with each other. The
observations of the error term are drawn independently from each other. If
a systematic correlation exists between one observation of the error term
and another, then it will be more difficult for OLS to get accurate estimates
of the standard errors of the coefficients. For example, if the fact that the
from one observation is positive increases the probability that the from
another observation also is positive, then the two observations of the error
term are positively correlated. Such a correlation would violate Classical As-
sumption IV.
In economic applications, this assumption is most important in time-
series models. In such a context, Assumption IV says that an increase in the
error term in one time period (a random shock, for example) does not
show up in or affect in any way the error term in another time period.
�
�
�
THE CLASSICAL MODEL
101

In some cases, though, this assumption is unrealistic, since the effects of a
random shock sometimes last for a number of time periods. For example,
a natural disaster like Hurricane Katrina will have a negative impact on a
region far after the time period in which it was truly a random event. If,
over all the observations of the sample, is correlated with then the
error term is said to be serially correlated (or autocorrelated), and Assump-
tion IV is violated.
V. The error term has a constant variance. The variance (or dispersion) of
the distribution from which the observations of the error term are drawn is
constant. That is, the observations of the error term are assumed to be drawn
continually from identical distributions (for example, the one pictured in
Figure 1). The alternative would be for the variance of the distribution of
the error term to change for each observation or range of observations. In
Figure 2, for example, the variance of the error term is shown to increase as
�t,�t11
THE CLASSICAL MODEL
Y
0 Z
Small �s
Associated with
Small Zs
Large �s Associated
with Large Zs
�0E(Y|X) = + �1Z
Figure 2 An Error Term Whose Variance Increases as Z Increases
(Heteroskedasticity)
One example of Classical Assumption V not being met is when the variance of the error
term increases as Z increases. In such a situation (called heteroskedasticity), the obser-
vations are on average farther from the true regression line for large values of Z than
they are for small values of Z.
102

the variable Z increases; such a pattern violates Classical Assumption V. The
actual values of the error term are not directly observable, but the lack of a
constant variance for the distribution of the error term causes OLS to gener-
ate inaccurate estimates of the standard error of the coefficients.
In economic applications, Assumption V is likely to be violated in cross-
sectional data sets. For example, suppose that you’re studying the amount of
money that the 50 states spend on education. Since New York and California
are much bigger than New Hampshire and Nevada, it’s probable that the
variance of the stochastic error term for big states is larger than it is for small
states. The amount of unexplained variation in educational expenditures
seems likely to be larger in big states like New York than in small states like
New Hampshire. The violation of Assumption V is referred to as heteroske-
dasticity.
VI. No explanatory variable is a perfect linear function of any other ex-
planatory variable(s). Perfect collinearity between two independent vari-
ables implies that they are really the same variable, or that one is a multiple
of the other, and/or that a constant has been added to one of the variables.
That is, the relative movements of one explanatory variable will be matched
exactly by the relative movements of the other even though the absolute
size of the movements might differ. Because every movement of one of the
variables is matched exactly by a relative movement in the other, the OLS
estimation procedure will be incapable of distinguishing one variable from
the other.
Many instances of perfect collinearity (or multicollinearity if more than
two independent variables are involved) are the result of the researcher not
accounting for identities (definitional equivalences) among the independent
variables. This problem can be corrected easily by dropping one of the per-
fectly collinear variables from the equation.
What’s an example of perfect multicollinearity? Suppose that you decide
to build a model of the profits of tire stores in your city and you include an-
nual sales of tires (in dollars) at each store and the annual sales tax paid by
each store as independent variables. Since the tire stores are all in the same
city, they all pay the same percentage sales tax, so the sales tax paid will be a
constant percentage of their total sales (in dollars). If the sales tax rate is 7%,
then the total taxes paid will be exactly 7% of sales for each and every tire
store. Thus sales tax will be a perfect linear function of sales, and you’ll have
perfect multicollinearity!
Perfect multicollinearity also can occur when two independent variables
always sum to a third or when one of the explanatory variables doesn’t
change within the sample. With perfect multicollinearity, the OLS computer
program (or any other estimation technique) will be unable to estimate the
THE CLASSICAL MODEL
103

coefficients of the collinear variables (unless there is a rounding error).
While it’s quite unusual to encounter perfect multicollinearity in practice,
even imperfect multicollinearity can cause problems for estimation.
VII. The error term is normally distributed. Although we have already
assumed that observations of the error term are drawn independently
(Assumption IV) from a distribution that has a zero mean (Assumption II)
and that has a constant variance (Assumption V), we have said little about the
shape of that distribution. Assumption VII states that the observations of the
error term are drawn from a distribution that is normal (that is, bell-shaped,
and generally following the symmetrical pattern portrayed in Figure 3).
This assumption of normality is not required for OLS estimation. Its major
application is in hypothesis testing , which uses the estimated regression co-
efficient to investigate hypotheses about economic behavior. One example of
such a test is deciding whether a particular demand curve is elastic or inelas-
tic in a particular range.
THE CLASSICAL MODEL
0 2.0 4.022.0
Probability
Distribution A
μ = 0
σ2 = 1
Distribution B
μ = 2
σ2 = 0.5
Figure 3 Normal Distributions
Although all normal distributions are symmetrical and bell-shaped, they do not neces-
sarily have the same mean and variance. Distribution A has a mean of 0 and a variance
of 1, whereas distribution B has a mean of 2 and a variance of 0.5. As can be seen, the
whole distribution shifts when the mean changes, and the distribution gets fatter as the
variance increases.
104

THE CLASSICAL MODEL
Even though Assumption VII is optional, it’s usually advisable to add the
assumption of normality to the other six assumptions for two reasons:
1. The error term can be thought of as the sum of a number of minor
influences or errors. As the number of these minor influences gets
larger, the distribution of the error term tends to approach the normal
distribution.3
2. The t-statistic and the F-statistic are not truly applicable unless the
error term is normally distributed (or the sample is quite large).
A quick look at Figure 3 shows how normal distributions differ when the
means and variances are different. In normal distribution A (a Standard
Normal Distribution), the mean is 0 and the variance is 1; in normal distri-
bution B, the mean is 2, and the variance is 0.5. When the mean is different,
the entire distribution shifts. When the variance is different, the distribution
becomes fatter or skinnier.
The Sampling Distribution of
“It cannot be stressed too strongly how important it is for students to un-
derstand the concept of a sampling distribution.”4
Just as the error term follows a probability distribution, so too do the estimates
of �. In fact, each different sample of data typically produces a different esti-
mate of �. The probability distribution of these values across different sam-
ples is called the sampling distribution of .
Recall that an estimator is a formula, such as the OLS formula, while an
estimate is the value of computed by the formula for a given sample.
Since researchers usually have only one sample, beginning econometri-
cians often assume that regression analysis can produce only one estimate
of � for a given population. In reality, however, each different sample
from the same population will produce a different estimate of �.
The collection of all the possible samples has a distribution, with a
�̂
�̂
�̂
�̂2
�i
3. This is because of the Central Limit Theorem, which states that:
The mean (or sum) of a number of independent, identically distributed random vari-
ables will tend to be normally distributed, regardless of their distribution, if the number
of different random variables is large enough.
4. Peter Kennedy, A Guide to Econometrics (Malden, MA: Blackwell, 2008), p. 403.
105

mean and a variance, and we need to discuss the properties of this sampling
distribution of , even though in most real applications we will encounter
only a single draw from it. Be sure to remember that a sampling distribution
refers to the distribution of different values of across different samples, not
within one. These usually are assumed to be normally distributed because
the normality of the error term implies that the OLS estimates of � are nor-
mally distributed as well.
Let’s look at an example of a sampling distribution of . Suppose you de-
cide to build a regression model to explain the starting salaries of last year’s
graduates of your school as a function of their GPAs at your school:
�
SALARYi � f(GPAi) � �0 � �1GPAi � i (8)
For the time being, let’s focus on the sampling distribution of 1. If you select
a sample of 25 students and get data for their salaries and grades, you can es-
timate Equation 8 with OLS and get an estimate of �1. So far, so good.
But what will happen if you select a second sample of students and do the
same thing? Will you get the same exact 1 that you got from the first sam-
ple? Nope! Your estimate obviously depends on the sample you pick. If your
random sample includes by accident quite a few of the highest-paid gradu-
ates, the estimate will be fairly high. If another sample by chance includes an
underemployed student, then the estimate will be low. As a result, you’re al-
most certain to get a different 1 for every different sample you draw, because
different samples are likely to have different students with different character-
istics. In essence, there is a distribution of all the possible estimates that will
have a mean and a variance, just as the distribution of observations of the
error term does.
So, if you collect five different samples, you’re extremely likely to get five
different 1s. For instance, you might get:
First sample: � 8,612
Second sample: � 8,101
Third sample: � 11,355
Fourth sample: � 6,934
Fifth sample: � 7,994
Average � 8,599
Each sample yields an estimate of the true population � (which is, let’s say,
8,400), and the distribution of the of all the possible samples has its own�̂s
�̂
�̂1
�̂1
�̂1
�̂1
�̂1
�̂
�̂
�̂
�̂
�
�̂
�̂s
�̂
�̂
THE CLASSICAL MODEL
106

mean and variance. For a “good” estimation technique, we’d want the mean
of the sampling distribution of the s to be equal to our true population � of
8,400. This is called unbiasedness. Although the mean for our five samples is
8,599, it’s likely that if we took enough samples and calculated enough ,
the average would eventually approach 8,400.
Therefore the estimated by OLS for Equation 8 form a distribution of
their own. Each sample of observations will produce a different , and the
distribution of these estimates for all possible samples has a mean and a vari-
ance like any distribution. When we discuss the properties of estimators in
the next section, it will be important to remember that we are discussing the
properties of the distribution of estimates generated from a number of sam-
ples (a sampling distribution).
Properties of the Mean
A desirable property of a distribution of estimates is that its mean equals the
true mean of the variable being estimated. An estimator that yields such esti-
mates is called an unbiased estimator.
�̂
�̂s
�̂
�̂s
�̂
�̂
THE CLASSICAL MODEL
An estimator is an unbiased estimator if its sampling distribution has
as its expected value the true value of .
(9)E(�̂) 5 �
�
�̂
Only one value of is obtained in practice, but the property of unbiasedness
is useful because a single estimate drawn from an unbiased distribution is
more likely to be near the true value (assuming identical variances) than one
taken from a distribution not centered around the true value. If an estimator
produces that are not centered around the true the estimator is referred
to as a biased estimator.
We cannot ensure that every estimate from an unbiased estimator is better
than every estimate from a biased one, because a particular unbiased estimate5
could, by chance, be farther from the true value than a biased estimate might be.
�,�̂s
�̂
5. Technically, since an estimate has just one value, an estimate cannot be unbiased (or biased).
On the other hand, the phrase “estimate produced by an unbiased estimator” is cumbersome,
especially if repeated 10 times on a page. As a result, many econometricians use “unbiased esti-
mate” as shorthand for “a single estimate produced by an unbiased estimator.”
107

This could happen by chance or because the biased estimator had a smaller vari-
ance. Without any other information about the distribution of the estimates,
however, we would always rather have an unbiased estimate than a biased one.
Properties of the Variance
Just as we would like the distribution of the to be centered around the true
population so too would we like that distribution to be as narrow (or pre-
cise) as possible. A distribution centered around the truth but with an extremely
large variance might be of very little use because any given estimate would quite
likely be far from the true value. For a distribution with a small variance, the
estimates are likely to be close to the mean of the sampling distribution. To see
this more clearly, compare distributions A and B (both of which are unbiased)
in Figure 4. Distribution A, which has a larger variance than distribution B, is
less precise than distribution B. For comparison purposes, a biased distribution
(distribution C) is also pictured; note that bias implies that the expected value
of the distribution is to the right or left of the true �.
�̂�
�,
�̂s
THE CLASSICAL MODEL
True
�
Distribution B
(unbiased, small variance)
Distribution A
(unbiased, large variance)
Distribution C
(biased, medium variance)
Figure 4 Distributions of
Different distributions of can have different means and variances. Distributions A and
B, for example, are both unbiased, but distribution A has a larger variance than does dis-
tribution B. Distribution C has a smaller variance than distribution A, but it is biased.
�̂
�̂
108

The variance of the distribution of the can be decreased by increasing
the size of the sample. This also increases the degrees of freedom, since the
number of degrees of freedom equals the sample size minus the number of
coefficients or parameters estimated. As the number of observations in-
creases, other things held constant, the variance of the sampling distribution
tends to decrease. Although it is not true that a sample of 15 will always pro-
duce estimates closer to the true than a sample of 5, it is quite likely to do
so; such larger samples should be sought. Figure 5 presents illustrative sam-
pling distributions of for 15 and 5 observations for OLS estimators of
when the true equals 1. The larger sample does indeed produce a sampling
distribution that is more closely centered around .
In econometrics, general tendencies must be relied on. The element of
chance, a random occurrence, is always present in estimating regression coeffi-
cients, and some estimates may be far from the true value no matter how good
the estimating technique. However, if the distribution is centered around the
�
�
��̂s
�
�̂s
THE CLASSICAL MODEL
1 2 3
�
4
Probability
N = 15
N = 5
02122
Figure 5 Sampling Distribution of for Various Observations (N)
As the size of the sample increases, the variance of the distribution of s calculated
from that sample tends to decrease. In the extreme case (not shown), a sample equal to
the population would yield only an estimate equal to the mean of that distribution,
which (for unbiased estimators) would equal the true and the variance of the esti-
mates would be zero.
�,
�̂
�̂
109

true value and has as small a variance as possible, the element of chance is less
likely to induce a poor estimate. If the sampling distribution is centered around
a value other than the true (that is, if is biased) then a lower variance implies
that most of the sampling distribution of is concentrated on the wrong value.
However, if this value is not very different from the true value, which is usually
not known in practice, then the greater precision will still be valuable.
One method of deciding whether this decreased variance in the distribution
of the s is valuable enough to offset the bias is to compare different estimation
techniques by using a measure called the Mean Square Error (MSE). The Mean
Square Error is equal to the variance plus the square of the bias. The lower the
MSE, the better.
A final item of importance is that as the variance of the error term in-
creases, so too does the variance of the distribution of The reason for the
increased variance of is that with the larger variance of the more extreme
values of are observed with more frequency, and the error term becomes
more important in determining the values of Yi.
The Standard Error of
Since the standard error of the estimated coefficient, is the square root
of the estimated variance of the , it is similarly affected by the size of the
sample and the other factors we’ve mentioned. For example, an increase in
sample size will cause to fall; the larger the sample, the more precise
our coefficient estimates will be.
The Gauss–Markov Theorem and the Properties
of OLS Estimators
The Gauss–Markov Theorem proves two important properties of OLS estima-
tors. This theorem is proven in all advanced econometrics textbooks and
readers interested in the proof should see Exercise 8. For a regression user,
however, it’s more important to know what the theorem implies than to be
able to prove it. The Gauss–Markov Theorem states that:
3
SE(�̂)
�̂s
SE(�̂),
�̂
�i
�i,�̂
�̂.
�̂
�̂
�̂�
THE CLASSICAL MODEL
Given Classical Assumptions I through VI (Assumption VII, normality, is
not needed for this theorem), the Ordinary Least Squares estimator of
is the minimum variance estimator from among the set of all linear un-
biased estimators of for k � 0, 1, 2, . . . , K.�k,
�k
110

The Gauss–Markov Theorem is perhaps most easily remembered by stat-
ing that “OLS is BLUE” where BLUE stands for “Best (meaning minimum
variance) Linear Unbiased Estimator.” Students who might forget that “best”
stands for minimum variance might be better served by remembering “OLS is
MvLUE,” but such a phrase is hardly catchy or easy to remember.
If an equation’s coefficient estimation is unbiased (that is, if each of the es-
timated coefficients is produced by an unbiased estimator of the true popula-
tion coefficient), then:
Best means that each has the smallest variance possible (in this case, out
of all the linear unbiased estimators of An unbiased estimator with the
smallest variance is called efficient, and that estimator is said to have the
property of efficiency.
The Gauss–Markov Theorem requires that just the first six of the seven
classical assumptions be met. What happens if we add in the seventh as-
sumption, the assumption that the error term is normally distributed? In this
case, the result of the Gauss–Markov Theorem is strengthened because the
OLS estimator can be shown to be the best (minimum variance) unbiased es-
timator out of all the possible estimators, not just out of the linear estima-
tors. In other words, if all seven assumptions are met, OLS is “BUE.”
Given all seven classical assumptions, the OLS coefficient estimators can
be shown to have the following properties:
1. They are unbiased. That is, This means that the OLS estimates
of the coefficients are centered around the true population values of
the parameters being estimated.
2. They are minimum variance. The distribution of the coefficient estimates
around the true parameter values is as tightly or narrowly distributed as
is possible for an unbiased distribution. No other unbiased estimator
has a lower variance for each estimated coefficient than OLS.
3. They are consistent. As the sample size approaches infinity, the esti-
mates converge to the true population parameters. Put differently,
as the sample size gets larger, the variance gets smaller, and each
estimate approaches the true value of the coefficient being
estimated.
4. They are normally distributed. The Thus various
statistical tests based on the normal distribution may indeed be ap-
plied to these estimates.
�̂s are N(�, VARf�̂g).
E(�̂) is �.
�k).
�̂k
E(�̂k) 5 �k (k 5 0, 1, 2, . . . , K)
THE CLASSICAL MODEL
111

Standard Econometric Notation
This section presents the standard notation used throughout the economet-
rics literature. Table 1 presents various alternative notational devices used to
represent the different population (true) parameters and their corresponding
estimates (based on samples).
The measure of the central tendency of the sampling distribution of
which can be thought of as the mean of the is denoted as read as
“the expected value of beta-hat.” The variance of is the typical measure of
dispersion of the sampling distribution of The variance (or, alternatively,
the square root of the variance, called the standard deviation) has several
alternative notational representations, including read as
the “variance of beta-hat.”
VAR(�̂) and �2(�̂),
�̂.
�̂
E(�̂),�̂s,
�̂,
4
THE CLASSICAL MODEL
Table 1 Notation Conventions
Population Parameter Estimate
(True Values, but Unobserved) (Observed from Sample)
Name Symbol(s) Name Symbol(s)
Regression Estimated regression
coefficient coefficient
Expected value of
the estimated
coefficient
Variance of Estimated variance
the error of the error
term term
Standard Standard error of s or SE
deviation of the equation
the error term (estimate)
Variance of the Estimated variance
estimated of the estimated
coefficient coefficient
Standard deviation Standard error of
of the estimated the estimated
coefficient coefficient
Error or Residual (estimate ei
disturbance of error in a
term loose sense)
�i
�̂(�̂k) or SE(�̂k)��̂k or �(�̂k)
s2(�̂k) or VAR(�̂k)�
2(�̂k) or VAR(�̂k)
�
s2 or �̂2�2 or VAR(�i)
E(�̂k)
�̂k�k
112

The variance of the estimates is a population parameter that is never actu-
ally observed in practice; instead, it is estimated with also written as
Note, by the way, that the variance of the true is zero, since
there is only one true with no distribution around it. Thus, the estimated
variance of the estimated coefficient is defined and observed, the true vari-
ance of the estimated coefficient is unobservable, and the true variance of the
true coefficient is zero. The square root of the estimated variance of the coef-
ficient estimate, is the standard error of which we will use exten-
sively in hypothesis testing.
Summary
1. The seven Classical Assumptions state that the regression model is
linear with an additive error term that has a mean of zero, is uncorre-
lated with the explanatory variables and other observations of the error
term, has a constant variance, and is normally distributed (optional).
In addition, explanatory variables must not be perfect linear functions
of each other.
2. The two most important properties of an estimator are unbiasedness
and minimum variance. An estimator is unbiased when the expected
value of the estimated coefficient is equal to the true value. Minimum
variance holds when the estimating distribution has the smallest vari-
ance of all the estimators in a given class of estimators (for example,
unbiased estimators).
3. Given the Classical Assumptions, OLS can be shown to be the min-
imum variance, linear, unbiased estimator (or BLUE, for best linear
unbiased estimator) of the regression coefficients. This is the
Gauss–Markov Theorem. When one or more of the classical proper-
ties do not hold (excluding normality), OLS is no longer BLUE,
although it still may provide better estimates in some cases than
the alternative estimation techniques discussed in subsequent
chapters.
4. Because the sampling distribution of the OLS estimator of is BLUE,
it has desirable properties. Moreover, the variance, or the measure of
dispersion of the sampling distribution of decreases as the num-
ber of observations increases.
�̂k,
�̂k
5
�̂, SE(�̂k),
�k
�, �2(�),s2(�̂k).
�̂2(�̂k),
THE CLASSICAL MODEL
113

5. There is a standard notation used in the econometric literature. Table 1
presents this fairly complex set of notational conventions for use in
regression analysis. This table should be reviewed periodically as a
refresher.
EXERCISES
(The answer to Exercise 2 is at the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or to your notes), and compare your definition with the
version in the text for each:
a. the Classical Assumptions
b. classical error term
c. standard normal distribution
d. SE( )
e. unbiased estimator
f. BLUE
g. sampling distribution
2. Consider the following estimated regression equation (standard errors
in parentheses):
where: Yt � the corn yield (bushels/acre) in year t
Ft � fertilizer intensity (pounds/acre) in year t
Rt � rainfall (inches) in year t
a. Carefully state the meaning of the coefficients 0.10 and 5.33 in this
equation in terms of the impact of F and R on Y.
b. Does the constant term of �120 really mean that negative amounts
of corn are possible? If not, what is the meaning of that estimate?
c. Suppose you were told that the true value of is known to be 0.20.
Does this show that the estimate is biased? Why or why not?
d. Suppose you were told that the equation does not meet all the clas-
sical assumptions and, therefore, is not BLUE. Does this mean that
the true is definitely not equal to 5.33? Why or why not?�R
�F
(0.05) (1.00)
Yt
ˆ 5 2120 1 0.10Ft 1 5.33Rt R2 5 .50
�̂
THE CLASSICAL MODEL
114

3. Which of the following pairs of independent variables would violate
Assumption VI? (That is, which pairs of variables are perfect linear
functions of each other?)
a. right shoe size and left shoe size (of students in your class)
b. consumption and disposable income (in the United States over the
last 30 years)
c. Xi and 2Xi
d. Xi and (Xi)
2
4. The Gauss–Markov Theorem shows that OLS is BLUE, so we, of course,
hope and expect that our coefficient estimates will be unbiased and
minimum variance. Suppose, however, that you had to choose one or
the other.
a. If you had to pick one, would you rather have an unbiased non-
minimum variance estimate or a biased minimum variance one?
Explain your reasoning.
b. Are there circumstances in which you might change your answer to
part a? (Hint: Does it matter how biased or less-than-minimum vari-
ance the estimates are?)
c. Can you think of a way to systematically choose between estimates
that have varying amounts of bias and less-than-minimum variance?
5. Edward Saunders published an article that tested the possibility that
the stock market is affected by the weather on Wall Street. Using daily
data from 28 years, he estimated an equation with the following signif-
icant variables (standard errors in parentheses):6
where: DJt � the percentage change in the Dow Jones industrial
average on day t
Rt � the daily index capital gain or loss for day t
Jt � a dummy variable equal to 1 if the ith day was in
January, 0 otherwise
N 5 6,911 (daily) R2 5 .02
(0.01) (0.0006) (0.004) (0.0002)
DJt 5 �̂0 1 0.10Rt21 1 0.0010Jt 2 0.017Mt 1 0.0005Ct
THE CLASSICAL MODEL
6. Edward M. Saunders, Jr., “Stock Prices and Wall Street Weather,” American Economic Review,
Vol. 76, No. 1, pp. 1337–1346. Saunders also estimated equations for the New York and Amer-
ican Stock Exchange indices, both of which had much higher R2s than did this equation. Rt�1
was included in the equation “to account for nonsynchronous trading effects” (p. 1341).
115

THE CLASSICAL MODEL
Mt � a dummy variable equal to 1 if the ith day was a
Monday, 0 otherwise
Ct � a variable equal to 1 if the cloud cover was 20 per-
cent or less, equal to �1 if the cloud cover was 100
percent, 0 otherwise
a. Saunders did not include an estimate of the constant term in his
published regression results. Which of the Classical Assumptions
supports the conclusion that you shouldn’t spend much time ana-
lyzing estimates of the constant term? Explain.
b. Which of the Classical Assumptions would be violated if you de-
cided to add a dummy variable to the equation that was equal to 1 if
the ith day was a Tuesday, Wednesday, Thursday, or Friday, and equal
to 0 otherwise? (Hint: The stock market is not open on weekends.)
c. Carefully state the meaning of the coefficients of R and M, being
sure to take into account the fact that R is lagged (one time period
behind) in this equation for valid theoretical reasons.
d. The variable C is a measure of the percentage of cloud cover from
sunrise to sunset on the ith day and reflects the fact that approxi-
mately 85 percent of all New York’s rain falls on days with 100 per-
cent cloud cover. Is C a dummy variable? What assumptions (or
conclusions) did the author have to make to use this variable?
What constraints does it place on the equation?
e. Saunders concludes that these findings cast doubt on the hypothe-
sis that security markets are entirely rational. Based just on the
small portion of the author’s work that we include in this question,
would you agree or disagree? Why?
6. Complete the following exercises:
a. Write out the Classical Assumptions without looking at your book
or notes. (Hint: Don’t just say them to yourself in your head—put
pen or pencil to paper!)
b. After you’ve completed writing out all six assumptions, compare
your version with the text’s. What differences are there? Are they
important?
116

THE CLASSICAL MODEL
c. (Optional) Get together with a classmate and take turns explaining
the assumptions to each other. In this exercise, try to go beyond the
definition of the assumption to give your classmate a feeling for
the real-world meaning of each assumption.
7. W. Bowen and T. Finegan7 estimated the following regression equa-
tion for 78 cities (standard errors in parentheses):
where: Li � percent labor force participation (males ages 25 to 54)
in the ith city
Ui � percent unemployment rate in the ith city
Ei � average earnings (hundreds of dollars/year) in the ith
city
Ii � average other income (hundreds of dollars/year) in
the ith city
Si � average schooling completed (years) in the ith city
Ci � percent of the labor force that is nonwhite in the ith
city
Di � a dummy equal to 1 if the city is in the South, 0
otherwise
a. Interpret the estimated coefficients of C and D. What do they
mean?
b. How likely is perfect collinearity in this equation? Explain your
answer.
c. Suppose that you were told that the data for this regression were
old and that estimates on new data yielded a much different coeffi-
cient of the dummy variable. Would this imply that one of the esti-
mates was biased? If not, why not? If so, how would you determine
which year’s estimate was biased?
d. Comment on the following statement. “I know that these results
are not BLUE because the estimated coefficient of S is wrong. It’s
negative when it should be positive!” Do you agree or disagree?
Why?
N 5 78 R2 5 .51
(0.08) (0.06) (0.16) (0.18) (0.03) (0.53)
L̂i 5 94.2 2 0.24Ui 1 0.20Ei 2 0.69Ii 2 0.06Si 1 0.002Ci 2 0.80Di
7. W. G. Bowen and T. A. Finegan, “Labor Force Participation and Unemployment,” in Arthur
M. Ross (ed.), Employment Policy and Labor Markets (Berkeley: University of California Press,
1965), Table 2.
117

THE CLASSICAL MODEL
8. A typical exam question in a more advanced econometrics class is to
prove the Gauss–Markov Theorem. How might you go about start-
ing such a proof? What is the importance of such a proof?
9. For your first econometrics project you decide to model sales at the
frozen yogurt store nearest your school. The owner of the store is
glad to help you with data collection because she believes that stu-
dents from your school make up the bulk of her business. After
countless hours of data collection and an endless supply of frozen
yogurt, you estimate the following regression equation (standard errors
in parentheses):
where: Yt � the total number of frozen yogurts sold during the tth
two-week time period
Tt � average high temperature (in degrees F) during period t
Pt � the price of frozen yogurt (in dollars) at the store in
period t
At � a dummy variable equal to 1 if the owner places an ad
in the school newspaper during period t, 0 otherwise
Ct � a dummy variable equal to 1 if your school is in regu-
lar session in period t (early September through early
December and early January through late May), 0
otherwise
a. Does this equation appear to violate any of the Classical Assump-
tions? That is, do you see any evidence that a Classical Assump-
tion is or is not met in this equation?
b. What is the real-world economic meaning of the fact that the esti-
mated coefficient of At is 134.3? Be specific.
c. You and the owner are surprised at the sign of the coefficient of Ct.
Can you think of any reason for this sign? (Hint: Assume that your
school has no summer session.)
d. If you could add one variable to this equation, what would it be?
Be specific.
10. In Hollywood, most nightclubs hire “promoters,” or people who walk
around near the nightclub and try to convince passersby to enter
N 5 29 R2 5 .78
(0.7) (20.0) (108.0) (138.3)
5 262.5 1 3.9Tt 2 46.94Pt 1 134.3At 2 152.1CtŶt
118

THE CLASSICAL MODEL
the club. Recently, one of the nightclubs asked a marketing consultant to
estimate the effectiveness of such promoters in terms of their ability to
attract patrons to the club. The consultant did some research and
found that the main entertainment at the nightclubs were attractive
dancers and that the most popular nightclubs were on Hollywood
Boulevard or attached to hotels, so he hypothesized the following
model of nightclub attendance:
PEOPLEi � β0 � β1HOLLYi � β2PROMOi � β3HOTELi � β4GOGOi �
where: PEOPLEi � attendance at the ith nightclub at midnight on
Saturday 11/24/07
HOLLYi � equal to 1 if the ith nightclub is on Hollywood
Boulevard, 0 otherwise
PROMOi � number of promoters working at the ith night-
club that night
HOTELi � equal to 1 if the ith nightclub is part of a hotel,
0 otherwise
GOGOi � number of dancers working at the ith night-
club that night
He then collected data from 25 similarly sized nightclubs on or near
Hollywood Boulevard and came up with the following estimates
(standard errors in parentheses):
PEOPLEi � 162.8 � 47.4HOLLYi � 22.3PROMOi � 214.5HOTELi� 26.9GOGOi
(21.7) (11.8) (46.0) (7.2)
N � 25 � .57
Let’s work through the classical assumptions to see which assump-
tions might or might not be met by this model. As we analyze each as-
sumption, make sure that you can state the assumption from memory
and that you understand how the following questions help us under-
stand whether the assumption has been met.
a. Assumption I: Is the equation linear with an additive error term? Is
there a chance that there’s an omitted variable or an incorrect func-
tional form?
b. Assumption II: Is there a constant term in the equation to guaran-
tee that the expected value of the error term is zero?
R2
�i
119

THE CLASSICAL MODEL
c. Assumption III: Is there a chance that there’s an omitted variable
or that this equation is part of a simultaneous system?
d. Assumption IV: Is the model estimated with time-series data with
the chance that a random event in one time period could affect the
regression in subsequent time periods?
e. Assumption V: Is the model estimated with cross-sectional data
with dramatic variations in the size of the dependent variable?
f. Assumption VI: Is any independent variable a perfect linear func-
tion of any other independent variable?
g. Assume that dancers earn about as much per hour as promoters. If
the equation is accurate, should the nightclub hire one more pro-
moter or one more dancer if they want to increase attendance? Ex-
plain your answer.
11. In 2001, Donald Kenkel and Joseph Terza published an article in
which they investigated the impact on an individual’s alcohol con-
sumption of a physician’s advice to reduce drinking.8 In that article,
Kenkel and Terza used econometric techniques well beyond the scope
of this text to conclude that such physician advice can play a signifi-
cant role in reducing alcohol consumption.
We took a fifth (no pun intended) of the authors’ dataset9 and
estimated the following equation (standard errors in parentheses):
DRINKSi � 13.00 � 11.36ADVICEi � 0.20EDUCi � 2.85DIVSEPi � 14.20UNEMPi
(2.12) (0.31) (2.55) (5.16)
t � 5.37 – 0.65 1.11 2.75
N � 500 � .07
where: DRINKSi � drinks consumed by the ith individual in the
last two weeks
ADVICEi � 1 if a physician had advised the ith individual
to cut back on drinking alcohol, 0 otherwise
EDUCi � years of schooling of the ith individual
R2
8. Donald S. Kenkel and Joseph V. Terza, “The Effect of Physician Advice on Alcohol Consump-
tion: Count Regression with an Endogenous Treatment Effect,” Journal of Applied Econometrics,
2001, pp. 165–184.
9. The dataset, which is available on the JAE website, consists of more than 20 variables for
2467 males who participated in the 1990 National Health Interview Survey and who were cur-
rent drinkers with high blood pressure.
120

THE CLASSICAL MODEL
DIVSEPi � 1 if the ith individual was divorced or sepa-
rated, 0 otherwise
UNEMPi � 1 if the ith individual was unemployed, 0
otherwise
a. Carefully state the meaning of the estimated coefficients of
DIVSEP and UNEMP. Do the signs of the coefficients make sense
to you? Do the relative sizes of the coefficients make sense to
you? Explain.
b. Carefully state the meaning of the estimated coefficient of ADVICE.
Does the sign of the coefficient make sense to you? If so, explain. If
not, this unexpected sign might be related to a violation of one of
the Classical Assumptions. What Classical Assumption (other than
Assumption I) is this equation almost surely violating? (Hint:
Think about what might cause a physician to advise a patient to cut
back on alcohol drinking and then review the Classical Assump-
tions one more time.)
c. We broke up our sample of 500 observations into five different
samples of 100 observations each and calculated s for four of the
five samples. The results (for ADVICE) were:
1st sample: ADVICE � 10.43
2nd sample: ADVICE � 13.52
3rd sample: ADVICE � 14.39
4th sample: ADVICE � 8.01
The s are different! Explain in your own words how it’s possible
to get different s when you’re estimating identical specifications
on data that are drawn from the same source. What term would
you use to describe this group of s?
d. The data for the fifth sample of 100 observations are in Table 2.
Use these data to estimate DRINKS � f(ADVICE, EDUC, DIVSEP,
and UNEMP) with EViews, Stata, or another regression program.
What value do you get for ADVICE? How do your estimated coeffi-
cients compare to those of the entire sample of 500?
�̂
�̂
�̂
�̂
�̂
�̂
�̂
�̂
�̂
�̂
121

THE CLASSICAL MODEL
Table 2 Data for the Physician Advice Equation
obs DRINKS ADVICE EDUC DIVSEP UNEMP
1 24.0 0 13 0 1
2 10.0 0 14 0 0
3 0.0 0 14 0 0
4 24.0 1 7 0 0
5 0.0 0 12 0 0
6 1.5 1 13 0 0
7 45.0 1 15 0 0
8 0.0 0 12 0 0
9 0.0 0 16 0 0
10 0.0 0 10 0 0
11 2.0 0 16 0 0
12 13.5 0 9 0 0
13 8.0 1 12 0 0
14 0.0 0 14 1 0
15 25.0 0 13 0 0
16 11.3 0 12 1 0
17 0.0 0 17 0 0
18 0.0 0 16 0 0
19 7.0 0 14 0 0
20 40.0 1 16 0 0
21 28.0 0 14 0 0
22 1.0 1 15 0 0
23 0.0 0 10 0 0
24 0.0 0 10 0 0
25 56.0 1 16 0 0
26 0.0 0 16 1 0
27 24.0 1 12 1 0
28 5.0 0 13 0 0
29 28.0 0 7 0 0
30 14.0 0 12 0 0
31 3.0 0 18 0 0
32 0.0 0 7 0 0
33 0.0 0 18 0 0
34 0.0 0 11 0 0
35 3.0 0 12 0 0
36 10.0 0 16 0 0
37 42.0 1 17 0 0
38 1.0 0 12 0 0
39 14.0 0 15 1 0
40 9.0 0 18 0 0
41 0.0 0 18 0 0
42 15.0 0 14 0 0
(continued)
122

THE CLASSICAL MODEL
43 12.0 1 18 0 0
44 6.0 0 14 1 0
45 6.0 1 17 0 0
46 0.0 1 12 0 0
47 0.0 0 12 0 0
48 0.0 0 8 0 0
49 2.0 0 9 1 0
50 0.0 1 12 0 0
51 10.0 1 12 0 0
52 58.5 1 6 0 0
53 14.0 1 14 0 0
54 0.0 0 18 0 0
55 0.0 1 12 0 0
56 5.0 0 13 0 0
57 0.0 0 7 0 0
58 14.0 0 12 0 0
59 36.0 0 13 0 0
60 0.0 0 8 0 0
61 2.0 1 8 1 0
62 70.0 1 16 0 1
63 12.0 1 12 0 0
64 3.0 1 12 0 0
65 30.0 1 9 1 0
66 10.0 0 15 0 0
67 12.0 0 16 0 0
68 84.0 0 12 0 0
69 71.5 1 12 0 0
70 49.0 0 18 0 0
71 4.0 1 13 0 0
72 3.0 1 8 0 0
73 1.0 0 12 0 0
74 33.8 0 13 0 0
75 21.0 0 14 0 0
76 12.0 0 12 0 0
77 14.0 0 18 1 0
78 0.0 0 17 0 0
79 0.0 1 7 0 0
80 1.0 0 12 0 0
81 0.0 1 12 0 0
82 70.0 0 15 1 0
83 4.0 1 16 1 0
84 4.0 0 14 0 0
Table 2 (continued)
obs DRINKS ADVICE EDUC DIVSEP UNEMP
(continued)
123

THE CLASSICAL MODEL
Table 2 (continued)
obs DRINKS ADVICE EDUC DIVSEP UNEMP
85 21.0 1 14 1 0
86 2.0 0 16 0 0
87 30.0 1 10 0 0
88 10.0 1 13 0 0
89 16.0 1 9 1 0
90 36.0 0 13 0 0
91 0.0 1 11 0 0
92 0.0 0 12 0 0
93 108.0 1 12 1 0
94 0.0 0 12 0 0
95 0.0 1 12 0 0
96 11.0 0 13 1 0
97 28.5 0 0 0 0
98 56.0 0 13 0 0
99 3.0 0 12 0 0
100 2.0 0 12 0 0
Datafile � DRINKS4
Source: Donald S. Kenkel and Joseph V. Terza, “The Effect of Physician Advice on Alcohol
Consumption: Count Regression with an Endogenous Treatment Effect,” Journal of Applied
Econometrics, 2001, pp. 165–184.
124

THE CLASSICAL MODEL
Answers
Exercise 2
a. An additional pound of fertilizer per acre will cause corn yield to
increase by 0.10 bushels per acre, holding rainfall constant. An
additional inch of rain will increase corn yield by 5.33 bushels
per acre, holding fertilizer per acre constant.
b. No, for a couple of reasons. First, it’s hard to imagine zero inches
of rain falling in an entire year, so this particular intercept has no
real-world meaning. More generally, recall that the OLS estimate
of the intercept includes the nonzero mean of the error term in
order to meet Classical Assumption II, so even if rainfall were
zero, it wouldn’t make sense to attempt to analyze the OLS esti-
mate of the intercept.
c. No. An unbiased estimator will produce a distribution of esti-
mates that is centered around the true �, but individual estimates
can vary widely from that true value. 0.10 is the estimated coeffi-
cient for this sample, not for the entire population, so it could be
an unbiased estimate.
d. Not necessarily: 5.33 still could be close to or even equal to the
true value. More generally, an estimated coefficient produced by
an estimator that is not BLUE still could be accurate. For exam-
ple, the amount of the bias could be very small, or the variation
due to sampling could offset the bias.
125

126

1 What Is Hypothesis Testing?
2 The t-Test
3 Examples of t-Tests
4 Limitations of the t-Test
5 Summary and Exercises
6 Appendix: The F -Test
Hypothesis Testing
In this chapter, we return to the essence of econometrics—an effort to quan-
tify economic relationships by analyzing sample data—and ask what conclu-
sions we can draw from this quantification. Hypothesis testing goes beyond
calculating estimates of the true population parameters to a much more com-
plex set of questions. Hypothesis testing determines what we can learn about
the real world from a sample. Is it likely that our result could have been
obtained by chance? Can our theories be rejected using the results generated
by our sample? If our theory is correct, what is the probability that this par-
ticular sample would have been observed? This chapter starts with a brief
introduction to the topic of hypothesis testing. We then examine the t-test,
the statistical tool typically used for hypothesis tests of individual regression
coefficients.
Hypothesis testing and the t-test should be familiar topics to readers
with strong backgrounds in statistics, who are encouraged to skim this
chapter and focus on only those applications that seem somewhat new.
The development of hypothesis testing procedures is explained here in
terms of the regression model, however, so parts of the chapter may be in-
structive even to those already skilled in statistics. Students with a weak
background in statistics are encouraged to review that subject before be-
gining this chapter.
Our approach will be classical in nature, since we assume that the sample
data are our best and only information about the population. An alternative,
From Chapter 5 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
127

HYPOTHESIS TESTING
Bayesian statistics, uses a completely different definition of probability and
does not use the sampling distribution concept.1
What Is Hypothesis Testing?
Hypothesis testing is used in a variety of settings. The Food and Drug Admin-
istration (FDA), for example, tests new products before allowing their sale. If
the sample of people exposed to the new product shows some side effect sig-
nificantly more frequently than would be expected to occur by chance, the
FDA is likely to withhold approval of marketing that product. Similarly, econ-
omists have been statistically testing various relationships between consump-
tion and income for almost a century; theories developed by John Maynard
Keynes and Milton Friedman, among others, have been tested on macroeco-
nomic and microeconomic data sets.
Although researchers are always interested in learning whether the theory
in question is supported by estimates generated from a sample of real-world
observations, it’s almost impossible to prove that a given hypothesis is correct.
All that can be done is to state that a particular sample conforms to a particu-
lar hypothesis. Even though we cannot prove that a given theory is “correct”
using hypothesis testing, we often can reject a given hypothesis with a certain
level of significance. In such a case, the researcher concludes that it is very un-
likely that the sample result would have been observed if the hypothesized
theory were correct.
Classical Null and Alternative Hypotheses
The first step in hypothesis testing is to state the hypotheses to be tested. This
should be done before the equation is estimated because hypotheses devel-
oped after estimation run the risk of being justifications of particular results
rather than tests of the validity of those results.
The null hypothesis typically is a statement of the values that the re-
searcher does not expect. The notation used to specify the null hypothesis
is “H0:” followed by a statement of the range of values you do not expect.
1
1. Bayesians, by being forced to state explicitly their prior expectations, tend to do most of their
thinking before estimation, which is a good habit for a number of important reasons. For more
on this approach, see Peter Kennedy, A Guide to Econometrics (Malden, MA: Blackwell, 2008),
pp. 213–231. For more advanced coverage, see Tony Lancaster, An Introduction to Bayesian Econo-
metrics (Oxford: Blackwell Publishing, 2004).
128

HYPOTHESIS TESTING
For example, if you expect a positive coefficient, then you don’t expect a zero
or negative coefficient, and the null hypothesis is:
Null hypothesis H0: � � 0 (the values you do not expect)
The alternative hypothesis typically is a statement of the values that the
researcher expects. The notation used to specify the alternative hypothesis is
“HA:” followed by a statement of the range of values you expect. To continue
our previous example, if you expect a positive coefficient, then the alternative
hypothesis is:
Alternative hypothesis HA: � � 0 (the values you expect)
To test yourself, take a moment and think about what the null and alterna-
tive hypotheses will be if you expect a negative coefficient. That’s right,
they’re:
H0: � � 0
HA: � � 0
The above hypotheses are for a one-sided test because the alternative hy-
potheses have values on only one side of the null hypothesis. Another ap-
proach is to use a two-sided test (or a two-tailed test) in which the alternative
hypothesis has values on both sides of the null hypothesis. For a two-sided
test around zero, the null and alternative hypotheses are:
H0: � � 0
HA: � 2 0
We should note that there are a few rare cases in which we must violate
our rule that the value you expect goes in the alternative hypothesis. Classical
hypothesis testing requires that the null hypothesis contain the equal sign in
some form (whether it be �, �, or �). This requirement means that re-
searchers are forced to put the value they expect in the null hypothesis if their
expectation includes an equal sign. This typically happens when the re-
searcher specifies a specific value rather than a range. Luckily, such exceptions
are unusual in elementary applications.
With the exception of the unusual cases previously mentioned, economists
always put what they expect in the alternative hypothesis. This allows us to
make rather strong statements when we reject a null hypothesis. However, we
129

can never say that we accept the null hypothesis; we must always say that we
cannot reject the null hypothesis. As put by Jan Kmenta:
Just as a court pronounces a verdict as not guilty rather than
innocent, so the conclusion of a statistical test is do not reject rather
than accept.2
Type I and Type II Errors
The typical testing technique in econometrics is to hypothesize an expected
sign (or value) for each regression coefficient (except the constant term) and
then to determine whether to reject the null hypothesis. Since the regression
coefficients are only estimates of the true population parameters, it would be
unrealistic to think that conclusions drawn from regression analysis will al-
ways be right.
There are two kinds of errors we can make in such hypothesis testing:
Type I: We reject a true null hypothesis.
Type II: We do not reject a false null hypothesis.
We will refer to these errors as Type I and Type II Errors, respectively.
Suppose we have the following null and alternative hypotheses:
Even if the true parameter is not positive, the particular estimate ob-
tained by a researcher may be sufficiently positive to lead to the rejection of
the null hypothesis that This is a Type I Error; we have rejected the
truth! A Type I Error is graphed in Figure 1.
Alternatively, it’s possible to obtain an estimate of that is close enough
to zero (or negative) to be considered “not significantly positive.” Such a re-
sult may lead the researcher to “accept”3 the hypothesis that when in
truth This is a Type II Error; we have failed to reject a false null hy-
pothesis! A Type II Error is graphed in Figure 2. (The specific value of
was selected as the true value in that figure purely for illustrative purposes.)
� 5 1
� . 0.
� # 0
�
� # 0.
�
HA: � . 0
H0: � # 0
HYPOTHESIS TESTING
2. Jan Kmenta, Elements of Econometrics (Ann Arbor: University of Michigan Press, 1986), p. 112.
(Emphasis added.)
3. We will consistently put the word accept in quotes whenever we use it. In essence, “accept”
means do not reject.
130

HYPOTHESIS TESTING
0
Distribution of �s
Centered Around 0
�
� Quite Positive
Figure 1 Rejecting a True Null Hypothesis Is a Type I Error
If that is very positive, you might reject a true null
hypothesis, and conclude incorrectly that the alternative hypothesis
is true.HA: � . 0
H0: � # 0,
� 5 0, but you observe a �̂
1.0
Distribution of �s
Centered Around 1
0
�
� Negative
(But Close to 0)
Figure 2 Failure to Reject a False Null Hypothesis Is a Type II Error
If that is negative but close to zero, you might fail to reject
a false null hypothesis, and incorrectly ignore the fact that the alternative
hypothesis, , is true.HA: � . 0
H0: � # 0,
� 5 1, but you observe a �̂
131

As an example of Type I and Type II Errors, let’s suppose that you’re on a
jury in a murder case.4 In such a situation, the presumption of “innocent
until proven guilty” implies that:
H0: The defendant is innocent.
HA: The defendant is guilty.
What would a Type I Error be? Rejecting the null hypothesis would mean
sending the defendant to jail, so a Type I Error, rejecting a true null hypothe-
sis, would mean:
Type I Error � Sending an innocent defendant to jail.
Similarly,
Type II Error � Freeing a guilty defendant.
Most reasonable jury members would want both levels of error to be quite
small, but such certainty is almost impossible. After all, couldn’t there be a
mistaken identification or a lying witness? In the real world, decreasing the
probability of a Type I Error (sending an innocent defendant to jail) means in-
creasing the probability of a Type II Error (freeing a guilty defendant). If we
never sent an innocent defendant to jail, we’d be freeing quite a few murderers!
Decision Rules of Hypothesis Testing
A decision rule is a method of deciding whether to reject a null hypothesis.
Typically, a decision rule involves comparing a sample statistic with a pre-
selected critical value found in tables such as those in the end of this text.
A decision rule should be formulated before regression estimates are ob-
tained. The range of possible values of is divided into two regions, an
“acceptance” region and a rejection region, where the terms are expressed rela-
tive to the null hypothesis. To define these regions, we must determine a
critical value (or, for a two-tailed test, two critical values) of Thus, a critical
value is a value that divides the “acceptance” region from the rejection region
when testing a null hypothesis. Graphs of these “acceptance” and rejection
regions are presented in Figures 3 and 4.
To use a decision rule, we need to select a critical value. Let’s suppose that
the critical value is 1.8. If the observed is greater than 1.8, we can reject the�̂
�̂.
�̂
HYPOTHESIS TESTING
4. This example comes from and is discussed in much more detail in Ed Leamer, Specification
Searches (New York: John Wiley and Sons, 1978), pp. 93–98.
132

HYPOTHESIS TESTING
0
Distribution of �s
Probability of
Type I Error
1.8
�
“Acceptance” Region Rejection
Region
Figure 3 “Acceptance” and Rejection Regions for a One-Sided Test of
For a one-sided test of vs. the critical value divides the distribu-
tion of (centered around zero on the assumption that H0 is true) into “acceptance”
and rejection regions.
�̂
HA: � . 0,H0: � # 0
�
0
Distribution of �s
Probability of
Type I Error
�
“Acceptance” Region Rejection
Region
Rejection
Region
Figure 4 “Acceptance” and Rejection Regions for a Two-Sided Test of
For a two-sided test of vs. we divided the distribution of into an
“acceptance” region and two rejection regions.
�̂HA: � 2 0,H0: � 5 0
�
133

null hypothesis that is zero or negative. To see this, take a look at Figure 3.
Any above 1.8 can be seen to fall into the rejection region, whereas any
below 1.8 can be seen to fall into the “acceptance” region.
The rejection region measures the probability of a Type I Error if the null
hypothesis is true. Some students react to this news by suggesting that we
make the rejection region as small as possible. Unfortunately, decreasing
the chance of a Type I Error means increasing the chance of a Type II Error
(not rejecting a false null hypothesis). This is because if you make the rejec-
tion region so small that you almost never reject a true null hypothesis,
then you’re going to be unable to reject almost every null hypothesis,
whether they’re true or not! As a result, the probability of a Type II Error
will rise.
Given that, how do you choose between Type I and Type II Errors? The an-
swer is easiest if you know that the cost (to society or the decision maker) of
making one kind of error is dramatically larger than the cost of making the
other. If you worked for the FDA, for example, you’d want to be very sure that
you hadn’t released a product that had horrible side effects. We’ll discuss this
dilemma for the t-test later in this chapter.
The t-Test
The t-test is the test that econometricians usually use to test hypotheses
about individual regression slope coefficients. Tests of more than one coeffi-
cient at a time (joint hypotheses) are typically done with the F-test, pre-
sented in Section 6.
The t-test is easy to use because it accounts for differences in the units of
measurement of the variables and in the standard deviations of the esti-
mated coefficients. More important, the t-statistic is the appropriate test to
use when the stochastic error term is normally distributed and when the
variance of that distribution must be estimated. Since these usually are the
case, the use of the t-test for hypothesis testing has become standard prac-
tice in econometrics.
The t-Statistic
For a typical multiple regression equation:
(1)
we can calculate t-values for each of the estimated coefficients in the equa-
tion. The t-tests are usually done only on the slope coefficients; for these, the
relevant form of the t-statistic for the kth coefficient is
Yi 5 �0 1 �1X1i 1 �2X2i 1 �i
2
�̂�̂
�
HYPOTHESIS TESTING
134

(2)tk 5
(�̂
k
2 �
H
0
)
SE(�̂k)
  (k 5 1, 2, . . . , K)
HYPOTHESIS TESTING
where: � the estimated regression coefficient of the kth variable
� the border value (usually zero) implied by the null
hypothesis for
� the estimated standard error of (that is, the square
root of the estimated variance of the distribution of the
note that there is no “hat” attached to SE because SE
is already defined as an estimate)
How do you decide what border is implied by the null hypothesis? Some null
hypotheses specify a particular value. For these, is simply that value; if
Other null hypotheses involve ranges, but we are
concerned only with the value in the null hypothesis that is closest to the
border between the “acceptance” region and the rejection region. This border
value then becomes the then
the value in the null hypothesis closest to the border is zero, and
Since most regression hypotheses test whether a particular regression co-
efficient is significantly different from zero, is typically zero, and the
most-used form of the t-statistic becomes
which simplifies to
(3)
or the estimated coefficient divided by the estimate of its standard error. This
is the t-statistic formula used by most computer programs.
For an example of this calculation, let’s consider this equation for the
check volume at Woody’s restaurants:
(4)
N 5 33  R2 5 .579
t 5 24.42 4.88 2.37
(2053) (0.0727) (0.543)
Ŷi 5 102,192 2 9075Ni 1 0.3547Pi 1 1.288Ii
tk 5
�̂
k
SE(�̂k)
  (k 5 1, 2, . . . , K)
tk 5
(�̂
k
2 0)
SE(�̂k)
  (k 5 1, 2, . . . , K)
�H0
�H0
5 0.
�H0
. For example, if H0: � $ 0 and HA: � , 0,
H0: � 5 S, then �H0
5 S.
�H0
�̂k;
�̂kSE(�̂k)
�k
�H0
�̂k
135

In Equation 4, the numbers in parentheses underneath the estimated regression
coefficients are the estimated standard errors of the estimated and the num-
bers below them are t-values calculated according to Equation 3. The format
used to document Equation 4 is the one we’ll use whenever possible through-
out this text. Note that the sign of the t-value is always the same as that of the es-
timated regression coefficient, and the standard error is always positive.
Using the regression results in Equation 4, let’s calculate the t-value for the
estimated coefficient of P, the population variable. Given the values in Equa-
tion 4 of 0.3547 for and 0.0727 for SE , and given the rel-
evant t-value is indeed 4.88, as specified in Equation 4:
The larger in absolute value this t-value is, the greater the likelihood that the
estimated regression coefficient is significantly different from zero.
The Critical t-Value and the t-Test Decision Rule
To decide whether to reject or not to reject a null hypothesis based on a calcu-
lated t-value, we use a critical t-value. A critical t-value is the value that dis-
tinguishes the “acceptance” region from the rejection region. The critical
t-value, tc, is selected from a t-table (see the critical values of the t-Distribution
Table at the end of this chapter) depending on whether the test is one-sided or
two-sided, on the level of Type I Error you specify and on the degrees of freedom,
which we have defined as the number of observations minus the number of co-
efficients estimated (including the constant) or The level of Type I
Error in a hypothesis test is also called the level of significance of that test and will
be discussed in more detail later in this section. The t-table was created to save
time during research; it consists of critical t-values given specific areas under-
neath curves such as those in Figure 3 for Type I Errors. A critical t-value is thus a
function of the probability of Type I Error that the researcher wants to specify.
Once you have obtained a calculated t-value tk and a critical t-value tc, you
reject the null hypothesis if the calculated t-value is greater in absolute value
than the critical t-value and if the calculated t-value has the sign implied by HA.
Thus, the rule to apply when testing a single regression coefficient is that
you should:
N 2 K 2 1.
tP 5
�̂
P
SE(�̂P)
5
0.3547
0.0727
5 4.88
H0: �P # 0,(�̂P)�̂P
�̂s,
HYPOTHESIS TESTING
Reject H0 if |tk| � tc and if tk also has the sign implied by HA. Do not
reject H0 otherwise.
136

This decision rule works for calculated t-values and critical t-values for one-
sided hypotheses around zero:
for two-sided hypotheses around zero:
for one-sided hypotheses based on hypothesized values other than zero:
and for two-sided hypotheses based on hypothesized values other than zero:
The decision rule is the same: Reject the null hypothesis if the appropriately
calculated t-value, tk, is greater in absolute value than the critical t-value, tc, as
long as the sign of tk is the same as the sign of the coefficient implied in HA.
Otherwise, do not reject H0. Always use Equation 2 whenever the hypothe-
sized value is not zero.
Statistical Table B-1 contains the critical values tc for varying degrees of free-
dom and levels of significance. The columns indicate the levels of significance
according to whether the test is one-sided or two-sided, and the rows indicate
the degrees of freedom. For an example of the use of this table and the decision
rule, let’s return to the Woody’s restaurant example and, in particular, to the
t-value for calculated in the previous section. Recall that we hypothesized
that population’s coefficient would be positive, so this is a one-sided test:
HA: �p . 0
H0: �p # 0
�̂P
HA: �k 2 S
H0: �k 5 S
HA: �k , S
H0: �k $ S
HA: �k . S
H0: �k # S
HA: �k 2 0
H0: �k 5 0
HA: �k , 0
H0: �k $ 0
HA: �k . 0
H0: �k # 0
HYPOTHESIS TESTING
137

There are 29 degrees of freedom (equal to in this
regression, so the appropriate t-value with which to test the calculated
t-value is a one-tailed critical t-value with 29 degrees of freedom. To find this
value, pick a level of significance, say 5 percent, and turn to Statistical Table B-1.
Take a look for yourself. Do you agree that the number there is 1.699?
Given that, should you reject the null hypothesis? The decision rule is to
reject H0 if |tk| � tc and if tk has the sign implied by HA. Since the 5-percent,
one-sided, 29 degrees of freedom critical t-value is 1.699, and since the sign
implied by HA is positive, the decision rule (for this specific case) becomes:
Reject H0 if |tP| � 1.699 and if tP is positive
or, combining the two conditions:
Reject H0 if tP � 1.699
What is tP? In the previous section, we found that tP was �4.88, so we would
reject the null hypothesis and conclude that population does indeed tend to
have a positive relationship with Woody’s check volume (holding the other
variables in the equation constant).
Note from Statistical Table B-1 that the critical t-value for a one-tailed test
at a given level of significance is exactly equal to the critical t-value for a two-
tailed test at twice the level of significance as the one-tailed test. This relation-
ship between one-sided and two-sided tests is illustrated in Figure 5. The crit-
ical value tc � 1.699 is for a one-sided, 5-percent level of significance, but it
also represents a two-sided, 10-percent level of significance because if one tail
represents 5 percent, then both tails added together represent 10 percent.
Choosing a Level of Significance
To complete the previous example, it was necessary to pick a level of signifi-
cance before a critical t-value could be found in Statistical Table B-1. The
words “significantly positive” usually carry the statistical interpretation that
was rejected in favor of according to the pre-
established decision rule, which was set up with a given level of significance.
The level of significance indicates the probability of observing an estimated
t-value greater than the critical t-value if the null hypothesis were correct. It
measures the amount of Type I Error implied by a particular critical t-value. If
the level of significance is 10 percent and we reject the null hypothesis at that
level, then this result would have occurred only 10 percent of the time that
the null hypothesis was indeed correct.
HA (� . 0)H0 (� # 0)
N 2 K 2 1, or 33 2 3 2 1)
HYPOTHESIS TESTING
138

HYPOTHESIS TESTING
How should you choose a level of significance? Most beginning econome-
tricians (and many published ones, too) assume that the lower the level of
significance, the better. After all, they say, doesn’t a low level of significance
guarantee a low probability of making a Type I Error? Unfortunately, an ex-
tremely low level of significance also dramatically increases the probability of
making a Type II Error. Therefore, unless you’re in the unusual situation of
not caring about mistakenly “accepting” a false null hypothesis, minimizing
the level of significance is not good standard practice.
Instead, we recommend using a 5-percent level of significance except in
those circumstances when you know something unusual about the relative
costs of making Type I and Type II Errors. If you know that a Type II Error will
be extremely costly, for example, then it makes sense to consider using a 10-
percent level of significance when you determine your critical value. Such
judgments are difficult, however, so we encourage beginning researchers to
adopt a 5-percent level of significance as standard.
0 1.699
5% One-Sided
Level of Significance
21.699
10% Two-Sided Level of Significance
Figure 5 One-Sided and Two-Sided t-Tests
The tc for a one-sided test at a given level of significance is equal exactly to the tc for a
two-sided test with twice the level of significance of the one-sided test. For example,
tc � 1.699 for a 10-percent two-sided and for a 5-percent one-sided test (for 29 degrees
of freedom).
139

If we can reject a null hypothesis at the 5-percent level of significance, we
can summarize our results by saying that the coefficient is “statistically signif-
icant” at the 5-percent level. Since the 5-percent level is arbitrary, we shouldn’t
jump to conclusions about the value of a variable simply because its coeffi-
cient misses being significant by a small amount; if a different level of signif-
icance had been chosen, the result might have been different.
Some researchers avoid choosing a level of significance by simply stating
the lowest level of significance possible for each estimated regression coeffi-
cient. The use of the resulting significance levels, called p-values, is an alterna-
tive approach to the t-test. p-values are described later in this chapter.
Other researchers produce tables of regression results, typically without hy-
pothesized signs for their coefficients, and then mark “significant” coefficients
with asterisks. The asterisks indicate when the t-score is larger in absolute
value than the two-sided 10-percent critical value (which merits one asterisk),
the two-sided 5-percent critical value (**), or the two-sided 1-percent critical
value (***). Such a use of the t-value should be regarded as a descriptive
rather than a hypothesis-testing use of statistics.
Now and then researchers will use the phrase “degree of confidence” or
“level of confidence” when they test hypotheses. What do they mean? The level
of confidence is nothing more than 100 percent minus the level of significance.
Thus a t-test for which we use a 5-percent level of significance can also be said
to have a 95-percent level of confidence. Since the two terms have identical
meanings, we will use level of significance throughout this text. Another reason
we prefer the term level of significance to level of confidence is to avoid any
possible confusion with the related concept of confidence intervals.
Confidence Intervals
A confidence interval is a range that contains the true value of an item a
specified percentage of the time.5 This percentage is the level of confidence
associated with the level of significance used to choose the critical t-value in
the interval. For an estimated regression coefficient, the confidence interval
can be calculated using the two-sided critical t-value and the standard error of
the estimated coefficient:
(5)Confidence interval 5 �̂ 6 tc ? SE(�̂)
HYPOTHESIS TESTING
5. Technically, if we could take repeated samples, a 90-percent confidence interval would con-
tain the true value in 90 out of 100 of these repeated samples.
140

As an example, let’s return to Equation 4 and our t-test of the significance of
the estimate of the coefficient of population in that equation:
(4)
What would a 90 percent confidence interval for look like? Well,
and so all we need is a 90-percent two-sided
critical t-value for 29 degrees of freedom. As can be seen in Statistical Table
B-1, this tc � 1.699. Substituting these values into Equation 5, we get:
In other words, we are confident that the true coefficient will fall between
0.2312 and 0.4782 90 percent of the time.
What’s the relationship between confidence intervals and two-sided hy-
pothesis testing? It turns out that if a hypothesized border value, falls
within the 90-percent confidence interval for an estimated coefficient, then
we will not be able to reject the null hypothesis at the 10-percent level of
significance in a two-sided test. If, on the other hand, falls outside the
90-percent confidence interval, then we can reject the null hypothesis.
Perhaps the most important econometric use of confidence intervals is
in forecasting. Many decision makers find it practical to be given a fore-
cast of a range of values because they find that a specific point forecast
provides them with little information about the reliability or variability
of the forecast.
p-Values
There’s an alternative approach to the t-test. This alternative, based on a mea-
sure called the p-value, or marginal significance level, is growing in popularity.
A p-value for a t-score is the probability of observing a t-score that size or
larger (in absolute value) if the null hypothesis were true. Graphically, it’s the
area under the curve of the t-distribution between the actual t-score and in-
finity (assuming that the sign of is as expected).
A p-value is a probability, so it runs from 0 to 1. It tells us the lowest level
of significance at which we could reject the null hypothesis (assuming that
�̂
�H0
�H0
,
5 0.3547 6 0.1235
90-percent confidence interval around �̂p 5 0.3547 6 1.699 ? 0.0727
SE(�̂p) 5 0.0727,�̂p 5 0.3547
�̂p
N 5 33 R2 5 .579
t 5 24.42 4.88 2.37
(2053) (0.0727) (0.543)
Ŷi 5 102,192 2 9075Ni 1 0.3547Pi 1 1.288Ii
HYPOTHESIS TESTING
141

the estimate is in the expected direction). A small p-value casts doubt on the
null hypothesis, so to reject a null hypothesis, we need a low p-value.
How do we calculate a p-value? One option would be to comb through
pages and pages of statistical tables, looking for the level of significance that
exactly matches the regression result. That could take days! Luckily, standard
regression software packages calculate p-values automatically and print them
out for every estimated coefficient.6 You’re thus able to read p-values off your
regression output just as you would your s. Be careful, however, because
virtually every regression package prints out p-values for two-sided alternative
hypotheses. Such two-sided p-values include the area in both “tails,” so two-
sided p-values are twice the size of one-sided ones. If your test is one-sided,
you need to divide the p-value in your regression output by 2 before doing
any tests.
How would you use a p-value to run a t-test? If your chosen level of signif-
icance is 5 percent and the p-value is less than .05, then you can reject your
null hypothesis as long as the sign is in the expected direction. Thus the
p-value decision rule is:
�̂
HYPOTHESIS TESTING
Reject H0 if p-valueK � the level of significance and if has the sign
implied by HA.
�̂K
Let’s look at an example of the use of a p-value to run a t-test. If we return to
the Woody’s example of Equation 4 and run a one-sided test on the coefficient
of I, the income variable, we have the following null and alternative hypotheses:
H0: �I � 0
HA: �I � 0
As you can see from the regression output for the Woody’s equation on page
81 or 83 the p-value for is .0246. This is a two-sided p-value and we’re run-
ning a one-sided test, so we need to divide .0246 by 2, getting .0123. Since
.0123 is lower than our chosen level of significance of .05, and since the sign
of agrees with that in HA, we can reject H0. Not surprisingly, this is the
same result we’d get if we ran a conventional t-test.
�I
ˆ
�I
ˆ
6. Different software packages use different names for p-values. EViews, for example, uses the
term “Prob.” Stata, on the other hand, uses P � |t|. Note that such p-values are for H0: � � 0.
142

p-values have a number of advantages. They’re easy to use, and they allow
readers of research to choose their own levels of significance instead of being
forced to use the level chosen by the original researcher. In addition, p-values
convey information to the reader about the relative strength with which we
can reject a null hypothesis. Because of these benefits, many researchers use
p-values on a consistent basis.
Despite these advantages, we will not use p-values in this text. We think that
beginning researchers benefit from learning the standard t-test procedure, par-
ticularly since it’s more likely to force them to remember to hypothesize the
sign of the coefficient and to use a one-sided test when a particular sign can be
hypothesized. In addition, if you know how to use the standard t-test approach,
it’s easy to switch to the p-value approach, but the reverse isn’t necessarily true.
However, we acknowledge that practicing econometricians today spend far
more energy estimating models and coefficients than they spend testing hy-
potheses. This is because most researchers are more confident in their theo-
ries (say, that demand curves slope downward) than they are in the quality of
their data or their regression methods.7 In such situations, where the statisti-
cal tools are being used more for descriptive purposes than for hypothesis
testing purposes, it’s clear that the use of p-values saves time and conveys
more information than does the standard t-test procedure.
Examples of t-Tests
Examples of One-Sided t-Tests
The most common use of the one-sided t-test is to determine whether a re-
gression coefficient is significantly different from zero in the direction pre-
dicted by theory. Let’s face it: if you expect a positive sign for a coefficient and
you get a negative it’s hard to reject the possibility that the true might be
negative (or zero). On the other hand, if you expect a positive sign and get a
positive things get a bit tricky. If is positive but fairly close to zero, then a
one-sided t-test should be used to determine whether the is different
enough from zero to allow the rejection of the null hypothesis. Recall that in
order to be able to control the amount of Type I Error we make, such a theory
implies an alternative hypothesis of (the expected sign) and a null
hypothesis of Let’s look at some complete examples of these
kinds of one-sided t-tests.
H0: � # 0.
HA: � . 0
�̂
�̂�̂,
��̂,
3
HYPOTHESIS TESTING
7. With thanks to Frank Wykoff.
143

Consider a simple model of the aggregate retail sales of new cars that hy-
pothesizes that sales of new cars (Y) are a function of real disposable income
(X1) and the average retail price of a new car adjusted by the consumer price
index (X2). Suppose you spend some time reviewing the literature on the au-
tomobile industry and are inspired to test a new theory. You decide to add a
third independent variable, the number of sports utility vehicles sold (X3), to
take account of the fact that some potential new car buyers now buy car-like
trucks instead. You therefore hypothesize the following model:
(6)
is expected to be positive and negative. This makes sense, since
you’d expect higher incomes, lower prices, or lower numbers of sports utility
vehicles sold to increase new car sales, holding the other variables in the
equation constant. The four steps to use when working with the t-test are:
1. Set up the null and alternative hypotheses.
2. Choose a level of significance and therefore a critical t-value.
3. Run the regression and obtain an estimated t-value (or t-score).
4. Apply the decision rule by comparing the calculated t-value with the
critical t-value in order to reject or not reject the null hypothesis.
Let’s look at each step in more detail.
1. Set up the null and alternative hypotheses.8 From Equation 6, the one-
sided hypotheses are set up as:
HA: �3 , 0
3. H0: �3 $ 0
HA: �2 , 0
2. H0: �2 $ 0
HA: �1 . 0
1. H0: �1 # 0
�2 and �3�1
Y 5 f( X
1
1, X
2
2, X
2
3) 1 �
HYPOTHESIS TESTING
8. The null hypothesis can be stated either as or because the value used
to test is the value in the null hypothesis closest to the border between the accep-
tance and the rejection regions. When the amount of Type I Error is calculated, this border
value of is the one that is used, because over the whole range of , the value gives
the maximum amount of Type I Error. The classical approach limits this maximum amount to a
preassigned level—the chosen level of significance.
� 5 0� # 0�
H0: � # 0
H0: � 5 0H0: � # 0
144

Remember that a t-test typically is not run on the estimate of the con-
stant term
2. Choose a level of significance and therefore a critical t-value. Assume that
you have considered the various costs involved in making Type I and
Type II Errors and have chosen 5 percent as the level of significance
with which you want to test. There are 10 observations in the data set
that is going to be used to test these hypotheses, and so there are
degrees of freedom. At a 5-percent level of sig-
nificance, the critical t-value, tc, can be found in Statistical Table B-1 to
be 1.943. Note that the level of significance does not have to be the
same for all the coefficients in the same regression equation. It could
well be that the costs involved in an incorrectly rejected null hypothe-
sis for one coefficient are much higher than for another, so lower
levels of significance would be used. In this equation, though, for all
three variables:
tc � 1.943
3. Run the regression and obtain an estimated t-value. You now use the data
(annual from 2000 to 2009) to run the regression on your OLS com-
puter package, getting:
(7)
where: Y � new car sales (in hundreds of thousands of units) in
year t
X1 � real U.S. disposable income (in hundreds of billions
of dollars)
X2 � the average retail price of a new car in year t (in dollars)
X3 � the number of sports utility vehicles sold in year t
(in millions)
Once again, we use our standard documentation notation, so the
figures in parentheses are the estimated standard errors of the The
t-values to be used in these hypothesis tests are printed out by standard
OLS programs:
(3)tk 5
�̂
k
SE(�̂k)
(k 5 1, 2, . . . , K)
�̂s.
t 5 2.1 5.6 2 0.1
(2.38) (0.00022) (71.38)
Ŷt 5 1.30 1 4.91X1t 1 0.00123X2t 2 7.14X3t
10 2 3 2 1 5 6
�0.
HYPOTHESIS TESTING
145

For example, the estimated coefficient of X3 divided by its estimated stan-
dard error is Note that since standard errors are al-
ways positive, a negative estimated coefficient implies a negative t-value.
4. Apply the decision rule by comparing the calculated t-value with the critical
t-value in order to reject or not reject the null hypothesis. As stated in Sec-
tion 2, the decision rule for the t-test is to
Reject H0 if |tk| � tc and if tk also has the sign implied by HA.
Do not reject H0 otherwise.
What would these decision rules be for the three hypotheses, given the rele-
vant critical t-value (1.943) and the calculated t-values?
and if 2.1 is positive.
In the case of disposable income, you reject the null hypothesis that
since 2.1 is indeed greater than 1.943. The result (that is, is as
you expected on the basis of theory, since the more income in the country,
the more new car sales you’d expect.
if |5.6| � 1.943 and if 5.6 is negative.
For prices, the t-statistic is large in absolute value (being greater than 1.943)
but has a sign that is contrary to our expectations, since the alternative hy-
pothesis implies a negative sign. Since both conditions in the decision rule
must be met before we can reject H0, you cannot reject the null hypothesis
that That is, you cannot reject the hypothesis that prices have a zero
or positive effect on new car sales! This is an extremely small data set that cov-
ers a time period of dramatic economic swings, but even so, you’re surprised
by this result. Despite your surprise, you stick with your contention that prices
belong in the equation and that their expected impact should be negative.
Notice that the coefficient of X2 is quite small, 0.00123, but that this size
has no effect on the t-calculation other than its relationship to the standard
error of the estimated coefficient. In other words, the absolute magnitude of
any is of no particular importance in determining statistical significance
because a change in the units of measurement of X2 will change both
in exactly the same way, so the calculated t-value (the ratio of
the two) is unchanged.
For sales of sports utility vehicles, the coefficient is not statistically differ-
ent from zero, since and you cannot reject the null hypothesis|20.1| , 1.943,
�̂3
For �3: Reject H0 if |20.1| . 1.943 and if 20.1 is negative.
�̂2 and SE(�̂2)
�̂
�2 $ 0.
For �2: Reject H0:
HA: �1 . 0)
�1 # 0
For �1: Reject H0 if |2.1| . 1.943
27.14>71.38 5 20.1.
HYPOTHESIS TESTING
146

HYPOTHESIS TESTING
0
t
1.943 2.1
t�1
H0 : �1 < 0 HA : �1 > 0
Rejection
Region
“Acceptance”
Region
20.1
t
21.943 5.6
H0 : �2 > 0
HA : �2 < 0 Rejection Region “Acceptance” Region H0 : �3 > 0
HA : �3 < 0 t�3 t�2 Figure 6 One-Sided t-Tests of the Coefficients of the New Car Sales Model Given the estimates in Equation 7 and the critical t-value of 1.943 for a 5-percent level of significance, one-sided, 6 degrees of freedom t-test, we can reject the null hypothesis for , but not for �̂2 or �̂3.�̂1 that even though the estimated coefficient has the sign implied by the alternative hypothesis. After thinking this model over again, you come to the conclusion that you were hasty in adding the variable to the equation. Figure 6 illustrates all three of these outcomes by plotting the critical t-value and the calculated t-values for all three null hypotheses on a t-distribution that is centered around zero (the value in the null hypothesis closest to the border between the acceptance and rejection regions). Students are urged to analyze � $ 0 147 the results of tests on the estimated coefficients of Equation 7 assuming differ- ent numbers of observations and different levels of significance. Exercise 2 has a number of such specific combinations, with answers at the end of the chapter. The purpose of this example is to provide practice in testing hypotheses, and the results of such a poorly thought-out equation for such a small number of observations should not be taken too seriously. Given all that, however, it’s still instructive to note that you did not react the same way to your inability to re- ject the null hypotheses for the price and sports utility vehicle variables. That is, the failure of the sports utility vehicle variable’s coefficient to be significantly negative caused you to realize that perhaps the addition of this variable was ill- advised. The failure of the price variable’s coefficient to be significantly nega- tive did not cause you to consider the possibility that price has no effect on new car sales. Put differently, estimation results should never be allowed to cause you to want to adjust theoretically sound variables or hypotheses, but if they make you realize you have made a serious mistake, then it would be fool- hardy to ignore that mistake. What to do about the positive coefficient of price, on the other hand, is what the “art” of econometrics is all about. Surely a posi- tive coefficient is unsatisfactory, but throwing the price variable out of the equation seems even more so. Possible answers to such issues are addressed more than once in the chapters that follow. Examples of Two-Sided t-Tests Although most hypotheses in regression analysis should be tested with one- sided t-tests, two-sided t-tests are appropriate in particular situations. Researchers sometimes encounter hypotheses that should be rejected if estimated coefficients are significantly different from zero, or a specific non- zero value, in either direction. This situation requires a two-sided t-test. The kinds of circumstances that call for a two-sided test fall into two categories: 1. Two-sided tests of whether an estimated coefficient is significantly dif- ferent from zero, and 2. Two-sided tests of whether an estimated coefficient is significantly dif- ferent from a specific nonzero value. Let’s take a closer look at these categories: 1. Testing whether a is statistically different from zero. The first case for a two-sided test of arises when there are two or more conflicting hypotheses about the expected sign of a coefficient. For example, in the Woody’s restaurant equation, the impact of the average income of an area on the expected number of Woody’s customers in �̂ �̂ HYPOTHESIS TESTING 148 HYPOTHESIS TESTING 0 t +2.045 Critical Value 22.045 Critical Value +2.37 Estimated t-Value H0 : �I = 0 HA : �I = 0 Rejection Region Rejection Region “Acceptance” Region t�I Figure 7 Two-Sided t-Test of the Coefficient of Income in the Woody’s Model Given the estimates of Equation 4 and the critical t-values of for a 5-percent level of significance, two-sided, 29 degrees of freedom t-test, we can reject the null hypothesis that �I 5 0. 62.045 that area is ambiguous. A high-income neighborhood might have more total customers going out to dinner, but those customers might decide to eat at a more formal restaurant than Woody’s. As a result, you might run a two-sided t-test around zero to determine whether the estimated coefficient of income is significantly different from zero in either direc- tion. In other words, since there are reasonable cases to be made for ei- ther a positive or a negative coefficient, it is appropriate to test the for income with a two-sided t-test: As Figure 7 illustrates, a two-sided test implies two different rejection regions (one positive and one negative) surrounding the acceptance region. A critical t-value, tc, must be increased in order to achieve the HA: �I 2 0 H0: �I 5 0 �̂ 149 same level of significance with a two-sided test as can be achieved with a one-sided test.9 As a result, there is an advantage to testing hypotheses with a one-sided test if the underlying theory allows because, for the same t-values, the possibility of Type I Error is half as much for a one-sided test as for a two-sided test. In cases where there are powerful theoretical argu- ments on both sides, however, the researcher has no alternative to using a two-sided t-test around zero. To see how this works, let’s follow through the Woody’s income variable example in more detail. a. Set up the null and alternative hypotheses. b. Choose a level of significance and therefore a critical t-value. You decide to keep the level of significance at 5 percent, but now this amount must be distributed between two rejection regions for 29 degrees of freedom. Hence, the correct critical t-value is 2.045 (found in Statis- tical Table B-1 for 29 degrees of freedom and a 5-percent, two-sided test). Note that, technically, there now are two critical t-values, �2.045 and c. Run the regression and obtain an estimated t-value. Since the value im- plied by the null hypothesis is still zero, the estimated t-value of �2.37 given in Equation 4 is applicable. d. Apply the decision rule by comparing the calculated t-value with the criti- cal t-value in order to reject or not reject the null hypothesis. We once again use the decision rule stated in Section 2, but since the alterna- tive hypothesis specifies either sign, the decision rule simplifies to: In this case, you reject the null hypothesis that equals zero because 2.37 is greater than 2.045 (see Figure 7). Note that the positive sign im- plies that, at least for Woody’s restaurants, income increases customer volume (holding constant population and competition). Given this re- sult, we might well choose to run a one-sided t-test on the next year’s Woody’s data set. For more practice with two-sided t-tests, see Exercise 6. �I For �I Reject H0 if |2.37| . 2.045 22.045. HA: �I 2 0 H0: �I 5 0 HYPOTHESIS TESTING 9. See Figure 5. In that figure, the same critical t-value has double the level of significance for a two-sided test as for a one-sided test. 150 2. Two-sided t-tests of a specific nonzero coefficient value. The second case for a two-sided t-test arises when there is reason to expect a specific nonzero value for an estimated coefficient. For example, if a previous researcher has stated that the true value of some coefficient almost surely equals a particular number, then that number would be the one to test by creating a two-sided t-test around the hypothesized value, To the extent that you feel that the hypothesized value is theoretically correct, you also violate the normal practice of using the null hypothesis to state the hypothesis you expect to reject.10 In such a case, the null and alternative hypotheses become: where is the specific nonzero value hypothesized. Since the hypothesized value is no longer zero, the formula with which to calculate the estimated t-value is Equation 2, repeated here: (2) This t-statistic is still distributed around zero if the null hypothesis is correct, because we have subtracted from the estimated regression coefficient whose expected value is supposed to be is true. Since the t-statistic is still centered around zero, the decision rule developed earlier still is applicable. For practice with this kind of t-test, see Exercise 6. Limitations of the t-Test A problem with the t-test is that it is easy to misuse; t-scores are printed out by computer regression packages and the t-test seems easy to work with, so beginning researchers sometimes attempt to use the t-test to “prove” things 4 �H0 when H0 �H0 tk 5 (�̂ k 2 � H 0 ) SE(�̂k) (k 5 1, 2, . . . , K) � �H0 HA: �k 2 �H0 H0: �k 5 �H0 �H0 . �H0 , HYPOTHESIS TESTING 10. Instead of being able to reject an incorrect theory based on the evidence, the researcher who violates the normal practice is reduced to “not rejecting” the value expected to be true. How- ever, there are many theories that are not rejected by the data, and the researcher is left with a regrettably weak conclusion. One way to accommodate such violations is to increase the level of significance, thereby increasing the likelihood of a Type I Error. � 151 that it was never intended to even test. For that reason, it’s probably just as important to know the limitations of the t-test11 as it is to know the appli- cations of that test. Perhaps the most important of these limitations is that the usefulness of the t-test diminishes rapidly as more and more specifica- tions are estimated and tested. The purpose of the present section is to give additional examples of how the t-test should not be used. The t-Test Does Not Test Theoretical Validity Recall that the purpose of the t-test is to help the researcher make inferences about a particular population coefficient based on an estimate obtained from a sample of that population. Some beginning researchers conclude that any statistically significant result is also a theoretically correct one. This is danger- ous because such a conclusion confuses statistical significance with theoreti- cal validity. Consider for instance, the following estimated regression that explains the consumer price index in the United Kingdom:12 (8) Apply the t-test to these estimates. Do you agree that the two slope coeffi- cients are statistically significant? As a quick check of Statistical Table B-1 shows, the critical t-value for 18 degrees of freedom and a 5-percent two- tailed level of significance is 2.101, so we can reject the null hypothesis of no effect in these cases and conclude that C and C2 are indeed statistically signif- icant variables in explaining P. The catch is that P is the consumer price index and C is the cumulative amount of rainfall in the United Kingdom! We have just shown that rain is statistically significant in explaining consumer prices; does that also show that the underlying theory is valid? Of course not. Why is the statistical result so significant? The answer is that by chance there is a common trend on both R2 5 .982 N 5 21 t 5 213.9 19.5 (0.23) (0.02) P̂ 5 10.9 2 3.2C 1 0.39C2 HYPOTHESIS TESTING 11. These limitations also apply to the use of p-values. For example, many beginning students conclude that the variable with the lowest p-value is the most important variable in an equa- tion, but this is just as false for p-values as it is for the t-test. 12. These results, and others similar to them, can be found in David F. Hendry, “Econometrics— Alchemy or Science?” Economica, Vol. 47, pp. 383–406. 152 sides of the equation. This common trend does not have any meaning. The moral should be clear: Never conclude that statistical significance, as shown by the t-test, is the same as theoretical validity. Occasionally, estimated coefficients will be significant in the direction op- posite from that hypothesized, and some beginning researchers may be tempted to change their hypotheses. For example, a student might run a regres- sion in which the hypothesized sign is positive, get a “statistically significant” negative sign, and be tempted to change the theoretical expectations to “ex- pect” a negative sign after “rethinking” the issue. Although it is admirable to be willing to reexamine incorrect theories on the basis of new evidence, that evi- dence should be, for the most part, theoretical in nature. If the evidence causes a researcher to go back to the theoretical underpinnings of a model and find a mistake, then the null hypothesis should be changed, but then this new hy- pothesis should be tested using a completely different data set. After all, we al- ready know what the result will be if the hypothesis is tested on the old one. The t-Test Does Not Test “Importance” One possible use of a regression equation is to help determine which inde- pendent variable has the largest relative effect (importance) on the dependent variable. Some beginning researchers draw the unwarranted conclusion that the most statistically significant variable in their estimated regression is also the most important in terms of explaining the largest portion of the move- ment of the dependent variable. Statistical significance indicates the likeli- hood that a particular sample result could have been obtained by chance, but it says little—if anything—about which variables determine the major portion of the variation in the dependent variable. To determine importance, a mea- sure such as the size of the coefficient multiplied by the average size of the in- dependent variable or the standard error of the independent variable would make much more sense. Consider the following hypothetical equation: (9) where: Y � mail-order sales of O’Henry’s Oyster Recipes X1 � hundreds of dollars of advertising expenditures in Gourmets’ Magazine X2 � hundreds of dollars of advertising expenditures on the Julia Adult TV Cooking Show R2 5 .90 N 5 30 t 5 10.0 8.0 (1.0) (25.0) Ŷ 5 300.0 1 10.0X1 1 200.0X2 HYPOTHESIS TESTING 153 (Assume that all other factors, including prices, quality, and competition, re- main constant during the estimation period.) Where should O’Henry be spending his advertising money? That is, which independent variable has the biggest impact per dollar on Y? Given that X2’s coefficient is 20 times X1’s coefficient, you’d have to agree that X2 is more important as defined, and yet which coefficient is more statistically significantly different from zero? With a t-score of 10.0, X1 is more statistically significant than X2 and its 8.0, but all that means is that we have more evidence that the co- efficient is positive, not that the variable itself is necessarily more important in determining Y. The t-Test Is Not Intended for Tests of the Entire Population The t-test helps make inferences about the true value of a parameter from an estimate calculated from a sample of the population (the group from which the sample is being drawn). As the size of the sample approaches the size of the population, an unbiased estimated coefficient approaches the true population value. If a coefficient is calculated from the entire popula- tion, then an unbiased estimate already measures the population value and a significant t-test adds nothing to this knowledge. One might forget this property and attach too much importance to t-scores that have been ob- tained from samples that approximate the population in size. All the t-test does is help decide how likely it is that a particular small sample will cause a researcher to make a mistake in rejecting hypotheses about the true popu- lation parameters. This point can perhaps best be seen by remembering that the t-score is the estimated regression coefficient divided by the standard error of the esti- mated regression coefficient. If the sample size is large enough to approach the population, then the standard error will fall close to zero because the dis- tribution of estimates becomes more and more narrowly distributed around the true parameter (if this is an unbiased estimate). The standard error will approach zero as the sample size approaches infinity. Thus, the t-score will eventually become: The mere existence of a large t-score for a huge sample has no real substan- tive significance, because if the sample size is large enough, you can reject al- most any null hypothesis! It is true that sample sizes in econometrics can t 5 �̂ 0 5 ` HYPOTHESIS TESTING 154 never approach infinity, but many are quite large; and others contain the en- tire population in one data set.13 Summary 1. Hypothesis testing makes inferences about the validity of specific eco- nomic (or other) theories from a sample of the population for which the theories are supposed to be true. The four basic steps of hypothe- sis testing (using a t-test as an example) are: a. Set up the null and alternative hypotheses. b. Choose a level of significance and, therefore, a critical t-value. c. Run the regression and obtain an estimated t-value. d. Apply the decision rule by comparing the calculated t-value with the critical t-value in order to reject or not reject the null hypothesis. 2. The null hypothesis states the range of values that the regression coef- ficient is expected to take on if the researcher’s theory is not correct. The alternative hypothesis is a statement of the range of values that the regression coefficient is expected to take if the researcher’s theory is correct. 3. The two kinds of errors we can make in such hypothesis testing are: Type I: We reject a null hypothesis that is true. Type II: We do not reject a null hypothesis that is false. 4. The t-test tests hypotheses about individual coefficients from regres- sion equations. The form for the t-statistic is In many regression applications, is zero. Once you have calcu- lated a t-value and chosen a critical t-value, you reject the null hypoth- esis if the t-value is greater in absolute value than the critical t-value and if the t-value has the sign implied by the alternative hypothesis. �H0 tk 5 (�̂ k 2 � H 0 ) SE(�̂k) (k 5 1, 2, . . . , K) 5 HYPOTHESIS TESTING 13. D. N. McCloskey, “The Loss Function Has Been Mislaid: The Rhetoric of Significance Tests,” American Economic Review, Vol. 75, No. 2, p. 204. 155 5. The t-test is easy to use for a number of reasons, but care should be taken when using the t-test to avoid confusing statistical significance with theoretical validity or empirical importance. EXERCISES (The answer to Exercise 2 is at the end of the chapter.) 1. Write the meaning of each of the following terms without referring to the book (or your notes), and compare your definition with the ver- sion in the text for each. a. null hypothesis b. alternative hypothesis c. Type I Error d. level of significance e. two-sided test f. decision rule g. critical value h. t-statistic i. confidence interval j. p-value 2. Return to Section 3 and test the hypotheses implied by Equation 6 with the results in Equation 7 for all three coefficients under the fol- lowing circumstances: a. 10 percent significance and 15 observations b. 10 percent significance and 28 observations c. 1 percent significance and 10 observations 3. Create null and alternative hypotheses for the following coefficients: a. the impact of height on weight b. all the coefficients in Equation A in Exercise 7, Chapter 2 c. all the coefficients in Y � f(X1, X2, and X3) where Y is total gasoline used on a particular trip, X1 is miles traveled, X2 is the weight of the car, and X3 is the average speed traveled d. the impact of the decibel level of the grunt of a shot-putter on the length of the throw involved (shot-putters are known to make loud noises when they throw, but there is little theory about the impact of this yelling on the length of the put). Assume all relevant “non- grunt” variables are included in the equation. HYPOTHESIS TESTING 156 4. Think of examples other than the ones in this chapter in which: a. It would be more important to keep the likelihood of a Type I Error low than to keep the likelihood of a Type II Error low. b. It would be more important to keep the likelihood of a Type II Error low than to keep the likelihood of a Type I Error low. 5. Return to Section 2 and test the appropriate hypotheses with the results in Equation 4 for all three coefficients under the following cir- cumstances: a. 5 percent significance and 6 degrees of freedom b. 10 percent significance and 29 degrees of freedom c. 1 percent significance and 2 degrees of freedom 6. Using the techniques of Section 3, test the following two-sided hy- potheses: a. For Equation 9, test the hypothesis that: at the 5-percent level of significance. b. For Equation 4, test the hypothesis that: at the 1-percent level of significance. c. For Equation 7, test the hypothesis that: at the 5-percent level of significance. 7. For all three tests in Exercise 6, under what circumstances would you worry about possible violations of the principle that the null hypoth- esis contains that which you do not expect to be true? In particular, what would your theoretical expectations have to be in order to avoid violating this principle in Exercise 6a? 8. Consider the following hypothetical equation for a sample of di- vorced men who failed to make at least one child support payment in the last four years (standard errors in parentheses): (0.10) (20.0) (1.00) (3.0) (0.05) P̂i 5 2.0 1 0.50Mi 1 25.0Yi 1 0.80Ai 1 3.0Bi 2 0.15Ci HA: �2 2 0 H0: �2 5 0 HA: �3 2 0 H0: �3 5 0 HA: �2 2 160.0 H0: �2 5 160.0 HYPOTHESIS TESTING 157 where: Pi � the number of monthly child support payments that the ith man missed in the last four years Mi � the number of months the ith man was unemployed in the last four years Yi � the percentage of disposable income that goes to child support payments for the ith man Ai � the age in years of the ith man Bi � the religious beliefs of the ith man (a scale of 1 to 4, with 4 being the most religious) Ci � the number of children the ith man has fathered a. Your friend expects the coefficients of M and Y to be positive. Test these hypotheses. (Use the 5-percent level and N � 20.) b. Test the hypothesis that the coefficient of A is different from zero. (Use the 1-percent level and N � 25.) c. Develop and test hypotheses for the coefficients of B and C. (Use the 10-percent level and N � 17.) 9. Suppose that you estimate a model of house prices to determine the impact of having beach frontage on the value of a house.14 You do some research, and you decide to use the size of the lot instead of the size of the house for a number of theoretical and data availability rea- sons. Your results (standard errors in parentheses) are: PRICEi � 40 � 35.0 LOTi 2.0 AGEi � 10.0 BEDi 4.0 FIREi � 100 BEACHi (5.0) (1.0) (10.0) (4.0) (10) N � 30 2 � .63 where: PRICEi � the price of the ith house (in thousands of dollars) LOTi � the size of the lot of the ith house (in thousands of square feet) AGEi � the age of the ith house in years BEDi � the number of bedrooms in the ith house FIREi � a dummy variable for a fireplace (1 � yes for the ith house) BEACHi � a dummy for having beach frontage (1 � yes for the ith house) R HYPOTHESIS TESTING 14. This hypothetical result draws on Rachelle Rush and Thomas H. Bruggink, “The Value of Ocean Proximity on Barrier Island Houses,” The Appraisal Journal, April 2000, pp. 142–150. 158 a. You expect the variables LOT, BED, and BEACH to have positive co- efficients. Create and test the appropriate hypotheses to evaluate these expectations at the 5-percent level. b. You expect AGE to have a negative coefficient. Create and test the appropriate hypotheses to evaluate these expectations at the 10-percent level. c. At first you expect FIRE to have a positive coefficient, but one of your friends says that fireplaces are messy and are a pain to keep clean, so you’re not sure. Run a two-sided t-test around zero to test these expectations at the 5-percent level. d. What problems appear to exist in your equation? (Hint: Do you have any unexpected signs? Do you have any coefficients that are not significantly different from zero?) e. Which of the problems that you outline in part d is the most worri- some? Explain your answer. f. What explanation or solution can you think of for this problem? 10. Suppose that you’ve been asked by the San Diego Padres baseball team to evaluate the economic impact of their new stadium by ana- lyzing the team’s attendance per game in the last year at their old sta- dium. After some research on the topic, you build the following model (standard errors in parentheses): ATTi � 25000 � 15000 WINi � 4000 FREEi 3000 DAYi 12000 WEEKi (15000) (2000) (3000) (3000) N � 35 2 � .41 where: ATTi � the attendance at the ith game WINi � the winning percentage of the opponent in the ith game FREEi � a dummy variable equal to 1 if the ith game was a “promotion” game at which something was given free to each fan, 0 otherwise DAYi � a dummy variable equal to 1 if the ith game was a day game and equal to 0 if the game was a night or twilight game WEEKi � a dummy variable equal to 1 if the ith game was during the week and equal to 0 if it was on the weekend a. You expect the variables WIN and FREE to have positive coeffi- cients. Create and test the appropriate hypotheses to evaluate these expectations at the 5-percent level. R HYPOTHESIS TESTING 159 b. You expect WEEK to have a negative coefficient. Create and test the appropriate hypotheses to evaluate these expectations at the 1-percent level. c. You’ve included the day game variable because your boss thinks it’s important, but you’re not sure about the impact of day games on attendance. Run a two-sided t-test around zero to test these expec- tations at the 5-percent level. d. What problems appear to exist in your equation? (Hint: Do you have any unexpected signs? Do you have any coefficients that are not significantly different from zero?) e. Which of the problems that you outlined in part d is the most wor- risome? Explain your answer. f. What explanation or solution can you think of for this problem? (Hint: You don’t need to be a sports fan to answer this question. If you like music, think about attendance at outdoor concerts.) 11. Thomas Bruggink and David Rose15 estimated a regression for the an- nual team revenue for Major League Baseball franchises: where: Ri � team revenue from attendance, broadcasting, and concessions (in thousands of dollars) Pi � the ith team’s winning rate (their winning percentage multiplied by a thousand, 1,000 � high) Mi � the population of the ith team’s metropolitan area (in millions) Si � a dummy equal to 1 if the ith team’s stadium was built before 1940, 0 otherwise Ti � a dummy equal to 1 if the ith team’s city has two Major League Baseball teams, 0 otherwise a. Develop and test appropriate hypotheses about the individual co- efficients at the 5 percent level. (Hint: You do not have to be a sports fan to do this question correctly.) R2 5 .682 N 5 78 (198421986) t 5 5.8 6.3 1.0 23.3 (9.1) (233.6) (1363.6) (2255.7) R̂i 5 21522.5 1 53.1Pi 1 1469.4Mi 1 1322.7Si 2 7376.3Ti HYPOTHESIS TESTING 15. Thomas H. Bruggink and David R. Rose, Jr., “Financial Restraint in the Free Agent Labor Market for Major League Baseball: Players Look at Strike Three,” Southern Economic Journal, Vol. 56, pp. 1029–1043. 160 b. The authors originally expected a negative coefficient for S. Their explanation for the unexpected positive sign was that teams in older stadiums have greater revenue because they’re better known and have more faithful fans. Since this is just one observation from the sampling distribution of do you think they should have changed their expected sign? c. On the other hand, Keynes reportedly said, “When I’m wrong, I change my mind; what do you do?” If one lets you realize an error, shouldn’t you be allowed to change your expectation? How would you go about resolving this difficulty? d. Assume that your team is in last place with P � 350. According to this regression equation, would it be profitable to pay $7 million a year to a free agent who would raise the team’s winning rate (P) to 500? Be specific. 12. To get some practice with the t-test, let’s return to the model of iPod prices on eBay that was developed in Exercise 11 in Chapter 3. That equation was: PRICEi � 109.24 � 54.99NEWi 20.44SCRATCHi � 0.73BIDRSi (5.34) (5.11) (0.59) t � 10.28 4.00 1.23 N � 215 where: PRICEi � the price at which the ith iPod sold on eBay NEWi � a dummy variable equal to 1 if the ith iPod was new, 0 otherwise SCRATCHi � a dummy variable equal to 1 if the ith iPod had a minor cosmetic defect, 0 otherwise BIDRSi � the number of bidders on the ith iPod a. Create and test hypothesis for the coefficients of NEW and SCRATCH at the 5-percent level. (Hint: Use the critical value for 120 degrees of freedom.) b. In theory, the more bidders there are on a given iPod, the higher the price should be. Create and test hypotheses at the 1-percent level to see if this theory can be supported by the results. c. Based on the hypothesis tests you conducted in parts a and b, are there any variables that you think should be dropped from the equation? Explain. d. If you could add one variable to this equation, what would it be? Explain. (Hint: All the iPods in the sample are silver-colored, 4 GB Apple iPod minis.) �̂ �̂s, �̂ HYPOTHESIS TESTING 161 13. To get more experience with the t-test, let’s return to the model of al- cohol consumption that we developed in Exercise 11 of Chapter 4. That equation was: DRINKSi � 13.00 � 11.36ADVICEi 0.20EDUCi � 2.85DIVSEPi � 14.20UNEMPi (2.12) (0.31) (2.55) (5.16) t � 5.37 0.65 1.11 2.75 N � 500 � .07 where: DRINKSi � drinks consumed by the ith individual in the last two weeks ADVICEi � 1 if a physician had advised the ith individual to cut back on drinking alcohol, 0 otherwise EDUCi � years of schooling of the ith individual DIVSEPi � 1 if the ith individual was divorced or sepa- rated, 0 otherwise UNEMPi � 1 if the ith individual was unemployed, 0 otherwise a. It seems reasonable to expect positive coefficients for DIVSEP and UNEMP. Create and test appropriate hypotheses for the coefficients of DIVSEP and UNEMP at the 5-percent level. (Hint: Use the criti- cal value for 120 degrees of freedom.) b. Create and run a two-sided hypothesis test around zero of the coef- ficient of EDUC at the 1-percent level. Why might a two-sided test be appropriate for this coefficient? c. Most physicians would expect that if they urged patients to drink less alcohol, that’s what the patients actually would do (holding con- stant the other variables in the equation). Create and test appropri- ate hypotheses for the coefficient of ADVICE at the 10-percent level. d. Does your answer to part c cause you to wonder if perhaps you should change your hypotheses in part c? Explain. 14. Frederick Schut and Peter VanBergeijk16 published an article in which they attempted to see if the pharmaceutical industry practiced inter- national price discrimination by estimating a model of the prices of pharmaceuticals in a cross section of 32 countries. The authors felt R2 HYPOTHESIS TESTING 16. Frederick T. Schut and Peter A. G. VanBergeijk, “International Price Discrimination: The Pharmaceutical Industry,” World Development, Vol. 14, No. 9, pp. 1141–1150. The estimated co- efficients we list are those produced by EViews using the original data and differ slightly from those in the original article. 162 that if price discrimination existed, then the coefficient of per capita income in a properly specified price equation would be strongly posi- tive. The reason they felt that the coefficient of per capita income would measure price discrimination went as follows: the higher the ability to pay, the lower (in absolute value) the price elasticity of de- mand for pharmaceuticals and the higher the price a price discrimina- tor could charge. In addition, the authors expected that prices would be higher if pharmaceutical patents were allowed and that prices would be lower if price controls existed, if competition was encour- aged, or if the pharmaceutical market in a country was relatively large. Their estimates were (standard errors in parentheses): (10) where: Pi � the pharmaceutical price level in the ith country divided by that of the United States GDPNi � per capita domestic product in the ith country divided by that of the United States CVNi � per capita volume of consumption of pharma- ceuticals in the ith country divided by that of the United States PPi � a dummy variable equal to 1 if patents for phar- maceutical products are recognized in the ith country, 0 otherwise DPCi � a dummy variable equal to 1 if the ith country applied strict price controls, 0 otherwise IPCi � a dummy variable equal to 1 if the ith country encouraged price competition, 0 otherwise a. Develop and test appropriate hypotheses concerning the regression coefficients using the t-test at the 5-percent level. b. Set up 90-percent confidence intervals for each of the estimated slope coefficients. c. Do you think Schut and VanBergeijk concluded that international price discrimination exists? Why or why not? d. How would the estimated results have differed if the authors had not divided each country’s prices, per capita income, and per capita N 5 32 R2 5 .775 t 52 2.25 2 1.59 (6.93) (7.16) 2 15.63DPCi 2 11.38IPCi t 5 6.69 22.66 ̨1.19 (0.21) (0.22) (6.12) P̂i 5 38.22 1 1.43GDPNi 2 0.6CVNi 1 7.31PPi HYPOTHESIS TESTING 163 pharmaceutical consumption by that of the United States? Explain your answer. e. Reproduce their regression results by using the EViews computer program (datafile DRUGS5) or your own computer program and the data from Table 1. HYPOTHESIS TESTING Table 1 Data for the Pharmaceutical Price Discrimination Exercise Country P GDPN CV N CVN PP IPC DPC Malawi 60.83 4.9 0.014 2.36 0.6 1 0 0 Kenya 50.63 6.56 0.07 6.27 1.1 1 0 0 India 31.71 6.56 18.66 282.76 6.6 0 0 1 Pakistan 38.76 8.23 3.42 32.9 10.4 0 1 1 Sri Lanka 15.22 9.3 0.42 6.32 6.7 1 1 1 Zambia 96.58 10.3 0.05 2.33 2.2 1 0 0 Thailand 48.01 13.0 2.21 19.60 11.3 0 0 0 Philippines 51.14 13.2 0.77 19.70 3.9 1 0 0 South Korea 35.10 20.7 2.20 16.52 13.3 0 0 0 Malaysia 70.74 21.5 0.50 5.58 8.9 1 0 0 Colombia 48.07 22.4 1.56 11.09 14.1 0 1 0 Jamaica 46.13 24.0 0.21 0.96 22.0 1 0 0 Brazil 63.83 25.2 10.48 50.17 21.6 0 1 0 Mexico 69.68 34.7 7.77 28.16 27.6 0 0 0 Yugoslavia 48.24 36.1 3.83 9.42 40.6 0 1 1 Iran 70.42 37.7 3.27 15.33 21.3 0 0 0 Uruguay 65.95 39.6 0.44 1.30 33.8 0 0 0 Ireland 73.58 42.5 0.57 1.49 38.0 1 0 0 Hungary 57.25 49.6 2.36 4.94 47.8 0 1 1 Poland 53.98 50.1 8.08 15.93 50.7 0 1 1 Italy 69.01 53.8 12.02 26.14 45.9 0 0 1 Spain 69.68 55.9 9.01 16.63 54.2 0 0 0 United Kingdom 71.19 63.9 9.96 26.21 38.0 1 1 1 Japan 81.88 68.4 28.58 52.24 54.7 0 0 1 Austria 139.53 69.6 1.24 3.52 35.2 0 0 0 Netherlands 137.29 75.2 1.54 6.40 24.1 1 0 0 Belgium 101.73 77.7 3.49 4.59 76.0 1 0 1 France 91.56 81.9 25.14 24.70 101.8 1 0 1 Luxembourg 100.27 82.0 0.10 0.17 60.5 1 0 1 Denmark 157.56 82.4 0.70 2.35 29.5 1 0 0 Germany, West 152.52 83.0 24.29 28.95 83.9 1 0 0 United States 100.00 100.0 100.00 100.00 100.0 1 1 0 Source: Frederick T. Schut and Peter A. G. VanBergeijk, “International Price Discrimination: The Pharmaceutical Industry,” World Development, Vol. 14, No. 9, p. 1144. Datafile � DRUGS5 164 Appendix: The F -Test Although the t-test is invaluable for hypotheses about individual regression coefficients, it can’t be used to test multiple hypotheses simultaneously. Such a limitation is unfortunate because many interesting ideas involve a number of hypotheses or involve one hypothesis about multiple coefficients. For ex- ample, suppose that you want to test the null hypothesis that there is no sea- sonal variation in a quarterly regression equation that has dummy variables for the seasons. To test such a hypothesis, most researchers would use the F-test. What Is the F -Test? The F-test is a formal hypothesis test that is designed to deal with a null hy- pothesis that contains multiple hypotheses or a single hypothesis about a group of coefficients.17 Such “joint” or “compound” null hypotheses are ap- propriate whenever the underlying economic theory specifies values for mul- tiple coefficients simultaneously. The way in which the F-test works is fairly ingenious. The first step is to translate the particular null hypothesis in question into constraints that will be placed on the equation. The resulting constrained equation can be thought of as what the equation would look like if the null hypothesis were correct; you substitute the hypothesized values into the regression equation in order to see what would happen if the equation were constrained to agree with the null hypothesis. As a result, in the F-test the null hypothesis always leads to a constrained equation, even if this violates our standard practice that the alter- native hypothesis contains what we expect is true. The second step in an F-test is to estimate this constrained equation with OLS and compare the fit of this constrained equation with the fit of the un- constrained equation. If the fits of the constrained equation and the uncon- strained equation are not significantly different, the null hypothesis should not be rejected. If the fit of the unconstrained equation is significantly better than that of the constrained equation, then we reject the null hypothesis. The fit of the constrained equation is never superior to the fit of the uncon- strained equation, as we’ll explain next. 6 HYPOTHESIS TESTING 17. As you will see, the F-test works by placing constraints or restrictions on the equation to be tested. Because of this, it’s equivalent to say that the F-test is for tests that involve multiple linear restrictions. 165 The fits of the equations are compared with the general F-statistic: (11) where: RSS � residual sum of squares from the unconstrained equation RSSM � residual sum of squares from the constrained equation M � number of constraints placed on the equation (usually equal to the number of eliminated from the unconstrained equation) � degrees of freedom in the unconstrained equation RSSM is always greater than or equal to RSS; imposing constraints on the co- efficients instead of allowing OLS to select their values can never decrease the summed squared residuals. (Recall that OLS selects that combination of val- ues of the coefficients that minimizes RSS.) At the extreme, if the uncon- strained regression yields exactly the same estimated coefficients as does the constrained regression, then the RSS are equal, and the F-statistic is zero. In this case, H0 is not rejected because the data indicate that the constraints ap- pear to be correct. As the difference between the constrained coefficients and the unconstrained coefficients increases, the data indicate that the null hy- pothesis is less likely to be true. Thus, when F gets larger than the critical F-value, the hypothesized restrictions specified in the null hypothesis are re- jected by the test. The decision rule to use in the F-test is to reject the null hypothesis if the calculated F-value (F) from Equation 11 is greater than the appropriate criti- cal F-value (Fc): (N 2 K 2 1) �s F 5 (RSS M 2 RSS)>M
RSS>(N 2 K 2 1)
HYPOTHESIS TESTING
Reject
Do not reject H0 if F # Fc
H0 if F . Fc
The critical F-value, Fc, is determined from Statistical Table B-2 or B-3, found at
the end of the chapter, depending on a level of significance chosen by the re-
searcher and on the degrees of freedom. The F-statistic has two types of degrees of
freedom: the degrees of freedom for the numerator of Equation 11 (M, the num-
ber of constraints implied by the null hypothesis) and the degrees of freedom
166

for the denominator of Equation 11 the degrees of freedom in
the regression equation). The underlying principle here is that if the calcu-
lated F-value (or F-ratio) is greater than the critical value, then the estimated
equation’s fit is significantly better than the constrained equation’s fit, and
we can reject the null hypothesis of no effect.
The F -Test of Overall Significance
Although R2 and measure the overall degree of fit of an equation, they
don’t provide a formal hypothesis test of that overall fit. Such a test is pro-
vided by the F-test. The null hypothesis in an F-test of overall significance is
that all the slope coefficients in the equation equal zero simultaneously. For
an equation with K independent variables, this means that the null and alter-
native hypotheses would be18:
To show that the overall fit of the estimated equation is statistically signifi-
cant, we must be able to reject this null hypothesis using the F-test.
For the F-test of overall significance, Equation 11 simplifies to:
(12)
This is the ratio of the explained sum of squares (ESS) to the residual sum of
squares (RSS), adjusted for the number of independent variables (K) and the
number of observations in the sample (N). In this case, the “constrained
equation” to which we’re comparing the overall fit is:
(13)
which is nothing more than saying . Thus the F-test of overall signifi-
cance is really testing the null hypothesis that the fit of the equation isn’t sig-
nificantly better than that provided by using the mean alone.
Ŷi 5 Y
Yi 5 �0 1 �i
F 5
ESS>K
RSS>(N 2 K 2 1)
5
g (Ŷi 2 Y)
2>K
g e
2
i >(N 2 K 2 1)
HA: H0 is not true
H0: �1 5 �2 5
c 5 �K 5 0
R2
(N 2 K 2 1,
HYPOTHESIS TESTING
18. Note that we don’t hypothesize that This would imply that Note also
that for the test of overall significance, M � K.
E(Y) 5 0.�0 5 0.
167

To see how this works, let’s test the overall significance of the Woody’s
restaurant model of Equation 4 from Chapter 3. Since there are three inde-
pendent variables, the null and alternative hypotheses are:
To decide whether to reject or not reject this null hypothesis, we need to
calculate Equation 12 from Chapter 12 for the Woody’s example. There are
three constraints in the null hypothesis, so K � 3. If we check the EViews
computer output for the Woody’s equation in Chapter 3, we can see that
N � 33 and RSS � 6,130,000,000. In addition, it can be calculated that ESS
equals 9,929,450,000.19 Thus the appropriate F-ratio is:
(14)
In practice, this calculation is never necessary, since virtually every computer
regression package routinely provides the computed F-ratio for a test of over-
all significance as a matter of course. On the Woody’s computer output, the
value of the F-statistic can be found in the right-hand column.
Our decision rule tells us to reject the null hypothesis if the calculated
F-value is greater than the critical F-value. To determine that critical F-value,
we need to know the level of significance and the degrees of freedom. If we
assume a 5-percent level of significance, the appropriate table to use is the
F-Distribution Table at the end of this chapter. The numerator degrees of
freedom equal 3 (K), and the denominator degrees of freedom equal 29
so we need to look in Statistical Table B-2 for the critical F-
value for 3 and 29 degrees of freedom. As the reader can verify,20 Fc � 2.93 is
well below the calculated F-value of 15.65, so we can reject the null hypothe-
sis and conclude that the Woody’s equation does indeed have a significant
overall fit.
(N 2 K 2 1),
F 5
ESS>K
RSS>(N 2 K 2 1)
5
9,929,450,000>3
6,130,000,000>29
5 15.65
HA: H0 is not true
H0: �N 5 �P 5 �I 5 0
HYPOTHESIS TESTING
19. To do this calculation, note that If you substitute
the second equation into the first and solve for ESS, you obtain
Since both RSS and R2 are included in the computer output, you can then calculate ESS.
20. Note that this critical F-value must be interpolated. The critical value for 30 denominator
degrees of freedom is 2.92, and the critical value for 25 denominator degrees of freedom is
2.99. Since both numbers are well below the calculated F-value of 15.65, however, the interpo-
lation isn’t necessary to reject the null hypothesis. As a result, many researchers don’t bother
with such interpolations unless the calculated F-value is inside the range of the interpolation.
RSS ? (R2)>(1 2 R2).ESS 5
R2 5 ESS>TSS and that TSS 5 ESS 1 RSS.
168

Just as p-values provide an alternative approach to the t-test, so too can
p-values provide an alternative approach to the F-test of overall significance.
Most standard regression estimation programs report not only the F-value for
the test of overall significance but also the p-value associated with that test.
Other Uses of the F -Test
There are many other uses of the F-test besides the test of overall significance.
For example, let’s look at a Cobb–Douglas production function.
(15)
where: Qt � the natural log of total output in the United States in year t
Lt � the natural log of labor input in the United States in year t
Kt � the natural log of capital input in the United States in year t
� a well-behaved stochastic error term
This is a double-log functional form, and one of the properties of a dou-
ble-log equation is that the coefficients of Equation 15 can be used to test
for constant returns to scale. (Constant returns to scale refers to a situation
in which a given percentage increase in inputs translates to exactly that per-
centage increase in output.) It can be shown that a Cobb–Douglas produc-
tion function with constant returns to scale is one where add up
to exactly 1, so the null hypothesis to be tested is:
To test this null hypothesis with the F-test, we must run regressions on the
unconstrained Equation 15 and an equation that is constrained to conform
to the null hypothesis. To create such a constrained equation, we solve the
null hypothesis for and substitute it into Equation 15, obtaining:
(16)
5 �0 1 �1(Lt 2 Kt) 1 Kt 1 �t
Qt 5 �0 1 �1Lt 1 (1 2 �1)Kt 1 �t
�2
HA: otherwise
H0: �1 1 �2 5 1
�1 and �2
�t
Qt 5 �0 1 �1Lt 1 �2Kt 1 �t
HYPOTHESIS TESTING
169

If we move Kt to the left-hand side of the equation, we obtain our con-
strained equation:
(17)
Equation 17 is the equation that would hold if our null hypothesis were
correct.
To run an F-test on our null hypothesis of constant returns to scale, we
need to run regressions on the constrained Equation 17 and the uncon-
strained Equation 15 and compare the fits of the two equations with the
F-ratio from Equation 14. If we use annual U.S. data, we obtain an uncon-
strained equation of:
(18)
If we run the constrained equation and substitute the appropriate RSS into
Equation 14, with M � 1, we obtain F � 16.26. When this F is compared to a
5-percent critical F-value of only 4.32 (for 1 and 21 degrees of freedom) we
must reject the null hypothesis that constant returns to scale characterize the
U.S. economy. Note that M � 1 and the degrees of freedom in the numerator
equal one because only one coefficient ( ) has been eliminated from the
equation by the constraint.
Interestingly, the estimate of indicates
drastically increasing returns to scale. However, since and since
economic theory suggests that the slope coefficient of a Cobb–Douglas
production function should be between 0 and 1, we should be extremely
cautious. There are problems in the equation that need to be resolved before
we can feel comfortable with this conclusion.
Finally, let’s take a look at the problem of testing the significance of sea-
sonal dummies. Seasonal dummies are dummy variables that are used to
account for seasonal variation in the data in time-series models. In a quarterly
model, if:
X1t 5 e
1 in quarter 1
0 otherwise
�1
ˆ 5 1.28,
�̂1 1 �̂2 5 1.28 1 0.72 5 2.00
�2
N 5 24 (annual U.S. data) R2 5 .997 F 5 4,118.9
t 5 4.24 13.29
(0.30) (0.05)
Q̂t 5 2 38.08 1 1.28Lt 1 0.72Kt
(Qt 2 Kt) 5 �0 1 �1(Lt 2 Kt) 1 �t
HYPOTHESIS TESTING
170

then:
(19)
where X4 is a nondummy independent variable and t is quarterly. Notice that
only three dummy variables are required to represent four seasons. In this
formulation shows the extent to which the expected value of Y in the first
quarter differs from its expected value in the fourth quarter, the omitted con-
dition. can be interpreted similarly.
Inclusion of a set of seasonal dummies “deseasonalizes” Y. This proce-
dure may be used as long as Y and X4 are not “seasonally adjusted” prior to
estimation. Many researchers avoid the type of seasonal adjustment done
prior to estimation because they think it distorts the data in unknown and
arbitrary ways, but seasonal dummies have their own limitations such as re-
maining constant for the entire time period. As a result, there is no unam-
biguously best approach to deseasonalizing data.
To test the hypothesis of significant seasonality in the data, one must test
the hypothesis that all the dummies equal zero simultaneously rather than
test the dummies one at a time. In other words, the appropriate test of sea-
sonality in a regression model using seasonal dummies involves the use of
the F-test instead of the t-test.
In this case, the null hypothesis is that there is no seasonality:
The constrained equation would then be To determine
whether the whole set of seasonal dummies should be included, the fit of
the estimated constrained equation would be compared to the fit of the esti-
mated unconstrained equation by using the F-test in equation 11. Note that
this example uses the F-test to test null hypotheses that include only a sub-
set of the slope coefficients. Also note that in this case M � 3, because three
coefficients ( ) have been eliminated from the equation.�2, and �3�1,
Y 5 �0 1 �4X4 1 �.
HA: H0 is not true
H0: �1 5 �2 5 �3 5 0
�2 and �3
�1
Yt 5 �0 1 �1X1t 1 �2X2t 1 �3X3t 1 �4X4t 1 �t
X3t 5 e
1 in quarter 3
0 otherwise
X2t 5 e
1 in quarter 2
0 otherwise
HYPOTHESIS TESTING
171

The exclusion of some seasonal dummies because their estimated coeffi-
cients have low t-scores is not recommended. Seasonal dummy coefficients
should be tested with the F-test instead of with the t-test because seasonality
is usually a single compound hypothesis rather than 3 individual hypotheses
(or 11 with monthly data) having to do with each quarter (or month). To the
extent that a hypothesis is a joint one, it should be tested with the F-test. If
the hypothesis of seasonal variation can be summarized into a single dummy
variable, then the use of the t-test will cause no problems. Often, where sea-
sonal dummies are unambiguously called for, no hypothesis testing at all is
undertaken.
HYPOTHESIS TESTING
172

HYPOTHESIS TESTING
Critical Values of the t-Distribution
Level of Significance
Degrees of One-Sided: 10% 5% 2.5% 1% 0.5%
Freedom Two-Sided: 20% 10% 5% 2% 1%
1 3.078 6.314 12.706 31.821 63.657
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8 1.397 1.860 2.306 2.896 3.355
9 1.383 1.833 2.262 2.821 3.250
10 1.372 1.812 2.228 2.764 3.169
11 1.363 1.796 2.201 2.718 3.106
12 1.356 1.782 2.179 2.681 3.055
13 1.350 1.771 2.160 2.650 3.012
14 1.345 1.761 2.145 2.624 2.977
15 1.341 1.753 2.131 2.602 2.947
16 1.337 1.746 2.120 2.583 2.921
17 1.333 1.740 2.110 2.567 2.898
18 1.330 1.734 2.101 2.552 2.878
19 1.328 1.729 2.093 2.539 2.861
20 1.325 1.725 2.086 2.528 2.845
21 1.323 1.721 2.080 2.518 2.831
22 1.321 1.717 2.074 2.508 2.819
23 1.319 1.714 2.069 2.500 2.807
24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750
40 1.303 1.684 2.021 2.423 2.704
60 1.296 1.671 2.000 2.390 2.660
120 1.289 1.658 1.980 2.358 2.617
(Normal)

1.282 1.645 1.960 2.326 2.576
Source: Reprinted from Table IV in Sir Ronald A. Fisher, Statistical Methods for Research Workers,
14th ed. (copyright © 1970, University of Adelaide) with permission of Hafner, a division of the
Macmillan Publishing Company, Inc.
173

HYPOTHESIS TESTING
v
2
�
D
e
g
re
e
s
o
f
F
re
e
d
o
m
f
o
r
D
e
n
o
m
in
a
to
r
Critical Values of the F-Statistic: 5-Percent Level of Significance
v1 � Degrees of Freedom for Numerator
1 2 3 4 5 6 7 8 10 12 20 �
1 161 200 216 225 230 234 237 239 242 244 248 254
2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.5
3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.79 8.74 8.66 8.53
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 5.96 5.91 5.80 5.63
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.74 4.68 4.56 4.36
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.06 4.00 3.87 3.67
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.64 3.57 3.44 3.23
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.35 3.28 3.15 2.93
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.14 3.07 2.94 2.71
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 2.98 2.91 2.77 2.54
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.85 2.79 2.65 2.40
12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.75 2.69 2.54 2.30
13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.67 2.60 2.46 2.21
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.60 2.53 2.39 2.13
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.54 2.48 2.33 2.07
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.49 2.42 2.28 2.01
17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.45 2.38 2.23 1.96
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.41 2.34 2.19 1.92
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.38 2.31 2.16 1.88
20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.35 2.28 2.12 1.84
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.32 2.25 2.10 1.81
22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.30 2.23 2.07 1.78
23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.27 2.20 2.05 1.76
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.25 2.18 2.03 1.73
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.24 2.16 2.01 1.71
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.16 2.09 1.93 1.62
40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.08 2.00 1.84 1.51
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 1.99 1.92 1.75 1.39
120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.91 1.83 1.66 1.25

3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.83 1.75 1.57 1.00
Source: Abridged from M. Merrington and C. M. Thompson, “Tables of percentage points of the
inverted beta (F ) distribution,” Biometrika, Vol. 33, 1943, p. 73, by permission of the Biometrika
trustees.
174

Answers
Exercise 2
For all three parts:
X1 X2 X3
H0: �1 � 0 �2 � 0 �3 � 0
HA: �1 � 0 �2 � 0 �3 � 0
t1 � 2.1 t2 � 5.6 t3 � 0.1
a. tc � 1.363. For �1, we reject H0, because |t1| � 1.363 and the sign
of t1 is that implied by HA. For �2, we cannot reject H0, even
though |t2| � 1.363, because the sign of t2 does not agree with HA.
For �3, we cannot reject H0, even though the sign of t3 agrees with
HA, because |t3| � 1.363.
b. tc � 1.318. The decisions are identical to those in part a, except
that tc � 1.318.
c. tc � 3.143. For �1, we cannot reject H0, even though the sign
of t1 is that implied by HA, because |t1| � 3.143. For �2 and
�3, the decisions are identical to those in parts a and b, except
that tc � 3.143.
HYPOTHESIS TESTING
175

176

6Specification: Choosing
the Independent Variables
1 Omitted Variables
2 Irrelevant Variables
3 An Illustration of the Misuse of Specification Criteria
4 Specification Searches
5 An Example of Choosing Independent Variables
6 Summary and Exercises
7 Appendix: Additional Specification Criteria
Before any equation can be estimated, it must be completely specified. Spec-
ifying an econometric equation consists of three parts: choosing the correct
independent variables, the correct functional form, and the correct form of
the stochastic error term.
A specification error results when any one of these choices is made incor-
rectly. This chapter is concerned with only the first of these, choosing the
variables.
That researchers can decide which independent variables to include in
regression equations is a source of both strength and weakness in economet-
rics. The strength is that the equations can be formulated to fit individual
needs, but the weakness is that researchers can estimate many different speci-
fications until they find the one that “proves” their point, even if many other
results disprove it. A major goal of this chapter is to help you understand
how to choose variables for your regressions without falling prey to the vari-
ous errors that result from misusing the ability to choose.
The primary consideration in deciding whether an independent variable
belongs in an equation is whether the variable is essential to the regression
on the basis of theory. If the answer is an unambiguous yes, then the vari-
able definitely should be included in the equation, even if it seems to be
lacking in statistical significance. If theory is ambivalent or less emphatic, a
From Chapter 6 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
177

SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
dilemma arises. Leaving a relevant variable out of an equation is likely to
bias the remaining estimates, but including an irrelevant variable leads
to higher variances of the estimated coefficients. Although we’ll develop
statistical tools to help us deal with this decision, it’s difficult in practice
to be sure that a variable is relevant, and so the problem often remains
unresolved.
We devote the fourth section of the chapter to specification searches and
the pros and cons of various approaches to such searches. For example,
poorly done specification searches often cause bias or make the usual tests of
significance inapplicable. Instead, we suggest trying to minimize the number
of regressions estimated and relying as much as possible on theory rather
than statistical fit when choosing variables. There are no pat answers, how-
ever, and so the final decisions must be left to each individual researcher.
Omitted Variables
Suppose that you forget to include one of the relevant independent variables
when you first specify an equation (after all, no one’s perfect!). Or suppose
that you can’t get data for one of the variables that you do think of. The result
in both these situations is an omitted variable, defined as an important
explanatory variable that has been left out of a regression equation.
Whenever you have an omitted (or left-out) variable, the interpretation and
use of your estimated equation become suspect. Leaving out a relevant vari-
able, like price from a demand equation, not only prevents you from getting
an estimate of the coefficient of price but also usually causes bias in the esti-
mated coefficients of the variables that are in the equation.
The bias caused by leaving a variable out of an equation is called
omitted variable bias (or, more generally, specification bias). In an equa-
tion with more than one independent variable, the coefficient represents
the change in the dependent variable Y caused by a one-unit increase in the
independent variable Xk, holding constant the other independent variables
in the equation. If a variable is omitted, then it is not included as an inde-
pendent variable, and it is not held constant for the calculation and inter-
pretation of This omission can cause bias: It can force the expected
value of the estimated coefficient away from the true value of the popula-
tion coefficient.
Thus, omitting a relevant variable is usually evidence that the entire esti-
mated equation is suspect, because of the likely bias in the coefficients
of the variables that remain in the equation. Let’s look at this issue in
more detail.
�̂k.
�k
1
178

The Consequences of an Omitted Variable
What happens if you omit an important variable from your equation (per-
haps because you can’t get the data for the variable or didn’t even think of the
variable in the first place)? The major consequence of omitting a relevant in-
dependent variable from an equation is to cause bias in the regression coeffi-
cients that remain in the equation. Suppose that the true regression model is:
(1)
where is a classical error term. If you omit X2 from the equation, then the
equation becomes:
(2)
where equals:
(3)
because the stochastic error term includes the effects of any omitted variables.
From Equations 2 and 3, it might seem as though we could get unbiased esti-
mates of even if we left X2 out of the equation. Unfortunately, this
is not the case,1 because the included coefficients almost surely pick up some
of the effect of the omitted variable and therefore will change, causing bias.
To see why, take another look at Equations 2 and 3. Most pairs of variables
are correlated to some degree, even if that correlation is random, so X1 and X2
almost surely are correlated. When X2 is omitted from the equation, the im-
pact of X2 goes into , so and X2 are correlated. Thus if X2 is omitted from
the equation and X1 and X2 are correlated, both X1 and will change when
X2 changes, and the error term will no longer be independent of the explana-
tory variable. That violates Classical Assumption III!
In other words, if we leave an important variable out of an equation, we
violate Classical Assumption III (that the explanatory variables are indepen-
dent of the error term), unless the omitted variable is uncorrelated with all
the included independent variables (which is extremely unlikely). In gen-
eral, when there is a violation of one of the Classical Assumptions, the
Gauss–Markov Theorem does not hold, and the OLS estimates are not
BLUE. Given linear estimators, this means that the estimated coefficients are
�*
�*�*
�0 and �1
�*i 5 �i 1 �2X2i
�*i
Yi 5 �0 1 �1X1i 1 �*i
�i
Yi 5 �0 1 �1X1i 1 �2X2i 1 �i
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
1. To avoid bias, X1 and X2 must be perfectly uncorrelated—an extremely unlikely result.
179

no longer unbiased or are no longer minimum variance (for all linear unbi-
ased estimators), or both. In such a circumstance, econometricians first deter-
mine the exact property (unbiasedness or minimum variance) that no longer
holds and then suggest an alternative estimation technique that might be
better than OLS.
An omitted variable causes Classical Assumption III to be violated in a way
that causes bias. Estimating Equation 2 when Equation 1 is the truth will
cause bias. This means that:
(4)
Instead of having an expected value equal to the true the estimate will
compensate for the fact that X2 is missing from the equation. If X1 and X2 are
correlated and X2 is omitted from the equation, then the OLS estimation pro-
cedure will attribute to X1 variations in Y actually caused by X2, and a biased
estimate of will result.
To see how a left-out variable can cause bias, picture a production function
that states that output (Y) depends on the amount of labor (X1) and capital
(X2) used. What would happen if data on capital were unavailable for some
reason and X2 was omitted from the equation? In this case, we would be leav-
ing out the impact of capital on output in our model. This omission would
almost surely bias the estimate of the coefficient of labor because it is likely
that capital and labor are positively correlated (an increase in capital usually
requires at least some labor to utilize it and vice versa). As a result, the OLS
program would attribute to labor the increase in output actually caused by
capital to the extent that labor and capital were correlated. Thus the bias
would be a function of the impact of capital on output and the correla-
tion between capital and labor.
To generalize for a model with two independent variables, the expected
value of the coefficient of an included variable (X1) when a relevant variable
(X2) is omitted from the equation equals:
(5)
where is the slope coefficient of the secondary regression that relates X2
to X1:
(6)
where ui is a classical error term. can be expressed as a function of the cor-
relation between X1 and X2, the included and excluded variables, or f(r12).
�1
X2i 5 �0 1 �1X1i 1 ui
�1
E(�1) 5 �1 1 �2 ? �1
(�2)
�̂1
�1,
E(�̂1) 2 �1
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
180

Let’s take a look at Equation 5. It states that the expected value of the in-
cluded variable’s coefficient is equal to its true value plus the omitted variable’s
true coefficient times a function of the correlation between the included (in)
and omitted (om) variables.2 Since the expected value of an unbiased estimate
equals the true value, the right-hand term in Equation 5 measures the omitted
variable bias in the equation:
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
(7)Bias 5 �2�1 or Bias 5 �omf(rin,om)
In general terms, the bias thus equals the coefficient of the omitted vari-
able, times , a function of the correlation between the included and
omitted variables.
This bias exists unless:
1. the true coefficient equals zero, or
2. the included and omitted variables are uncorrelated.
The term is the amount of specification bias introduced into
the estimate of the coefficient of the included variable by leaving out the
omitted variable. Although it’s true that there is no bias if the included and
excluded variables are uncorrelated, there almost always is some correlation
between any two variables in the real world (even if it’s just random), and so
bias is almost always caused by the omission of a relevant variable. Although
the omission of a relevant variable almost always produces bias in the esti-
mators of the coefficients of the included variables, the variances of these
estimators are generally lower than they otherwise would be.
An Example of Specification Bias
As an example of specification bias, let’s take a look at a simple model of the
annual consumption of chicken in the United States. There are a variety of
variables that might make sense in such an equation, and at least three vari-
ables seem obvious. We’d expect the demand for chicken to be a negative
�omf(rin,om)
f(rin,om)
�om,
2. Equations 5 and 7 hold when there are exactly two independent variables, but the more gen-
eral equations are quite similar.
181

SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
function of the price of chicken and a positive function of the price of beef
(its main substitute) and income:
� � �
Yt � f(PCt PBt YDt) �
where: Yt � per capita chicken consumption (in pounds) in year t
PCt � the price of chicken (in cents per pound) in year t
PBt � the price of beef (in cents per pound) in year t
YDt � U.S. per capita disposable income (in hundreds of dollars)
in year t
If we collect data for these variables for the years 1974 through 2002, we
can estimate the following equation. (The data for this example are included
in Exercise 5; t-scores differ because of rounding.)
(8)
(0.03) (0.02) (0.01)
t � �3.38 �1.86 �15.7
How does our estimated equation look? The overall fit of Equation 8 is
excellent, and each of the individual regression coefficients is significantly
different from zero in the expected direction. The price of chicken does in-
deed have a significant negative effect (holding the price of beef and dispos-
able income constant), and the price of beef and disposable income do in-
deed have positive effects (holding the other independent variables constant).
If we estimate this equation without the price of the substitute, we obtain:
� 30.7 � 0.09PCt � 0.25YDt (9)
(0.03) (0.005)
t � �2.76 �46.1
� .9895 N � 29 (annual 1974–2002)
Let’s compare Equations 8 and 9 to see if dropping the beef price variable
had an impact on the estimated equations. If you compare the overall fit, for
example, you can see that fell from .9904 to .9895 when PB was dropped,
exactly what we’d expect to occur when a relevant variable is omitted.
More important, from the point of view of showing that an omitted variable
causes bias, let’s see if the coefficient estimates of the remaining variables
changed. Sure enough, dropping PB caused to go from to and
caused to go from 0.23 to 0.25. The direction of this bias, by the way, is con-
sidered positive because the biased coefficient of PC is more positive
(less negative) than the suspected unbiased one and the biased coeffi-
cient of YD (0.25) is more positive than the suspected unbiased one of (0.23).
(20.09)
(20.11)
�̂YD
20.0920.11�̂PC
R2
R2
Yt
R2 5 .9904 N 5 29 (annual 1974–2002)
Yt 5 27.7 2 0.11PCt 1 0.03PBt 1 0.23YDt
�t
182

The fact that the bias is positive could have been guessed before any regres-
sions were run if Equation 7 had been used. The specification bias caused by
omitting the price of beef is expected3 to be positive because the expected
sign of the coefficient of PB is positive and because the expected correlation
between the price of beef and the price of chicken itself is positive:
Similarly for YD:
Note that both correlation coefficients are anticipated to be (and actually
are) positive. To see this, think of the impact of an increase in the price of
chicken on the price of beef and then follow through the impact of any in-
crease in income on the price of beef.
To sum, if a relevant variable is left out of a regression equation,
1. there is no longer an estimate of the coefficient of that variable in the
equation, and
2. the coefficients of the remaining variables are likely to be biased.
Although the amount of the bias might not be very large in some cases
(when, for instance, there is little correlation between the included and ex-
cluded variables), it is extremely likely that at least a small amount of specifi-
cation bias will be present in all such situations.
Correcting for an Omitted Variable
In theory, the solution to a problem of specification bias seems easy: add the
omitted variable to the equation! Unfortunately, that’s easier said than
done, for a couple of reasons.
First, omitted variable bias is hard to detect. As mentioned earlier, the
amount of bias introduced can be small and not immediately detectable.
Expected bias in �̂YD 5 �PB ? f(rYD,PB) 5 (1) ? (1) 5 (1)
Expected bias in �̂PC 5 �PB ? f(rPC,PB) 5 (1) ? (1) 5 (1)
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
3. It is important to note the distinction between expected bias and any actual observed differences
between coefficient estimates. Because of the random nature of the error term (and hence the
the change in an estimated coefficient brought about by dropping a relevant variable from the
equation will not necessarily be in the expected direction. Biasedness refers to the central tendency
of the sampling distribution of the not to every single drawing from that distribution. However,
we usually (and justifiably) rely on these general tendencies. Note also that Equation 8 has three
independent variables, whereas Equation 7 was derived for use with equations with exactly two.
However, Equation 7 represents a general tendency that is still applicable.
�̂s,
�̂s),
183

This is especially true when there is no reason to believe that you have mis-
specified the model. Some indications of specification bias are obvious (such
as an estimated coefficient that is significant in the direction opposite from
that expected), but others are not so clear. Could you tell from Equation 9
alone that a variable was missing? The best indicators of an omitted relevant
variable are the theoretical underpinnings of the model itself. What variables
must be included? What signs do you expect? Do you have any notions about
the range into which the coefficient values should fall? Have you accidentally
left out a variable that most researchers would agree is important? The best
way to avoid omitting an important variable is to invest the time to think
carefully through the equation before the data are entered into the computer.
A second source of complexity is the problem of choosing which variable
to add to an equation once you decide that it is suffering from omitted vari-
able bias. That is, a researcher faced with a clear case of specification bias
(like an estimated that is significantly different from zero in the unex-
pected direction) will often have no clue as to what variable could be causing
the problem. Some beginning researchers, when faced with this dilemma,
will add all the possible relevant variables to the equation at once, but this
process leads to less precise estimates, as will be discussed in the next sec-
tion. Other beginning researchers will test a number of different variables
and keep the one in the equation that does the best statistical job of appear-
ing to reduce the bias (by giving plausible signs and satisfactory t-values).
This technique, adding a “left-out” variable to “fix” a strange-looking regres-
sion result, is invalid because the variable that best corrects a case of specifi-
cation bias might do so only by chance rather than by being the true solution
to the problem. In such an instance, the “fixed” equation may give superb
statistical results for the sample at hand but then do terribly when applied
to other samples because it does not describe the characteristics of the true
population.
Dropping a variable will not help cure omitted variable bias. If the sign of
an estimated coefficient is different from expected, it cannot be changed to the
expected direction by dropping a variable that has a t-score lower (in absolute
value) than the t-score of the coefficient estimate that has the unexpected sign.
Furthermore, the sign in general will not likely change even if the variable to
be deleted has a large t-score.4
If an unexpected result leads you to believe that you have an omitted
variable, one way to decide which variable to add to the equation is to use
�̂
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
4. Ignazio Visco, “On Obtaining the Right Sign of a Coefficient Estimate by Omitting a Variable
from the Regression,” Journal of Econometrics, Vol. 7, No. 1, pp. 115–117.
184

expected bias analysis. Expected bias is the likely bias that omitting a par-
ticular variable would have caused in the estimated coefficient of one of the
included variables. It can be estimated with Equation 7:
(7)
If the sign of the expected bias is the same as the sign of your unexpected
result, then the variable might be the source of the apparent bias. If the sign
of the expected bias is not the same as the sign of your unexpected result,
however, then the variable is extremely unlikely to have caused your unex-
pected result. Expected bias analysis should be used only when you’re choos-
ing between theoretically sound potential variables.
As an example of expected bias analysis, let’s return to Equation 9, the
chicken demand equation without the beef price variable. Let’s assume
that you had expected the coefficient of and
that you were surprised by the unexpectedly positive coefficient of PC in
Equation 9.
This unexpectedly positive result could have been caused by an omitted
variable with positive expected bias. One such variable is the price of beef.
The expected bias in due to leaving out PB is positive, since both the
expected coefficient of PB and the expected correlation between PC and PB
are positive:
Hence the price of beef is a reasonable candidate to be an omitted variable in
Equation 9.
Although you can never actually observe bias (since you don’t know the
true the use of this technique to screen potential causes of specification
bias should reduce the number of regressions run and therefore increase the
statistical validity of the results.
A brief warning: It may be tempting to conduct what might be called
“residual analysis” by examining a plot of the residuals in an attempt to find
patterns that suggest variables that have been accidentally omitted. A major
problem with this approach is that the coefficients of the estimated equation
will possibly have some of the effects of the left-out variable already altering
their estimated values. Thus, residuals may show a pattern that only vaguely
resembles the pattern of the actual omitted variable. The chances are high
that the pattern shown in the residuals may lead to the selection of an incor-
rect variable. In addition, care should be taken to use residual analysis only
to choose between theoretically sound candidate variables rather than to
generate those candidates.
�),
Expected bias in �̂PC 5 �PB ? f(rPC,PB) 5 (1) ? (1) 5 (1)
�̂PC
�PC to be in the range of 21.0
Expected bias 5 �om ? f(rin,om)
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
185

Irrelevant Variables
What happens if you include a variable in an equation that doesn’t belong
there? This case, irrelevant variables, is the converse of omitted variables and
can be analyzed using the model we developed in Section 1. The addition of a
variable to an equation where it doesn’t belong does not cause bias, but it does
increase the variances of the estimated coefficients of the included variables.
Impact of Irrelevant Variables
If the true regression specification is:
(10)
but the researcher for some reason includes an extra variable,
(11)
the misspecified equation’s error term can be seen to be:
(12)
Such a mistake will not cause bias if the true coefficient of the extra (or irrele-
vant) variable is zero. That is, in Equation 11 is unbiased when
However, the inclusion of an irrelevant variable will increase the variance
of the estimated coefficients, and this increased variance will tend to decrease
the absolute magnitude of their t-scores. Also, an irrelevant variable usually
will decrease the (but not the R2).
Thus, although the irrelevant variable causes no bias, it causes problems
for the regression because it reduces the t-scores and .
Table 1 summarizes the consequences of the omitted variable and the in-
cluded irrelevant variable cases (unless r12 � 0).
R2
R2
�2 5 0.�̂1
�**i 5 �i 2 �2X2i
Yi 5 �0 1 �1X1i 1 �2X2i 1 �**i
Yi 5 �0 1 �1X1i 1 �i
2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
Table 1 Effect of Omitted Variables and Irrelevant Variables on the
Coefficient Estimates
Effect on Coefficient Estimates Omitted Variable Irrelevant Variable
Bias Yes No
Variance Decreases Increases
186

An Example of an Irrelevant Variable
Let’s return to the equation from Section 1 for the annual consumption of
chicken and see what happens when we add an irrelevant variable to the
equation. The original equation was:
(8)
(0.03) (0.02) (0.01)
t � �3.38 �1.86 �15.7
Suppose you hypothesize that the demand for chicken also depends on TEMP,
the average annual change in temperature in tenths of a degree (included, per-
haps, on the dubious theory that demand for chicken might heat up when
temperatures are rising). If you now estimate the equation with TEMP included,
you obtain:
� 26.9 � 0.11PCt � 0.03PBt � 0.23YDt � 0.02TEMPt (13)
(0.03) (0.02) (0.015) (0.02)
t � �3.38 �1.99 �14.99 �0.93
� .9903 N � 29 (annual 1974–2002)
A comparison of Equations 8 and 13 will make the theory in Section 2 come
to life. First of all, has fallen slightly, indicating the reduction in fit adjusted
for degrees of freedom. Second, none of the regression coefficients from the
original equation changed; compare these results with the larger differences
between Equations 8 and 9. Further, the standard errors of the estimated co-
efficients increased or remained constant. Finally, the t-score for the potential
variable (TEMP) is small, indicating that it is not significantly different from
zero. Given the theoretical shakiness of the new variable, these results indi-
cate that it is irrelevant and never should have been included in the
regression.
Four Important Specification Criteria
We have now discussed at least four valid criteria to help decide whether a
given variable belongs in the equation. We think these criteria are so impor-
tant that we urge beginning researchers to work through them every time a
variable is added or subtracted.
R2
R2
Yt
R2 5 .9904 N 5 29 (annual 1974–2002)
Yt 5 27.7 2 0.11PCt 1 0.03PBt 1 0.23YDt
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
187

If all these conditions hold, the variable belongs in the equation; if none
of them do, the variable is irrelevant and can be safely excluded from the
equation. When a typical omitted relevant variable is included in the equa-
tion, its inclusion probably will increase and change at least one other
coefficient. If an irrelevant variable, on the other hand, is included, it will
reduce have an insignificant t-score, and have little impact on the other
variables’ coefficients.
In many cases, all four criteria do not agree. It is possible for a variable to
have an insignificant t-score that is greater than one, for example. In such a
case, it can be shown that will go up when the variable is added to the
equation and yet the t-score still will be insignificant.
Whenever our four specification criteria disagree, the econometrician
must use careful judgment and should not rely on a single criterion like
to determine the specification. Researchers should not misuse this freedom
by testing various combinations of variables until they find the results that
appear to statistically support the point they want to make. All such deci-
sions are a bit easier when you realize that the single most important deter-
minant of a variable’s relevance is its theoretical justification. No amount of
statistical evidence should make a theoretical necessity into an “irrelevant”
variable. Once in a while, a researcher is forced to leave a theoretically impor-
tant variable out of an equation for lack of data; in such cases, the usefulness
of the equation is limited.
An Illustration of the Misuse of Specification Criteria
At times, the four specification criteria outlined in the previous section will
lead the researcher to an incorrect conclusion if those criteria are applied to a
problem without proper concern for economic principles or common sense.
3
R2
R2
R2,
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
1. Theory: Is the variable’s place in the equation unambiguous and
theoretically sound?
2. t-Test: Is the variable’s estimated coefficient significant in the expected
direction?
3. : Does the overall fit of the equation (adjusted for degrees of free-
dom) improve when the variable is added to the equation?
4. Bias: Do other variables’ coefficients change significantly when the
variable is added to the equation?
R2
188

In particular, a t-score can often be insignificant for reasons other than the
presence of an irrelevant variable. Since economic theory is the most impor-
tant test for including a variable, an example of why a variable should not
be dropped from an equation simply because it has an insignificant t-score
is in order.
Suppose you believe that the demand for Brazilian coffee in the United
States is a negative function of the real price of Brazilian coffee (Pbc) and a
positive function of both the real price of tea (Pt) and real disposable income
in the United States (Yd).
5 Suppose further that you obtain the data, run the
implied regression, and observe the following results:
(14)
The coefficients of the second and third variables, Pt and Yd, appear to be
fairly significant in the direction you hypothesized, but the first variable, Pbc,
appears to have an insignificant coefficient with an unexpected sign. If you
think there is a possibility that the demand for Brazilian coffee is perfectly
price-inelastic (that is, its coefficient is zero), you might decide to run the
same equation without the price variable, obtaining:
(15)
By comparing Equations 14 and 15, we can apply our four specification crite-
ria for the inclusion of a variable in an equation that were outlined in the
previous section:
1. Theory: Since the demand for coffee could possibly be perfectly price-
inelastic, the theory behind dropping the variable seems plausible.
2. t-Test: The t-score of the possibly irrelevant variable is 0.5, insignificant
at any level.
R2 5 .61 N 5 25
t 5 2.6 4.0
(1.0) (0.0009)
COFFEE 5 9.3 1 2.6Pt 1 0.0036Yd
R2 5 .60 N 5 25
t 5 0.5 2.0 3.5
(15.6) (1.2) (0.0010)
COFFEE 5 9.1 1 7.8Pbc 1 2.4Pt 1 0.0035Yd
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
5. This example was inspired by a similar one concerning Ceylonese tea published in Potluri
Rao and Roger LeRoy Miller, Applied Econometrics (Belmont, CA: Wadsworth, 1971), pp. 38–40.
This wonderful book is now out of print.
189

3. : increases when the variable is dropped, indicating that the variable
is irrelevant. (Since the t-score is less than 1, this is to be expected.)
4. Bias: The remaining coefficients change only a small amount when Pbc
is dropped, suggesting that there is little—if any—bias caused by exclud-
ing the variable.
Based upon this analysis, you might conclude that the demand for Brazilian
coffee is indeed perfectly price-inelastic and that the variable is therefore irrele-
vant and should be dropped from the model. As it turns out, this conclusion
would be unwarranted. Although the elasticity of demand for coffee in general
might be fairly low (actually, the evidence suggests that it is inelastic only over
a particular range of prices), it is hard to believe that Brazilian coffee is
immune to price competition from other kinds of coffee. Indeed, one would
expect quite a bit of sensitivity in the demand for Brazilian coffee with respect
to the price of, for example, Colombian coffee. To test this hypothesis, the price
of Colombian coffee (Pcc) should be added to the original Equation 14:
(16)
By comparing Equations 14 and 16, we can once again apply our four speci-
fication criteria:
1. Theory: Both prices should always have been included in the model;
their logical justification is quite strong.
2. t-Test: The t-score of the new variable, the price of Colombian coffee,
is 2.0, significant at most levels.
3. : increases with the addition of the variable, indicating that the
variable was an omitted variable.
4. Bias: Although two of the coefficients remain virtually unchanged, indi-
cating that the correlations between these variables and the price of
Colombian coffee variable are low, the coefficient for the price of Brazil-
ian coffee does change significantly, indicating bias in the original result.
The moral to be drawn is that theoretical considerations never should be
discarded, even in the face of statistical insignificance. If a variable known to
be extremely important from a theoretical point of view turns out to be sta-
tistically insignificant in a particular sample, that variable should be left in
the equation despite the fact that it makes the results look bad.
R2R2
R2 5 .65 N 5 25
t 5 2.0 2 2.8 2.0 3.0
(4.0) (2.0) (1.3) (0.0010)
COFFEE 5 10.0 1 8.0Pcc 2 5.6Pbc 1 2.6Pt 1 0.0030Yd
R2R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
190

Don’t conclude that the particular path outlined in this example is the cor-
rect way to specify an equation. Trying a long string of possible variables until
you get the particular one that makes the coefficient of Pbc turn negative and
significant is not the way to obtain a result that will stand up well to other
samples or alternative hypotheses. The original equation should never have
been run without the Colombian coffee price variable. Instead, the problem
should have been analyzed enough so that such errors of omission were un-
likely before any regressions were attempted at all. The more thinking that’s
done before the first regression is run, and the fewer alternative specifications
that are estimated, the better the regression results are likely to be.
Specification Searches
One of the weaknesses of econometrics is that a researcher potentially can ma-
nipulate a data set to produce almost any result by specifying different regres-
sions until estimates with the desired properties are obtained. Because the
integrity of all empirical work is thus open to question, the subject of how to
search for the best specification is quite controversial among econometricians.6
Our goal in this section isn’t to summarize or settle this controversy; instead, I
hope to provide some guidance and insight for beginning researchers.
Best Practices in Specification Searches
The issue of how best to choose a specification from among alternative possi-
bilities is a difficult one, but our experience leads us to make the following
recommendations:
4
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
6. For an excellent summary of this controversy and the entire subject of specification, see Peter
Kennedy, A Guide to Econometrics (Malden, MA: Blackwell), pp. 71–92.
1. Rely on theory rather than statistical fit as much as possible when
choosing variables, functional forms, and the like.
2. Minimize the number of equations estimated (except for sensitivity
analysis, to be discussed later in this section).
3. Reveal, in a footnote or appendix, all alternative specifications
estimated.
191

If theory, not or t-scores, is the most important criterion for the inclu-
sion of a variable in a regression equation, then it follows that most of the
work of specifying a model should be done before you attempt to estimate the
equation. Since it’s unreasonable to expect researchers to be perfect, there will
be times when additional specifications must be estimated. However, these
new estimates should be few in number and should be thoroughly grounded
in theory. In addition, they should be explicitly taken into account when test-
ing for significance and/or summarizing results. In this way, the danger of
misleading the reader about the statistical properties of the final equation will
be reduced.
Sequential Specification Searches
Most econometricians tend to specify equations by estimating an initial
equation and then sequentially dropping or adding variables (or changing
functional forms) until a plausible equation is found with “good statistics.”
Faced with knowing that a few variables are relevant (on the basis of theory)
but not knowing whether other additional variables are relevant, inspecting
and t-tests for all variables for each specification appears to be the gener-
ally accepted practice. Indeed, casual reading of the previous section might
make it seem as if such a sequential specification search is the best way to go
about finding the “truth.” Instead, as we shall see, there is a vast difference
between a sequential specification search and our recommended approach.
The sequential specification search technique allows a researcher to esti-
mate an undisclosed number of regressions and then present a final choice
(which is based upon an unspecified set of expectations about the signs and
significance of the coefficients) as if it were the only specification estimated.
Such a method misstates the statistical validity of the regression results for
two reasons:
1. The statistical significance of the results is overestimated because the
estimations of the previous regressions are ignored.
2. The expectations used by the researcher to choose between various
regression results rarely, if ever, are disclosed. Thus the reader has no
way of knowing whether all the other regression results had opposite
signs or insignificant coefficients for the important variables.
Unfortunately, there is no universally accepted way of conducting sequen-
tial searches, primarily because the appropriate test at one stage in the proce-
dure depends on which tests previously were conducted, and also because
the tests have been very difficult to invent.
R2
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
192

Instead we recommend trying to keep the number of regressions estimated
as low as possible; to focus on theoretical considerations when choosing
variables or functional forms; and to document all the various specifications
investigated. That is, we recommend combining parsimony (using theory
and analysis to limit the number of specifications estimated) with disclosure
(reporting all the equations estimated).
Not everyone agrees with our advice. Some researchers feel that the true
model will show through if given the chance and that the best statistical
results (including signs of coefficients, etc.) are most likely to have come
from the true specification. In addition, reasonable people often disagree as
to what the “true” model should look like. As a result, different researchers
can look at the same data set and come up with very different “best” equa-
tions. Because this can happen, the distinction between good and bad econo-
metrics is not always as clear-cut as is implied by the previous paragraphs. As
long as researchers have a healthy respect for the dangers inherent in specifi-
cation searches, they are very likely to proceed in a reasonable way.
Bias Caused by Relying on the t-Test to Choose Variables
In the previous section, we stated that sequential specification searches are
likely to mislead researchers about the statistical properties of their results. In
particular, the practice of dropping a potential independent variable simply
because its coefficient has a low t-score will cause systematic bias in the esti-
mated coefficients (and their t-scores) of the remaining variables.
Let’s say the hypothesized model is:
(17)
Assume further that, on the basis of theory, we are certain that X1 belongs in
the equation but that we are not as certain that X2 belongs. Many inexperi-
enced researchers use only the t-test on to determine whether X2 should
be included. If this preliminary t-test indicates that is significantly differ-
ent from zero, then these researchers leave X2 in the equation. If, however,
the t-test does not indicate that is significantly different from zero, then
such researchers drop X2 from the equation and consider Y to be a function
of X1.
Two kinds of mistakes can be made using such a system. First, X2 some-
times can be left in the equation when it does not belong there, but such a
mistake does not change the expected value of
Second, X2 sometimes can be dropped from the equation when it belongs.
In this second case, the estimated coefficient of X1 will be biased. In other
�̂1.
�̂2
�̂2
�̂2
Yi 5 �0 1 �1X1i 1 �2X2i 1 �i
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
193

words, will be biased every time X2 belongs in the equation and is left out,
and X2 will be left out every time that its estimated coefficient is not signifi-
cantly different from zero. We will have systematic bias in our equation!
To summarize, the t-test is biased by sequential specification searches.
Since most researchers consider a number of different variables before set-
tling on the final model, someone who relies on the t-test alone is likely to
encounter this problem systematically.
Sensitivity Analysis
We’ve encouraged you to estimate as few specifications as possible and to
avoid depending on fit alone to choose between those specifications. If you
read the current economics literature, however, it won’t take you long to find
well-known researchers who have estimated five or more specifications and
then have listed all their results in an academic journal article. What’s
going on?
In almost every case, these authors have employed a technique called sen-
sitivity analysis.
Sensitivity analysis consists of purposely running a number of alterna-
tive specifications to determine whether particular results are robust (not
statistical flukes). In essence, we’re trying to determine how sensitive a
potential “best” equation is to a change in specification because the true
specification isn’t known. Researchers who use sensitivity analysis run (and
report on) a number of different reasonable specifications and tend to dis-
count a result that appears significant in some specifications and insignif-
icant in others. Indeed, the whole purpose of sensitivity analysis is to gain
confidence that a particular result is significant in a variety of alternative
specifications, functional forms, variable definitions, and/or subsets of
the data.
Data Mining
In contrast to sensitivity analysis, which consists of estimating a variety of
alternative specifications after a potential “best” equation has been identi-
fied, data mining involves estimating a variety of alternative specifications
before that “best” equation has been chosen. Readers of this text will not be
surprised to hear that we urge extreme caution when data mining. Improp-
erly done data mining is worse than doing nothing at all.
Done properly, data mining involves exploring a data set not for the pur-
pose of testing hypotheses or finding a specification, but for the purpose of
�̂1
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
194

uncovering empirical regularities that can inform economic theory.7 After all,
we can’t expect economic theorists to think of everything!
Be careful, however! If you develop a hypothesis using data mining tech-
niques, you must test that hypothesis on a different data set (or in a different
context) than the one you used to develop the hypothesis. A new data set must
be used because our typical statistical tests have little meaning if the new
hypothesis is tested on the data set that was used to generate it. After all, the
researcher already knows ahead of time what the results will be! The use of dual
data sets is easiest when there is a plethora of data. This sometimes is the case
in cross-sectional research projects but rarely is the case for time series research.
Data mining without using dual data sets is almost surely the worst way
to choose a specification. In such a situation, a researcher could estimate vir-
tually every possible combination of the various alternative independent
variables, could choose the results that “look” the best, and then could report
the “best” equation as if no data mining had been done. This improper use
of data mining ignores the fact that a number of specifications have been
examined before the final one is reported.
In addition, data mining will cause you to choose specifications that reflect
the peculiarities of your particular data set. How does this happen? Suppose
you have 100 true null hypotheses and you run 100 tests of these hypotheses.
At the 5-percent level of significance, you’d expect to reject about five true null
hypotheses and thus make about five Type I Errors. By looking for high
t-values, a data mining search procedure will find these Type I Errors and incor-
porate them into your specification. As a result, the reported t-scores will over-
state the statistical significance of the estimated coefficients.
In essence, improper data mining to obtain desired statistics for the final
regression equation is a potentially unethical empirical research method.
Whether the improper data mining is accomplished by estimating one equa-
tion at a time or by estimating batches of equations or by techniques like step-
wise regression procedures,8 the conclusion is the same. Hypotheses developed
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
7. For an excellent presentation of this approach, see Lawrence H. Summers, “The Scientific
Illusion in Empirical Macroeconomics,” Scandinavian Journal of Economics, Vol. 93, No. 2,
pp. 129–148.
8. A stepwise regression involves the use of an automated computer program to choose the
independent variables in an equation. The researcher specifies a “shopping list” of possible inde-
pendent variables, and then the computer estimates a number of equations until it finds the
one that maximizes . Such stepwise techniques are deficient in the face of multicollinearity
and they run the risk that the chosen specification will have little theoretical justification
and/or will have coefficients with unexpected signs. Because of these pitfalls, econometricians
avoid stepwise procedures.
R2
195

by data mining should always be tested on a data set different from the
one that was used to develop the hypothesis. Otherwise, the researcher
hasn’t found any scientific evidence to support the hypothesis; rather, a
specification has been chosen in a way that is essentially misleading. As
put by one econometrician, “if you torture the data long enough, they will
confess.”9
An Example of Choosing Independent Variables
It’s time to get some experience choosing independent variables. After all,
every equation so far in the text has come with the specification already deter-
mined, but once you’ve finished this course you’ll have to make all such spec-
ification decisions on your own. We’ll use a technique called “interactive re-
gression learning exercises” to allow you to make your own actual
specification choices and get feedback on your choices. To start, though, let’s
work through a specification together.
To keep things as simple as possible, we’ll begin with a topic near and dear
to your heart—your GPA! Suppose a friend who attends a small liberal arts
college surveys all 25 members of her econometrics class, obtains data on the
variables listed here, and asks for your help in choosing a specification:
GPAi � the cumulative college grade point average on the ith student on
a four-point scale
HGPAi � the cumulative high school grade point average of the ith student
on a four-point scale
MSATi � the highest score earned by the ith student on the math section
of the SAT test (800 maximum)
VSATi � the highest score earned by the ith student on the verbal section
of the SAT test (800 maximum)
SATi � MSATi � VSATi
GREKi � a dummy variable equal to 1 if the ith student is a member of a
fraternity or sorority, 0 otherwise
HRSi � the ith student’s estimate of the average number of hours spent
studying per course per week in college
5
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
9. Thomas Mayer, “Economics as a Hard Science: Realistic Goal or Wishful Thinking?” Economic
Inquiry, Vol. 18, No. 2, p. 175. (This quote also has been attributed to Ronald Coase.)
196

PRIVi � a dummy variable equal to 1 if the ith student graduated from a
private high school, 0 otherwise
JOCKi � a dummy variable equal to 1 if the ith student is or was a member
of a varsity intercollegiate athletic team for at least one season,
0 otherwise
lnEXi � the natural log of the number of full courses that the ith student
has completed in college.
Assuming that GPAi is the dependent variable, which independent vari-
ables would you choose? Before you answer, think through the possibilities
carefully. What does the literature tell us on this subject? (Is there literature?)
What are the expected signs of each of the coefficients? How strong is the the-
ory behind each variable? Which variables seem obviously important? Which
variables seem potentially irrelevant or redundant? Are there any other vari-
ables that you wish your friend had collected?
To get the most out of this example, you should take the time to write down
the exact specification that you would run:
GPAi
It’s hard for most beginning econometricians to avoid the temptation of
including all of these variables in a GPA equation and then dropping any
variables that have insignificant t-scores. Even though we mentioned in the
previous section that such a specification search procedure will result in
biased coefficient estimates, most beginners don’t trust their own judgment
and tend to include too many variables. With this warning in mind, do you
want to make any changes in your proposed specification?
No? OK, let’s compare notes. We believe that grades are a function of a stu-
dent’s ability, how hard the student works, and the student’s experience tak-
ing college courses. Consequently, our specification would be:
We can already hear you complaining! What about SATs, you say? Everyone
knows they’re important. How about jocks and Greeks? Don’t they have
lower GPAs? Don’t prep schools grade harder and prepare students better
than public high schools?
Before we answer, it’s important to note that we think of specification
choice as choosing which variables to include, not which variables to exclude.
That is, we don’t assume automatically that a given variable should be
GPAi 5 f(HG
1
PAi, HR
1
Si, ln
1
EXi) 1 �
5 f(?, ?, ?, ?, ?) 1 �
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
197

included in an equation simply because we can’t think of a good reason for
dropping it.
Given that, however, why did we choose the variables we did? First, we
think that the best predictor of a student’s college GPA is his or her high
school GPA. We have a hunch that once you know HGPA, SATs are redun-
dant, at least at a liberal arts college where there are few multiple choice tests.
In addition, we’re concerned that possible racial and gender bias in the SAT
test makes it a questionable measure of academic potential, but we recognize
that we could be wrong on this issue.
As for the other variables, we’re more confident. For example, we feel that
once we know how many hours a week a student spends studying, we couldn’t
care less what that student does with the rest of his or her time, so JOCK and
GREK are superfluous once HRS is included. In addition, the higher LnEX is,
the better student study habits are and the more likely students are to be
taking courses in their major. Finally, while we recognize that some private
schools are superb and that some public schools are not, we’d guess that
PRIV is irrelevant; it probably has only a minor effect.
If we estimate this specification on the 25 students, we obtain:
(18)
Since we prefer this specification on theoretical grounds, since the overall fit
seems reasonable, and since each coefficient meets our expectations in terms
of sign, size, and significance, we consider this an acceptable equation. The
only circumstance under which we’d consider estimating a second specifica-
tion would be if we had theoretical reasons to believe that we had omitted a
relevant variable. The only variable that might meet this description is SATi
(which we prefer to the individual MSAT and VSAT):
(19)
N 5 25 R2 5 .583
t 5 3.12 0.93
(0.14) (0.00064)
1 0.44lnEXi 1 0.00060SATi
t 5 2.12 2.50
(0.22) (0.02)
GPAi 5 2 0.92 1 0.47HGPAi 1 0.05HRSi
N 5 25 R2 5 .585
t 5 2.33 3.00 3.00
(0.21) (0.02) (0.14)
GPAi 5 2 0.26 1 0.49HGPAi 1 0.06HRSi 1 0.42lnEXi
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
198

Let’s use our four specification criteria to compare Equations 18 and 19:
1. Theory: As discussed previously, the theoretical validity of SAT tests is a
matter of some academic controversy, but they still are one of the most-
cited measures of academic potential in this country.
2. t-Test: The coefficient of SAT is positive, as we’d expect, but it’s not sig-
nificantly different from zero.
3. : As you’d expect (since SAT ’s t-score is under 1), falls slightly when
SAT is added.
4. Bias: None of the estimated slope coefficients changes significantly when
SAT is added, though some of the t-scores do change because of the in-
crease in the caused by the addition of SAT.
Thus, the statistical criteria support our theoretical contention that SAT is
irrelevant.
Finally, it’s important to recognize that different researchers could come
up with different final equations on this topic. A researcher whose prior
expectation was that SAT unambiguously belonged in the equation would
have estimated Equation 19 and accepted that equation without bothering to
estimate Equation 18. Other researchers, in the spirit of sensitivity analysis,
would report both equations.
Summary
1. The omission of a variable from an equation will cause bias in the esti-
mates of the remaining coefficients to the extent that the omitted vari-
able is correlated with included variables.
2. The bias to be expected from leaving a variable out of an equation
equals the coefficient of the excluded variable times a function of the
simple correlation coefficient between the excluded variable and the
included variable in question.
3. Including a variable in an equation in which it is actually irrelevant
does not cause bias, but it will usually increase the variances of the in-
cluded variables’ estimated coefficients, thus lowering their t-values
and lowering R2.
6
SE(�̂)s
R2R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
199

4. Four useful criteria for the inclusion of a variable in an equation are:
a. theory
b. t-test
c.
d. bias
5. Theory, not statistical fit, should be the most important criterion for
the inclusion of a variable in a regression equation. To do otherwise
runs the risk of producing incorrect and/or disbelieved results.
EXERCISES
(The answer to Exercise 2 appears at the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or your notes), and compare your definition with the ver-
sion in the text for each:
a. omitted variable
b. irrelevant variable
c. specification bias
d. sequential specification search
e. specification error
f. the four specification criteria
g. expected bias
h. sensitivity analysis
2. You’ve been hired by “Indo,” the new Indonesian automobile manufac-
turer, to build a model of U.S. car prices in order to help the company
undercut U.S. prices. Allowing Friedmaniac zeal to overwhelm any patri-
otic urges, you build the following model of the price of 35 different
American-made 2004 U.S. sedans (standard errors in parentheses):
where: Pi � the list price of the ith car (thousands of dollars)
Wi � the weight of the ith car (hundreds of pounds)
Ti � a dummy equal to 1 if the ith car has an automatic
transmission, 0 otherwise
R2 5 .92
(0.07) (0.4) (2.9) (0.20)
Model A: P̂i 5 3.0 1 0.28Wi 1 1.2Ti 1 5.8Ci 1 0.19Li
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
200

Ci � a dummy equal to 1 if the ith car has cruise control,
0 otherwise
Li � the size of the engine of the ith car (in liters)
a. Your firm’s pricing expert hypothesizes positive signs for all the slope
coefficients in Model A. Test her expectations at the 5-percent level.
b. What econometric problems appear to exist in Model A? In particu-
lar, does the size of the coefficient of C cause any concern? Why?
What could be the problem?
c. You decide to test the possibility that L is an irrelevant variable by
dropping it and rerunning the equation, obtaining the following
Model T equation. Which model do you prefer? Why? (Hint: Be
sure to use our four specification criteria.)
3. Consider the following annual model of the death rate (per million
population) due to coronary heart disease in the United States (Yt):
where: Ct � per capita cigarette consumption (pounds of tobacco)
in year t
Et � per capita consumption of edible saturated fats
(pounds of butter, margarine, and lard) in year t
Mt � per capita consumption of meat (pounds) in year t
a. Create and test appropriate hypotheses at the 10-percent level. What,
if anything, seems to be wrong with the estimated coefficient of M?
b. The most likely cause of a coefficient that is significant in the unex-
pected direction is omitted variable bias. Which of the following
variables could possibly be an omitted variable that is causing
unexpected sign? Explain. (Hint: Be sure to analyze expected bias in
your explanation.)
Bt � per capita consumption of hard liquor (gallons) in year t
Ft � the average fat content (percentage) of the meat that was
consumed in year t
�̂M’s
N 5 31 (197522005) R2 5 .678
t 5 4.0 4.0 2 2.0
(2.5) (1.0) (0.5)
Ŷt 5 140 1 10.0Ct 1 4.0Et 2 1.0Mt
R2 5 .93
(0.07) (0.30) (2.9)
Model T: P̂ 5 18 1 0.29Wi 1 1.2Ti 1 5.9Ci
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
201

Wt � per capita consumption of wine and beer (gallons) in year t
Rt � per capita number of miles run in year t
Ht � per capita open-heart surgeries in year t
Ot � per capita amount of oat bran eaten in year t
c. If you had to choose a variable not listed in part b to add to the
equation, what would it be? Explain your answer.
4. Assume that you’ve been hired by the surgeon general of the United
States to study the determinants of smoking behavior and that you
estimate the following cross-sectional model based on data for all
50 states (standard errors in parentheses):10
(20)
where: Ci � the number of cigarettes consumed per day per person
in the ith state
Ei � the average years of education for persons over 21 in
the ith state
Ii � the average income in the ith state (thousands of dollars)
Ti � the tax per package of cigarettes in the ith state (cents)
Vi � the number of video ads against smoking aired on the
three major networks in the ith state.
Ri � the number of radio ads against smoking aired on the
five largest radio networks in the ith state
a. Develop and test (at the 5-percent level) appropriate hypotheses
for the coefficients of the variables in this equation.
b. Do you appear to have any irrelevant variables? Do you appear to
have any omitted variables? Explain your answer.
c. Let’s assume that your answer to part b was yes to both. Which
problem is more important to solve first—irrelevant variables or
omitted variables? Why?
d. One of the purposes of running the equation was to determine the
effectiveness of antismoking advertising on television and radio.
What is your conclusion?
R2 5 .40 N 5 50 (states)
t 5 2 3.0 1.0 2 1.0 2 3.0 3.0
(3.0) (1.0) (0.04) (1.0) (0.5)
Ĉi 5 100 2 9.0Ei 1 1.0Ii 2 0.04Ti 2 3.0Vi 1 1.5Ri
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
10. This question is generalized from a number of similar studies, including John A. Bishop and
Jang H. Yoo, “Health Scare, Excise Taxes, and Advertising Ban in the Cigarette Demand and Sup-
ply,” Southern Economic Journal, Vol. 52, No. 1, pp. 402–411.
202

e. The surgeon general decides that tax rates are irrelevant to cigarette
smoking and orders you to drop the variable from your equation.
Given the following results, use our four specification criteria to de-
cide whether you agree with her conclusion. Carefully explain your
reasoning (standard errors in parentheses).
(21)
f. In answering part e, you surely noticed that the figures were
identical. Did this surprise you? Why or why not?
5. The data set in Table 2 is the one that was used to estimate the
chicken demand examples of Sections 1 and 2.
a. Use these data to reproduce the specifications in the chapter
(datafile � CHICK6).
b. Find data in Table 2 for the price of pork (another substitute for
chicken) and add that variable to Equation 8. Analyze your results.
In particular, apply the four criteria for the inclusion of a variable to
determine whether the price of pork is irrelevant or previously was
an omitted variable.
6. You have been retained by the “Expressive Expresso” company to help
them decide where to build their next “Expressive Expresso” store.
You decide to run a regression on the sales of the 30 existing “Expres-
sive Expresso” stores as a function of the characteristics of the loca-
tions they are in and then use the equation to predict the sales at the
various locations you are considering for the newest store. You end up
estimating (standard errors in parentheses):
where: Yi � average daily sales (in hundreds of dollars) of the
ith store
X1i � the number of cars that pass the ith location per hour
X2i � average income in the area of the ith store
X3i � the number of tables in the ith store
X4i � the number of competing shops in the area of the
ith store
(0.02) (0.01) (1.0) (1.0)
Ŷi 5 30 1 0.1X1i 1 0.01X2i 1 10.0X3i 1 3.0X4i
R2
R2 5 .40 N 5 50 (states)
(3.0) (0.9) (1.0) (0.5)
Ĉi 5 101 2 9.1Ei 1 1.0Ii 2 3.5Vi 1 1.6Ri
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
203

a. Hypothesize expected signs, calculate the correct t-scores, and test
the significance at the 1-percent level for each of the coefficients.
b. What problems appear to exist in the equation? What evidence of
these problems do you have?
c. What suggestions would you make for a possible second run of this
admittedly hypothetical equation? (Hint: Before recommending the
inclusion of a potentially omitted variable, consider whether the ex-
clusion of the variable could possibly have caused any observed bias.)
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
Table 2 Data for the Chicken Demand Equation
Year Y PC PB YD TEMP PRP
1974 39.70 42.30 143.80 50.10 �16 107.80
1975 38.69 49.40 152.20 54.98 �4 134.60
1976 42.02 45.50 145.70 59.72 �24 134.00
1977 42.71 45.30 145.90 65.17 16 125.40
1978 44.75 49.30 178.80 72.24 5 143.60
1979 48.35 50.00 222.40 79.67 13 152.50
1980 48.47 53.50 233.60 88.22 21 147.50
1981 50.37 53.80 234.70 97.65 49 161.20
1982 51.52 51.50 238.40 104.26 4 185.60
1983 52.55 56.00 234.10 111.31 35 179.70
1984 54.61 61.50 235.50 123.19 11 171.40
1985 56.42 56.20 228.60 130.37 4 170.80
1986 57.70 63.10 226.80 136.49 18 188.80
1987 61.94 53.10 238.40 142.41 35 199.40
1988 63.80 62.10 250.30 152.97 46 194.00
1989 66.88 64.20 265.70 162.57 32 193.50
1990 70.34 60.50 281.00 171.31 64 224.90
1991 73.26 57.70 288.30 176.09 52 224.20
1992 76.39 59.00 284.60 184.94 18 209.50
1993 78.27 27.10 293.40 188.72 27 209.10
1994 79.65 26.20 282.90 195.55 48 209.50
1995 79.27 26.90 284.30 202.87 71 206.10
1996 80.61 28.00 280.20 210.91 36 233.70
1997 83.10 33.20 279.50 219.40 60 245.00
1998 83.76 33.40 277.10 231.61 89 242.70
1999 88.98 39.50 287.80 239.68 60 241.40
2000 90.08 43.00 306.40 254.69 62 258.20
2001 89.71 43.40 337.70 262.24 74 269.40
2002 94.37 43.90 331.50 271.45 85 265.80
Sources: U.S. Department of Agriculture. Agricultural Statistics; U.S. Bureau of the Census.
Historical Statistics of the United States, U.S. Bureau of the Census. Statistical Abstract of the
United States. (Datafile � CHICK6)
204

SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
7. Discuss the topic of specification searches with various members of
your econometrics class. What is so wrong with not mentioning previ-
ous (probably incorrect) estimates? Why should readers be suspicious
when researchers attempt to find results that support their hypothe-
ses? Who would try to do the opposite? Do these concerns have any
meaning in the world of business? In particular, if you’re not trying to
publish a paper, couldn’t you use any specification search techniques
you want to find the best equation?
8. For each of the following situations, determine the sign (and, if possi-
ble, comment on the likely size) of the expected bias introduced by
omitting a variable:
a. In an equation for the demand for peanut butter, the impact on the
coefficient of disposable income of omitting the price of peanut
butter variable. (Hint: Start by hypothesizing signs.)
b. In an earnings equation for workers, the impact on the coefficient
of experience of omitting the variable for age.
c. In a production function for airplanes, the impact on the coeffi-
cient of labor of omitting the capital variable.
d. In an equation for daily attendance at outdoor concerts, the impact
on the coefficient of the weekend dummy variable (1 � weekend)
of omitting a variable that measures the probability of precipita-
tion at concert time.
9. Most of the examples so far have been demand-side equations or pro-
duction functions, but economists often also have to quantify supply-
side equations that are not true production functions. These equations
attempt to explain the production of a product (for example, Brazilian
coffee) as a function of the price of the product and various other at-
tributes of the market that might have an impact on the total output of
growers.
a. What sign would you expect the coefficient of price to have in a
supply-side equation? Why?
b. What other variables can you think of that might be important in a
supply-side equation?
c. Many agricultural decisions are made months (if not a full year or
more) before the results of those decisions appear in the market.
How would you adjust your hypothesized equation to take account
of these lags?
d. Using the information given so far, carefully specify the exact equa-
tion you would use to attempt to explain Brazilian coffee produc-
tion. Be sure to hypothesize the expected signs, be specific with
respect to lags, and try to make sure that you have not omitted an
important independent variable.
205

10. If you think about the previous question, you’ll realize that the same
dependent variable (quantity of Brazilian coffee) can have different
expected signs for the coefficient of the same independent variable
(the price of Brazilian coffee), depending on what other variables are
in the regression.
a. How is this possible? That is, how is it possible to expect different
signs in demand-side equations from what you would expect in
supply-side ones?
b. What can be done to avoid getting the price coefficient from the
demand equation in the supply equation and vice versa?
c. What can you do to systematically ensure that you do not have
supply-side variables in your demand equation or demand-side
variables in your supply equation?
11. Let’s use the model of financial aid awards at a liberal arts
college. We estimate the following equation (standard errors in
parentheses):
FINAIDi � 8927 � 0.36 PARENTi � 87.4 HSRANKi (22)
(0.03) (20.7)
t � �11.26 4.22
� 0.73 N � 50
where: FINAIDi � the financial aid (measured in dollars of
grant) awarded to the ith applicant
PARENTi � the amount (in dollars) that the parents of
the ith student are judged able to contribute
to college expenses
HSRANKi � the ith student’s GPA rank in high school,
measured as a percentage (ranging from a
low of 0 to a high of 100)
a. Create and test hypotheses for the coefficients of the independent
variables.
b. What econometric problems do you see in the equation? Are there
any signs of an omitted variable? Of an irrelevant variable? Explain
your answer.
c. Suppose that you now hear a charge that financial aid awards at the
school are unfairly tilted toward males, so you decide to attempt to
test this charge by adding a dummy variable for gender (MALEi � 1
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
206

if the ith student is a male, 0 if female) to your equation, getting
the following results:
FINAIDi � 9813 � 0.34 PARENTi � 83.3 HSRANKi � 1570 MALEi (23)
(0.03) (20.1) (784)
t � �10.88 4.13 �2.00
� 0.75 N � 50
d. Carefully explain the real-world meaning of the estimated coeffi-
cient of MALE.
e. Which equation is better, Equation 22 or Equation 23? Carefully
use our four specification criteria to make your decision, being sure
to state which criteria support which equation and why.
12. Determine the sign (and, if possible, comment on the likely size) of
the bias introduced by leaving a variable out of an equation in each
of the following cases:
a. In an annual equation for corn yields per acre (in year t), the impact
on the coefficient of rainfall in year t of omitting average temperature
that year. (Hint: Drought and cold weather both hurt corn yields.)
b. In an equation for daily attendance at Los Angeles Lakers’ home bas-
ketball games, the impact on the coefficient of the winning percentage
of the opponent (as of the game in question) of omitting a dummy
variable that equals 1 if the opponent’s team includes a superstar.
c. In an equation for annual consumption of apples in the United
States, the impact on the coefficient of the price of bananas of
omitting the price of oranges.
d. In an equation for student grades on the first midterm in this class,
the impact on the coefficient of total hours studied (for the test) of
omitting hours slept the night before the test.
13. Suppose that you run a regression to determine whether gender or race
has any significant impact on scores on a test of the economic under-
standing of children.11 You model the score of the ith student on the
test of elementary economics (Si) as a function of the composite score
on the Iowa Tests of Basic Skills of the ith student, a dummy variable
equal to 1 if the ith student is female (0 otherwise), the average num-
ber of years of education of the parents of the ith student, and a
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
11. These results have been jiggled to meet the needs of this question, but this research actually
was done. See Stephen Buckles and Vera Freeman, “Male-Female Differences in the Stock and
Flow of Economic Knowledge,” Review of Economics and Statistics, Vol. 65, No. 2, pp. 355–357.
207

dummy variable equal to 1 if the ith student is nonwhite (0 other-
wise). Unfortunately, a rainstorm floods the computer center and makes
it impossible to read the part of the computer output that identifies
which variable is which. All you know is that the regression results are
(standard errors in parentheses):
a. Attempt to identify which result corresponds to which variable. Be
specific.
b. Explain the reasoning behind your answer to part a.
c. Assuming that your answer is correct, create and test appropriate hy-
potheses (at the 5-percent level) and come to conclusions about the
effects of gender and race on the test scores of this particular sample.
d. Did you use a one-tailed or two-tailed test in part c? Why?
14. Let’s use the model of the auction price of iPods on eBay. In
this model, we use datafile IPOD3 to estimate the following
equation:
PRICEi � 109.24 � 54.99NEWi � 20.44SCRATCHi � 0.73BIDRSi (24)
(5.34) (5.11) (0.59)
t � 10.28 �4.00 1.23
N � 215
where: PRICEi � the price at which the ith iPod sold on eBay
NEWi � a dummy variable equal to 1 if the ith iPod
was new, 0 otherwise
SCRATCHi � a dummy variable equal to 1 if the ith iPod
had a minor cosmetic defect, 0 otherwise
BIDRSi � the number of bidders on the ith iPod
The dataset also includes a variable (PERCENTi ) that measures the per-
centage of customers of the seller of the ith iPod who gave that seller a
positive rating for quality and reliability in previous transactions.12 In
theory, the higher the rating of a seller, the more a potential bidder
N 5 24 R2 5 .54
(0.63) (0.88) (0.08) (0.10)
Ŝi 5 5.7 2 0.63X1i 2 0.22X2i 1 0.16X3i 1 1.20X4i
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
12. For more on this dataset and this variable, see Leonardo Rezende, “Econometrics of Auctions
by Least Squares,” Journal of Applied Econometrics, November/December 2008, pp. 925–948.
208

would trust that seller, and the more that potential bidder would be
willing to bid. If you add PERCENT to the equation, you obtain
PRICEi � 82.67 � 55.42NEWi � 20.95SCRATCHi � 0.63BIDRSi � 0.28PERCENTi
(5.34) (5.12) (0.59) (0.20)
t � 10.38 �4.10 1.07 1.40 (25)
N � 215
a. Use our four specification criteria to decide whether you think
PERCENT belongs in the equation. Be specific. (Hint: isn’t
given, but you’re capable of determining which equation had the
higher .)
b. Do you think that PERCENT is an accurate measure of the quality and
reliability of the seller? Why or why not? (Hint: Among other things,
consider the case of a seller with very few previous transactions.)
c. (optional) With datafile IPOD3, use EViews, Stata, or your own
regression program to estimate the equation with and without
PERCENT. What are the figures for the two specifications? Were
you correct in your determination (in part a) as to which equation
had the higher ?
15. Look back at Exercise 14 in Chapter 5, the equation on international
price discrimination in pharmaceuticals. In that cross-sectional study,
Schut and VanBergeijk estimated two equations in addition to the one
cited in the exercise.13 These two equations tested the possibility that
CVi, total volume of consumption of pharmaceuticals in the ith coun-
try, and Ni, the population of the ith country, belonged in the original
equation, Equation 5.10, repeated here:
N 5 32 R2 5 .775
t 5 22.25 21.59
(6.93) (7.16)
215.63DPCi 2 11.38IPCi
t 5 6.69 22.66 1.19
(0.21) (0.22) (6.12)
P̂i 5 38.22 1 1.43GDPNi 2 0.6CVNi 1 7.31PPi
R2
R2
R2
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
13. Frederick T. Schut and Peter A. G. VanBergeijk, “International Price Discrimination: The
Pharmaceutical Industry,” World Development, Vol. 14, No. 9, pp. 1141–1150.
209

where: Pi � the pharmaceutical price level in the ith country
divided by that of the United States
GDPNi � per capita domestic product in the ith country
divided by that of the United States
CVNi � per capita volume of consumption of pharma-
ceuticals in the ith country divided by that of
the United States
PPi � a dummy variable equal to 1 if patents for phar-
maceutical products are recognized in the ith
country, 0 otherwise
DPCi � a dummy variable equal to 1 if the ith country
applied strict price controls, 0 otherwise
IPCi � a dummy variable equal to 1 if the ith country
encouraged price competition, 0 otherwise
a. Using EViews, Stata (or your own computer program), and datafile
DRUG5, estimate:
i. Equation 10 from Chapter 5 with CVi added, and
ii. Equation 10 from Chapter 5 with Ni added
b. Use our four specification criteria to determine whether CV and N
are irrelevant or omitted variables. (Hint: The authors expected that
prices would be lower if market size were larger because of possible
economies of scale and/or enhanced competition.)
c. Why didn’t the authors run Equation 10 from Chapter 5 with both
CV and N included? (Hint: While you can estimate this equation
yourself, you don’t have to do so to answer the question.)
d. Why do you think that the authors reported all three estimated
specifications in their results when they thought that Equation 10
from Chapter 5 was the best?
16. You’ve just been promoted to be the product manager for “Amish Oats
Instant Oatmeal,” and your first assignment is to decide whether to
raise prices for next year. (Instant oatmeal is a product that can be
mixed with hot water to create a hot breakfast cereal in much less time
than it takes to make the same cereal using regular oatmeal.) In keeping
with your reputation as the econometric expert at Amish Oats, you de-
cide to build a model of the impact of price on sales, and you estimate
the following hypothetical equation (standard errors in parentheses):
(20) (6) (10) (0.0005)
R2 5 .78 N 5 29 (annual model)
t 5 1.00 3.00 3.00 3.00
OATt 5 30 1 20PRt 1 18PRCOMPt 1 30ADSt 1 0.0015YDt
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
210

where: OATt � U.S. sales of Amish Oats instant oatmeal in
year t
PRt � the U.S. price of Amish Oats instant oatmeal
in year t
PRCOMPt � the U.S. price of the competing instant oat-
meal in year t
ADSt � U.S. advertising for Amish Oats instant oat-
meal in year t
YDt � U.S. disposable income in year t
a. Create and test appropriate hypotheses about the slope coefficients
of this equation at the 5-percent level.
b. What econometric problems, if any, appear to be in this equation?
Do you see any indications that there is an omitted variable? Do
you see any indications that there is an irrelevant variable? Explain.
c. If you could add one variable to this equation, what would it be?
Explain your answer.
d. Suddenly it hits you! You’ve made a horrible mistake! What is it?
(Hint: Think about substitutes for OAT.)
Appendix: Additional Specification Criteria
So far in this chapter, we’ve suggested four criteria for choosing the indepen-
dent variables (economic theory, the t-test, and possible bias in the coeffi-
cients). Sometimes, however, these criteria don’t provide enough information
for a researcher to feel confident that a given specification is best. For instance,
there can be two different specifications that both have excellent theoretical
underpinnings. In such a situation, many econometricians use additional,
often more formal, specification criteria to provide comparisons of the prop-
erties of the alternative estimated equations.
The use of formal specification criteria is not without problems, however.
First, no test, no matter how sophisticated, can “prove” that a particular spec-
ification is the true one. The use of specification criteria, therefore, must be
tempered with a healthy dose of economic theory and common sense. A sec-
ond problem is that more than 20 such criteria have been proposed; how do
we decide which one(s) to use? Because many of these criteria overlap with
one another or have varying levels of complexity, a choice between the alter-
natives is a matter of personal preference.
In this section, we’ll describe the use of three of the most popular specifi-
cation criteria, J. B. Ramsey’s RESET test, Akaike’s Information Criterion, and
the Schwarz Criterion. Our inclusion of just these techniques does not imply
R2,
7
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
211

that other tests and criteria are not appropriate or useful. Indeed, the reader
will find that most other formal specification criteria have quite a bit in com-
mon with at least one of the techniques that we include. We think that you’ll
be better able to use and understand other formal specification criteria14
once you’ve mastered these three.
Ramsey’s Regression Specification Error Test (RESET)
One of the most-used formal specification criteria other than is the Ram-
sey Regression Specification Error Test (RESET).15 The Ramsey RESET test is
a general test that determines the likelihood of an omitted variable or some
other specification error by measuring whether the fit of a given equation can
be significantly improved by the addition of terms.
What’s the intuition behind RESET? The additional terms act as proxies for
any possible (unknown) omitted variables or incorrect functional forms. If
the proxies can be shown by the F-test to have improved the overall fit of the
original equation, then we have evidence that there is some sort of specifica-
tion error in our equation. The terms form a polynomial func-
tional form. Such a polynomial is a powerful curve-fitting device that has a
good chance of acting as a proxy for a specification error if one exists. If there
is no specification error, then we’d expect the coefficients of the added terms
to be insignificantly different from zero because there is nothing for them to
act as a proxy for.
The Ramsey RESET test involves three steps:
1. Estimate the equation to be tested using OLS:
(26)
2. Take the values from Equation 26 and create terms.
Then add these terms to Equation 26 as additional explanatory vari-
ables and estimate the new equation with OLS:
(27)Yi 5 �0 1 �1X1i 1 �2X2i 1 �3Ŷ
2
i 1 �4Ŷ
3
i 1 �5Ŷ
4
i 1 �i
Ŷ2i , Ŷ
3
i , and Ŷ
4
iŶi
Ŷi 5 �̂0 1 �̂1X1i 1 �̂2X2i
Ŷ2, Ŷ3, and Ŷ4
Ŷ2, Ŷ3, and Ŷ4
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
14. In particular, the likelihood ratio test can be used as a specification test. For an introductory
level summary of six other specification criteria, see Ramu Ramanathan, Introductory Economet-
rics (Fort Worth: Harcourt Brace Jovanovich, 1998, pp. 164–166).
15. J. B. Ramsey, “Tests for Specification Errors in Classical Linear Squares Regression Analysis,”
Journal of the Royal Statistical Society, Vol. 31, No. 2, pp. 350–371.
212

3. Compare the fits of Equations 26 and 27 using the F-test. If the two
equations are significantly different in overall fit, we can conclude that
it’s likely that Equation 26 is misspecified.
While the Ramsey RESET test is fairly easy to use, it does little more than
signal when a major specification error might exist. If you encounter a sig-
nificant Ramsey RESET test, then you face the daunting task of figuring out
exactly what the error is! Thus, the test often ends up being more useful in
“supporting” (technically, not refuting) a researcher’s contention that a given
specification has no major specification errors than it is in helping find an
otherwise undiscovered flaw.16
As an example of the Ramsey RESET test, let’s return to the chicken
demand model of this chapter to see if RESET can detect the known specifica-
tion error (omitting the price of beef) in Equation 9. Step one involves run-
ning the original equation without PB.
� 30.7 � 0.09PCt � 0.25YDt (9)
(0.03) (0.005)
t � �2.76 �46.1
� .9895 N � 29 (annual 1974–2002) RSS � 83.22
For step two, we take from Equation 9, calculate and then
reestimate Equation 9 with the three new terms added in:
(28)
(0.59) (1.77) (0.17)
(0.002) (0.000006)
In step three, we compare the fits of the two equations by using the F-test.
Specifically, we test the hypothesis that the coefficients of all three of the
added terms are equal to zero:
HA: otherwise
H0: �3 5 �4 5 �5 5 0
R2 5 .991 N 5 29 (annual 1974–2002) RSS 5 57.43
20.001Ŷ t
3 1 0.000002Ŷ t
4 1 et
Y t 5 241.4 1 0.40PCt 2 1.09YDt 1 0.11Ŷt
2
Ŷt
2, Ŷt
3, and Ŷt
4,Ŷt
R2
Yt
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
16. The particular version of the Ramsey RESET test we describe in this section is only one of a
number of possible formulations of the test. For example, some researchers delete the term from
Equation 27. In addition, versions of the Ramsey RESET test are useful in testing for functional
form errors and serial correlation.
Ŷ4
213

The appropriate F-statistic to use is one that is presented in Section 5.6.
(29)
where RSSM is the residual sum of squares from the restricted equation (Equa-
tion 9), RSS is the residual sum of squares from the unrestricted equation17
(Equation 28), M is the number of restrictions (3), and is the
number of degrees of freedom in the unrestricted equation (34):
The critical F-value to use, 3.03, is found in Statistical Table B-2 at the 5-percent
level of significance with 3 numerator and 23 denominator degrees of free-
dom. Since 3.44 is greater than 3.03, we can reject the null hypothesis that the
coefficients of the added variables are jointly zero, allowing us to conclude that
there is indeed a specification error in Equation 9. Such a conclusion is no sur-
prise, since we know that the price of beef was left out of the equation. Note,
however, that the Ramsey RESET test tells us only that a specification error is
likely to exist in Equation 9; it does not specify the details of that error.
Akaike’s Information Criterion and the Schwarz Criterion
A second category of formal specification criteria involves adjusting the
summed squared residuals (RSS) by one factor or another to create an index
of the fit of an equation. The most popular criterion of this type is but a
number of interesting alternatives have been proposed.
Akaike’s Information Criterion (AIC) and the Schwarz Criterion (SC)
are methods of comparing alternative specifications by adjusting RSS for the
sample size (N) and the number of independent variables (K).18 These crite-
ria can be used to augment our four basic specification criteria when we try
R2,
F 5
(83.22 2 57.43)>3
57.43>23
5 3.44
(N 2 K 2 1)
F 5
(RSS
M
2 RSS)>M
RSS>(N 2 K 2 1)
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
17. Because of the obvious correlation between the three values, Equation 28 (with most
RESET equations) suffers from extreme multicollinearity. Since the purpose of the RESET equa-
tion is to see whether the overall fit can be improved by adding in proxies for an omitted vari-
able (or other specification error), this extreme multicollinearity is not a concern.
18. Hirotogu Akaike, “Likelihood of a Model and Information Criteria,” Journal of Econometrics,
Vol. 16, No. 1, pp. 3–14 and G. Schwarz, “Estimating the Dimension of a Model,” The Annals of
Statistics, Vol. 6, pp. 461–464. The definitions of AIC and SC we use in Equations 30 and 31
produce slightly different numbers than the versions used by EViews, but the versions map on a
one-to-one basis and therefore produce identical conclusions.
Ŷ
214

to decide if the improved fit caused by an additional variable is worth the
decreased degrees of freedom and increased complexity caused by the addi-
tion. Their equations are:
(30)
(31)
To use AIC and SC, estimate two alternative specifications and calculate
AIC and SC for each equation. The lower AIC or SC are, the better the spec-
ification. Note that even though the two criteria were developed indepen-
dently to maximize different object functions, their equations are quite
similar. Both criteria tend to penalize the addition of another explanatory
variable more than does. As a result, AIC and SC will quite often19 be min-
imized by an equation with fewer independent variables than the ones that
maximize
Let’s apply Akaike’s Information Criterion and the Schwarz Criterion to the
same chicken demand example we used for Ramsey’s RESET. To see whether
AIC and/or SC can detect the specification error we already know exists in
Equation 9 (the omission of the price of beef ), we need to calculate AIC and SC
for equations with and without the price of beef. The equation with the lower
AIC and SC values will, other things being equal, be our preferred specification.
The original chicken demand model, Equation 8, was:
(8)
(0.03) (0.02) (0.01)
t � �3.38 � 1.86 �15.7
RSS � 73.11
Plugging the numbers from Equation 8 into Equations 30 and 31, AIC and
SC can be seen to be:
SC 5 Log(73.11>29) 1 Log(29)*4>29 5 1.39
AIC 5 Log(73.11>29) 1 2(4)>29 5 1.20
R2 5 .9904 N 5 29 (annual 1974–2002)
Yt 5 27.7 2 0.11PCt 1 0.03PBt 1 0.23YDt
R2.
R2
SC 5 Log(RSS>N) 1 Log(N) (K 1 1)>N
AIC 5 Log(RSS>N) 1 2(K 1 1)>N
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
19. Using a Monte Carlo study, Judge et al. showed that (given specific simplifying assump-
tions) a specification chosen by maximizing is more than 50 percent more likely to include an
irrelevant variable than is one chosen by minimizing AIC or SC. See George C. Judge, R. Carter Hill,
W. E. Griffiths, Helmut Lutkepohl, and Tsoung-Chao Lee, Introduction to the Theory and Practice
of Econometrics (New York: Wiley, 1988), pp. 849–850. At the same time, minimizing AIC or SC
will omit a relevant variable more frequently than will maximizing R2.
R2
215

Equation 9 which omits the price of beef, has an RSS of 83.22 and two inde-
pendent variables, so:
For AIC, so both Akaike’s Information
Criterion and the Schwarz Criterion provide evidence that Equation 8 is
preferable to Equation 9. That is, the price of beef appears to belong in the
equation. In practice, these calculations may not be necessary because AIC
and SC are automatically calculated by some regression software packages,
including EViews.
As it turns out, then, all three new specification criteria indicate the pres-
ence of a specification error when we leave the price of beef out of the equa-
tion. This result is not surprising, since we purposely omitted a theoretically
justified variable, but it provides an example of how useful these specifica-
tion criteria could be when we’re less than sure about the underlying theory.
Note that AIC and SC require the researcher to come up with a particular
alternative specification, whereas Ramsey’s RESET does not. Such a distinc-
tion makes RESET easier to use, but it makes AIC and SC more informative if
a specification error is found. Thus our additional specification criteria serve
different purposes. RESET is useful as a general test of the existence of a spec-
ification error, whereas AIC and SC are useful as means of comparing two or
more alternative specifications.
1.20 , 1.26, and for SC, 1.39 , 1.40,
SC 5 Log(83.22>29) 1 Log(29)*3>29 5 1.40
AIC 5 Log(83.22>29) 1 2(3)>29 5 1.26
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
216

Answers
Exercise 2
a. Wi Ti Ci Li
H0 �1 � 0 �2 � 0 �3 � 0 �4 � 0
HA �1 � 0 �2 � 0 �3 � 0 �4 � 0
tW � 4 tT � 3 tC � 2 tL � 0.95
tc � 1.697 tc � 1.697 tc � 1.697 tc � 1.697
For the first three coefficients, we can reject the null hypothesis,
because the absolute value of tk is greater than tc and the sign of
tk is that specified in HA. For L, however, we cannot reject the null
hypothesis, even though the sign is as expected, because the ab-
solute value of tL is less than 1.697.
b. Almost any equation potentially could have an omitted variable,
and this one is no exception. In addition, Li might be an irrele-
vant variable. Finally, the coefficient of C seems far too large, sug-
gesting at least one omitted variable. C appears to be acting as a
proxy for other luxury options or the general quality of the car.
c. Theory: Bigger engines cost more, so the variable’s place in the
equation seems theoretically sound. However, sedans with large
engines tend to weigh more, so perhaps the two variables are
measuring more or less the same thing.
t-Test: The variable’s estimated coefficient is insignificant in the
expected direction.
: The overall fit of the equation (adjusted for degrees of free-
dom) improves when the variable is dropped from the equation.
Bias: When the variable is dropped from the equation, the esti-
mated coefficients remain virtually unchanged.
The last three criteria are evidence in favor of dropping Li and the
theoretical argument for keeping it isn’t overwhelming, so we prefer
Model T. However, a researcher who firmly believed in the theo-
retical importance of engine size would pick Model A.
R2
SPECIFICATION: CHOOSING THE INDEPENDENT VARIABLES
217

218

Even after you’ve chosen your independent variables, the job of specifying
the equation is not over. The next step is to choose the functional form of the
relationship between each independent variable and the dependent variable.
Should the equation go through the origin? Do you expect a curve instead of
a straight line? Does the effect of a variable peak at some point and then start
to decline? An affirmative answer to any of these questions suggests that an
equation other than the standard linear model might be appropriate. Such
alternative specifications are important for two reasons: a correct explanatory
variable may well appear to be insignificant or to have an unexpected sign if
an inappropriate functional form is used, and the consequences of an incor-
rect functional form for interpretation and forecasting can be severe.
Theoretical considerations usually dictate the form of a regression model.
The basic technique involved in deciding on a functional form is to choose the
shape that best exemplifies the expected underlying economic or business
principles and then to use the mathematical form that produces that shape.
To help with that choice, this chapter contains plots of the most commonly
used functional forms along with the mathematical equations that corre-
spond to each.
1 The Use and Interpretation of the Constant Term
2 Alternative Functional Forms
3 Lagged Independent Variables
4 Using Dummy Variables
5 Slope Dummy Variables
6 Problems with Incorrect Functional Forms
7 Summary and Exercises
Specification: Choosing
a Functional Form
From Chapter 7 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
219

The chapter begins with a brief discussion of the constant term. In particu-
lar, we suggest that the constant term should be retained in equations even if
theory suggests otherwise and that estimates of the constant term should not
be relied on for inference or analysis. The chapter concludes with a discussion
of dummy variables.
The Use and Interpretation of the Constant Term
In the linear regression model, is the intercept or constant term. It is the
expected value of Y when all the explanatory variables (and the error term)
equal zero. An estimate of has at least three components:
1. the true ,
2. the constant impact of any specification errors (an omitted variable,
for example), and
3. the mean of for the correctly specified equation (if not equal to zero).
Unfortunately, these components can’t be distinguished from one another
because we can observe only , the sum of the three components. The result
is that we have to analyze differently from the way we analyze the other
coefficients in the equation.1
At times, is of theoretical importance. Consider, for example, the fol-
lowing cost equation:
where Ci is the total cost of producing output Qi. The term represents
the total variable cost associated with output level Qi, and represents the
total fixed cost, defined as the cost when output Thus, a regression
equation might seem useful to a researcher who wanted to determine the
relative magnitudes of fixed and variable costs. This would be an example of
relying on the constant term for inference.
Qi 5 0.
�0
�1Qi
Ci 5 �0 1 �1Qi 1 �i
�0
�0
�0
�
�0
�0
�0
1
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
1. If the second and third components of �0 are small compared to the first component, then
this difference diminishes. See R. C. Allen and J. H. Stone, “Textbook Neglect of the Constant
Coefficient,” The Journal of Economic Education, Fall 2005, pp. 379–384.
220

On the other hand, the product involved might be one for which it is
known that there are few—if any—fixed costs. In such a case, a researcher
might want to eliminate the constant term; to do so would conform to the no-
tion of zero fixed costs and would conserve a degree of freedom (which would
presumably make the estimate of more precise). This would be an example
of suppressing the constant term.
Neither suppressing the constant term nor relying on it for inference is ad-
visable, however, and reasons for these conclusions are explained in the fol-
lowing sections.
Do Not Suppress the Constant Term
Suppressing the constant term leads to a violation of the Classical Assump-
tions. This is because Classical Assumption II (that the error term has an ex-
pected value of zero) can be met only if the constant term absorbs any nonzero
mean that the observations of the error might have in a given sample.2
If you omit the constant term, then the impact of the constant is forced
into the estimates of the other coefficients, causing potential bias. This is
demonstrated in Figure 1. Given the pattern of the X and Y observations, esti-
mating a regression equation with a constant term would likely produce an
estimated regression line very similar to the true regression line, which has a
constant term quite different from zero. The slope of this estimated line
is very low, and the t-score of the estimated slope coefficient may be very
close to zero.
However, if the researcher were to suppress the constant term, which im-
plies that the estimated regression line must pass through the origin, then the
estimated regression line shown in Figure 1 would result. The slope coeffi-
cient is biased upward compared with the true slope coefficient. The t-score is
biased upward as well, and it may very well be large enough to indicate that
the estimated slope coefficient is statistically significantly positive. Such a
conclusion would be incorrect.
Thus, even though some regression packages allow the constant term to be
suppressed (set to zero), the general rule is: Don’t, even if theory specifically
calls for it.
(�0)
�1
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
2. The only time that Classical Assumption II isn’t violated by omitting the constant term is
when the mean of the unobserved error term equals zero (exactly) over all the observations.
This result is extremely unlikely.
221

Do Not Rely on Estimates of the Constant Term
It would seem logical that if it’s a bad idea to suppress the constant term,
then the constant term must be an important analytical tool to use in evalu-
ating the results of the regression. Unfortunately, there are at least two rea-
sons that suggest that the intercept should not be relied on for purposes of
analysis or inference.
First, the error term is generated, in part, by the omission of a number of
marginal independent variables, the mean effect of which is placed in the
constant term. The constant term acts as a garbage collector, with an un-
known amount of this mean effect being dumped into it. The constant term’s
estimated coefficient may be different from what it would have been without
performing this task, which is done for the sake of the equation as a whole.
As a result, it’s meaningless to run a t-test on �̂0.
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
Y
0 X
Estimated Relationship
Suppressing the Intercept
True Relation
Observations �0
Figure 1 The Harmful Effect of Suppressing the Constant Term
If the constant (or intercept) term is suppressed, the estimated regression will go
through the origin. Such an effect potentially biases the and inflates their t-scores.
In this particular example, the true slope is close to zero in the range of the sample,
but forcing the regression through the origin makes the slope appear to be signifi-
cantly positive.
�̂s
222

Second, the constant term is the value of the dependent variable when all
the independent variables and the error term are zero, but the variables used
for economic analysis are usually positive. Thus, the origin often lies outside
the range of sample observations (as can be seen in Figure 1). Since the con-
stant term is an estimate of Y when the Xs are outside the range of the sample
observations, estimates of it are tenuous.
Alternative Functional Forms
The choice of a functional form for an equation is a vital part of the specifi-
cation of that equation. Before we can talk about those functional forms,
however, we need to make a distinction between an equation that is linear
in the coefficients and one that is linear in the variables.
An equation is linear in the variables if plotting the function in terms of X
and Y generates a straight line. For example, Equation 1:
(1)
is linear in the variables, but Equation 2:
(2)
is not linear in the variables, because if you were to plot Equation 2 it would
be a quadratic, not a straight line.
An equation is linear in the coefficients only if the coefficients (the s)
appear in their simplest form—they are not raised to any powers (other than
one), are not multiplied or divided by other coefficients, and do not themselves
include some sort of function (like logs or exponents). For example, Equation 1
is linear in the coefficients, but Equation 3:
(3)
is not linear in the coefficients Equation 3 is not linear because
there is no rearrangement of the equation that will make it linear in the s of
original interest, In fact, of all possible equations for a single ex-
planatory variable, only functions of the general form:
(4)f(Y) 5 �0 1 �1f(X)
�0 and �1.
�
�0 and �1.
Y 5 �0 1 X
�1
�
Y 5 �0 1 �1X
2 1 �
Y 5 �0 1 �1X 1 �
2
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
223

are linear in the coefficients Linear regression analysis can be ap-
plied to an equation that is nonlinear in the variables as long as the equation
is linear in the coefficients. Indeed, when econometricians use the phrase
“linear regression”(for example, in the Classical Assumptions) they usually
mean “regression that is linear in the coefficients.”
The use of OLS requires that the equation be linear in the coefficients,
but there is a wide variety of functional forms that are linear in the coef-
ficients while being nonlinear in the variables. We’ve already used several
equations that are linear in the coefficients and nonlinear in the vari-
ables, but we’ve said little about when to use such nonlinear equations.
The purpose of the current section is to present the details of the most
frequently used functional forms to help the reader develop the ability to
choose the correct one when specifying an equation.
The choice of a functional form almost always should be based on the un-
derlying theory and only rarely on which form provides the best fit. The logi-
cal form of the relationship between the dependent variable and the inde-
pendent variable in question should be compared with the properties of
various functional forms, and the one that comes closest to that underlying
theory should be chosen. To allow such a comparison, the paragraphs that
follow characterize the most frequently used forms in terms of graphs, equa-
tions, and examples. In some cases, more than one functional form will be
applicable, but usually a choice between alternative functional forms can be
made on the basis of the information we’ll present.
Linear Form
The linear regression model, used almost exclusively in this text thus far, is
based on the assumption that the slope of the relationship between the inde-
pendent variable and the dependent variable is constant:3
�Y
�Xk
5 �k k 5 1, 2, . . . , K
�0 and �1.
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
3. Throughout this section, the “delta” notation will be used instead of the calculus notation
to make for easier reading. The specific definition of is “change,” and it implies a small change
in the variable it is attached to. For example, the term should be read as “change in X.” Since a
regression coefficient represents the change in the expected value of Y brought about by a one-
unit increase in Xk (holding constant all other variables in the equation), then
Those comfortable with calculus should substitute partial derivative signs for � s.
�k 5 � Y> � Xk.
� X
�
( � )
224

If the hypothesized relationship between Y and X is such that the slope of the
relationship can be expected to be constant, then the linear functional form
should be used.
Since the slope is constant, the elasticity of Y with respect to X (the per-
centage change in the dependent variable caused by a 1-percent increase in
the independent variable, holding the other variables in the equation con-
stant) can be calculated fairly easily:
Unless theory, common sense, or experience justifies using some other
functional form, you should use the linear model. Because, in effect, it’s being
used by default, the linear model is sometimes referred to as the default func-
tional form.
Double-Log Form
The double-log form is the most common functional form that is nonlinear
in the variables while still being linear in the coefficients. Indeed, the double-
log form is so popular that some researchers use it as their default functional
form instead of the linear form. In a double-log functional form, the natural
log of Y is the dependent variable and the natural log of X is the independent
variable:
(5)
where lnY refers to the natural log of Y, and so on. For a brief review of the
meaning of a log, see the boxed feature on the following pages.
The double-log form, sometimes called the log-log form, often is used be-
cause a researcher has specified that the elasticities of the model are constant
and the slopes are not. This is in contrast to the linear model, in which the
slopes are constant but the elasticities are not.
In a double-log equation, an individual regression coefficient can be inter-
preted as an elasticity because:
(6)�k 5
� (lnY)
� (lnXk)
5
�Y>Y
�Xk>Xk
5 ElasticityY, Xk
lnY 5 �0 1 �1 lnX1 1 �2 lnX2 1 �
ElasticityY, Xk
5
�Y>Y
�Xk>Xk
5
�Y
�Xk
?
Xk
Y
5 �k
Xk
Y
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
225

Since regression coefficients are constant, the condition that the model have
a constant elasticity is met by the double-log equation.
The way to interpret in a double-log equation is that if Xk increases
by 1 percent while the other Xs are held constant, then Y will change by
percent. Since elasticities are constant, the slopes are now no longer
constant.
Figure 2 is a graph of the double-log function (ignoring the error
term). The panel on the left shows the economic concept of an isoquant or
an indifference curve. Isoquants from production functions show the dif-
ferent combinations of factors X1 and X2, probably capital and labor, that
can be used to produce a given level of output Y. The panel on the right of
Figure 2 shows the relationship between Y and X1 that would exist if X2
were held constant or were not included in the model. Note that the shape
of the curve depends on the sign and magnitude of coefficient If is
negative, a double-log functional form can be used to model a typical de-
mand curve.
Double-log models should be run only when the logged variables
take on positive values. Dummy variables, which can take on the value
of zero, should not be logged but still can be used in a double-log
�1�1.
�k
�k
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
What Is a Log?
What the heck is a log? If e (a constant equal to 2.71828) to the “bth power” pro-
duces x, then b is the log of x:
Thus, a log (or logarithm) is the exponent to which a given base must be taken in
order to produce a specific number. While logs come in more than one variety, we’ll
use only natural logs (logs to the base e) in this text. The symbol for a natural log is
“ln,” so ln(x) � b means that (2.71828)b � x or, more simply,
For example, since e2 � (2.71828)2 � 7.389, we can state that:
ln(7.389) � 2
Thus, the natural log of 7.389 is 2! Two is the power of e that produces 7.389. Let’s
look at some other natural log calculations:
ln(1000) 5 6.908
ln(100) 5 4.605
ln(x) 5 b means that eb 5 x
b is the log of x to the base e if: eb 5 x
226

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
Note that as a number goes from 100 to 1,000,000, its natural log goes from 4.605 to
only 13.816! Since logs are exponents, even a small change in a log can mean a big
change in impact. As a result, logs can be used in econometrics if a researcher wants
to reduce the absolute size of the numbers associated with the same actual meaning.
One useful property of natural logs in econometrics is that they make it easier to
figure out impacts in percentage terms. If you run a double-log regression, the mean-
ing of a slope coefficient is the percentage change in the dependent variable caused
by a one percentage point increase in the independent variable, holding the other
independent variables in the equation constant.4 It’s because of this percentage
change property that the slope coefficients in a double-log equation are elasticities.
4. This is because the derivative of a natural log of X equals which is the
same as percentage change.
dX>X (or � X>X),
ln(1000000) 5 13.816
ln(100000) 5 11.513
ln(10000) 5 9.210
X2
0 X1
Y1
Y2
lnY = �0 + �1lnX1 + �2lnX2
Y
0 X1
�1 > 1
�1 < 0 0 < �1 < 1 Figure 2 Double-Log Functions Depending on the values of the regression coefficients, the double-log functional form can take on a number of shapes. The left panel shows the use of a double-log function to depict a shape useful in describing the economic concept of an isoquant or an indif- ference curve. The right panel shows various shapes that can be achieved with a double- log function if X2 is held constant or is not included in the equation. (Holding X2 constant) 227 equation if they’re adjusted.5 For an example of a double-log equation, see Exercise 7. Semilog Form The semilog functional form is a variant of the double-log equation in which some but not all of the variables (dependent and independent) are ex- pressed in terms of their natural logs. For example, you might choose to use the logarithm of one of the original independent variables, as in: (7) In this case, the economic meanings of the two slope coefficients are differ- ent, since X2 is linearly related to Y while X1 is nonlinearly related to Y. The right-hand side of Figure 3 shows the relationship between Y and X1 in this kind of semilog equation when X2 is held constant. Note that if is greater than zero, the impact of changes in X1 on Y decreases as X1 gets big- ger. Thus, the semilog functional form should be used when the relationship between X1 and Y is hypothesized to have this “increasing at a decreasing rate” form. Applications of the semilog form are quite frequent. For example, most consumption functions tend to increase at a decreasing rate past some level of income. These Engel curves tend to flatten out because as incomes get higher, a smaller percentage of income goes to consumption and a greater percentage goes to saving. Consumption thus increases at a de- creasing rate. If Y is the consumption of an item and X1 is disposable income (with X2 standing for all the other independent variables), then the use of the semilog functional form is justified whenever the item’s con- sumption can be expected to increase at a decreasing rate as income increases. �1 Yi 5 �0 1 �1 ln X1i 1 �2X2i 1 �i SPECIFICATION: CHOOSING A FUNCTIONAL FORM 5. If it is necessary to take the log of a dummy variable, that variable needs to be transformed to avoid the possibility of taking the log of zero. The best way is to redefine the entire dummy variable so that instead of taking on the values of 0 and 1, it takes on the values of 1 and e (the base of the natural logarithm). The log of this newly defined dummy then takes on the values of 0 and 1, and the interpretation of remains the same as in a linear equation. Such a trans- formation changes the coefficient value but not the usefulness or theoretical validity of the dummy variable. � 228 For example, use the beef demand equation: (A) where: � per capita consumption of beef P � the price of beef in cents per pound Yd � U.S. disposable income in thousands of dollars If we substitute the log of disposable income (lnYd ) for disposable income in the above equation, we get: (8) R2 5 .750 N 5 28 (annual) t 5 2 6.93 8.90 (0.13) (11.11) BCt 5 271.75 2 0.87Pt 1 98.87lnYdt t CB R2 5 0.631 N 5 28 (annual) t 5 2 5.36 6.75 (0.16) (1.76) CBt 5 37.54 2 0.88Pt 1 11.9Ydt SPECIFICATION: CHOOSING A FUNCTIONAL FORM Y = (�0 + �2X2) + �1lnX1 Y 0 X1 (Holding X2 constant) �1 > 0
�1 < 0 �1 > 0
Y
0 X1
(Holding X2 constant)
�1 < 0 lnY= �0 + �1X1 + �2X2 Figure 3 Semilog Functions The semilog functional form on the right (ln X) can be used to depict a situation in which the impact of X1 on Y is expected to increase at a decreasing rate as X1 gets bigger as long as is greater than zero (holding X2 constant). The semilog functional form on the left (lnY) can be used to depict a situation in which an increase in X1 causes Y to increase at an increasing rate. �1 229 In Equation 8, the independent variables include the price of beef and the log of disposable income. Equation 8 would be appropriate if we hypothesize that as income rises, consumption will increase at a decreasing rate. For other prod- ucts, perhaps like yachts or summer homes, no such decreasing rate could be hypothesized, and the semilog function would not be appropriate. Not all semilog functions have the log on the right-hand side of the equa- tion, as in Equation 7. The alternative semilog form is to have the log on the left-hand side of the equation. This would mean that the natural log of Y would be a function of unlogged values of the Xs, as in: (9) This model has neither a constant slope nor a constant elasticity, but the co- efficients do have a very useful interpretation. If X1 increases by one unit, then Y will change in percentage terms. Specifically, Y will change by percent, holding X2 constant, for every unit that X1 increases. The left-hand side of Figure 3 shows such a semilog function. This fact means that the lnY semilog function of Equation 9 is perfect for any model in which the dependent variable adjusts in percentage terms to a unit change in an independent variable. The most common economic and business application of Equation 9 is in a model of the earnings of individu- als, where firms often give annual raises in percentage terms. In such a model Y would be the salary or wage of the ith employee, and X1 would be the expe- rience of the ith worker. Each year X1 would increase by one, and would measure the percentage raises given by the firm. For more on this example of a left-side semilog functional form, see Exercise 4 at the end of the chapter. Note that we now have two different kinds of semilog functional forms, cre- ating possible confusion. As a result, many econometricians use phrases like “right-side semilog” or “lin-log functional form” to refer to Equation 7 while using “left-side semilog” or “log-lin functional form” to refer to Equation 9. Polynomial Form In most cost functions, the slope of the cost curve changes sign as output changes. If the slopes of a relationship are expected to depend on the level of the variable itself, then a polynomial model should be considered. Polynomial functional forms express Y as a function of independent variables, some of which are raised to powers other than 1. For example, in a second-degree poly- nomial (also called a quadratic) equation, at least one independent variable is squared: (10)Yi 5 �0 1 �1X1i 1 �2(X1i) 2 1 �3X2i 1 �i �1 �1 ? 100 lnY 5 �0 1 �1X1 1 �2X2 1 � SPECIFICATION: CHOOSING A FUNCTIONAL FORM 230 Such a model can indeed produce slopes that change sign as the independent variables change. The slope of Y with respect to X1 in Equation 10 is: (11) Note that the slope depends on the level of X1. For small values of X1, might dominate, but for large values of X1, will always dominate. If this were a cost function, with Y being the average cost of production and X1 being the level of output of the firm, then we would expect to be negative and to be positive if the firm has the typical U-shaped cost curve depicted in the left half of Figure 4. For another example, consider a model of annual employee earnings as a function of the age of each employee and a number of other measures of pro- ductivity such as education. What is the expected impact of age on earnings? As a young worker gets older, his or her earnings will typically increase. Be- yond some point, however, an increase in age will not increase earnings by very much at all, and around retirement we expect earnings to start to fall �2 �1 �2 �1 �Y �X1 5 �1 1 2�2X1 SPECIFICATION: CHOOSING A FUNCTIONAL FORM Y = (�0 + �3X2) + (�1X1 + �2X1 ) 2 Y 0 X1(Holding X2 constant) �2 > 0
�1 < 0 Y 0 X1(Holding X2 constant) �2 < 0 �1 > 0
Figure 4 Polynomial Functions
Quadratic functional forms (polynomials with squared terms) take on U or inverted
U shapes, depending on the values of the coefficients (holding X2 constant). The left
panel shows the shape of a quadratic function that could be used to show a typical cost
curve; the right panel allows the description of an impact that rises and then falls (like
the impact of age on earnings).
231

abruptly with age. As a result, a logical relationship between earnings and age
might look something like the right half of Figure 4; earnings would rise,
level off, and then fall as age increased. Such a theoretical relationship could
be modeled with a quadratic equation:
(12)
What would the expected signs of be? Since you expect the impact
of age to rise and fall, you’d thus expect to be positive and to be nega-
tive (all else being equal). In fact, this is exactly what many researchers in
labor economics have observed.
With polynomial regressions, the interpretation of the individual regres-
sion coefficients becomes difficult, and the equation may produce unwanted
results for particular ranges of X. Great care must be taken when using a poly-
nomial regression equation to ensure that the functional form will achieve
what is intended by the researcher and no more.
Inverse Form
The inverse functional form expresses Y as a function of the reciprocal (or
inverse) of one or more of the independent variables (in this case, X1):
(13)
The inverse (or reciprocal) functional form should be used when the impact
of a particular independent variable is expected to approach zero as that
independent variable approaches infinity. To see this, note that as X1 gets
larger, its impact on Y decreases.
In Equation 13, X1 cannot equal zero, since if X1 equaled zero, dividing
it into anything would result in infinite or undefined values. The slope with
respect to X1 is:
(14)
The slopes for X1 fall into two categories, both of which are depicted in
Figure 5:
1. When is positive, the slope with respect to X1 is negative and de-
creases in absolute value as X1 increases. As a result, the relationship
between Y and X1 holding X2 constant approaches
increases (ignoring the error term).
�0 1 �2X2 as X1
�1
�Y
�X1
5
2�
1
X
2
1
Yi 5 �0 1 �1(1>X1i) 1 �2X2i 1 �i
�̂2�̂1
�̂1 and �̂2
Earningsi 5 �0 1 �1Agei 1 �2Age˛
2
i 1
c1 �i
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
232

2. When is negative, the relationship intersects the X1 axis at
and slopes upward toward the same horizontal line
(called an asymptote) that it approaches when is positive.
Applications of reciprocals or inverses exist in a number of areas in economic
theory and the real world. For example, the once-popular Phillips curve origi-
nally was estimated with an inverse function.
Choosing a Functional Form
The best way to choose a functional form for a regression model is to choose
a specification that matches the underlying theory of the equation. In a ma-
jority of cases, the linear form will be adequate, and for most of the rest,
common sense will point out a fairly easy choice from among the alterna-
tives outlined above. Table 1 contains a summary of the properties of the
various alternative functional forms.
�1
(�0 1 �2X2)
2�1>�1
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
Y = (�0 + �2X2) + �11/X1
(�0 + �2X2)
Y
0 X1
(Holding X2 constant)
�1 > 0
�1 < 0 Figure 5 Inverse Functions Inverse (or reciprocal) functional forms allow the impact of an X1 on Y to approach zero as X1 increases in size. The inverse function approaches the same value (the asymp- tote) from the top or bottom depending on the sign of �1. 233 Lagged Independent Variables Virtually all the regressions we’ve studied so far have been “instantaneous” in nature. In other words, they have included independent and dependent vari- ables from the same time period, as in: (15) where the subscript t is used to refer to a particular point in time. If all vari- ables have the same subscript, then the equation is instantaneous. However, not all economic or business situations imply such instantaneous relationships between the dependent and independent variables. In many cases time elapses between a change in the independent variable and the re- sulting change in the dependent variable. The length of this time between cause and effect is called a lag. Many econometric equations include one or more lagged independent variables like indi- cates that the observation of X1 is from the time period previous to time pe- riod t, as in the following equation: (16) In this equation, X1 has been lagged by one time period, but the relationship between Y and X2 is still instantaneous. For example, think about the process by which the supply of an agricultural product is determined. Since agricultural goods take time to grow, decisions Yt 5 �0 1 �1X1t21 1 �2X2t 1 �t X1t21, where the subscript t 2 1 Yt 5 �0 1 �1X1t 1 �2X2t 1 �t 3 SPECIFICATION: CHOOSING A FUNCTIONAL FORM Table 1 Summary of Alternative Functional Forms Functional Form Equation (one X only) The Meaning of Linear The slope of Y with respect to X Double-log The elasticity of Y with respect to X Semilog (lnX) The change in Y (in units) related to a 1 percent increase in X Semilog (lnY) The percent change in Y related to a one-unit increase in X Polynomial Roughly, the slope of Y with respect to X for small X Inverse Roughly, the inverse of the slope of Y with respect to X for small X Yi5 �0 1 �1a 1 Xi b 1 �i Yi5 �0 1 �1Xi1 �2X˛ 2 i 1 �i lnYi5 �0 1 �1Xi1 �i Yi5 �0 1 �1 lnXi1 �i lnYi5 �0 1 �1 lnXi1 �i Yi5 �0 1 �1Xi1 �i �1 234 on how many acres to plant or how many eggs to let hatch into egg-producing hens (instead of selling them immediately) must be made months, if not years, before the product is actually supplied to the consumer. Any change in an agricultural market, such as an increase in the price that the farmer can earn for providing cotton, has a lagged effect on the supply of that product: (17) where: Ct � the quantity of cotton supplied in year t � the price of cotton in year PFt �the price of farm labor in year Note that this equation hypothesizes a lag between the price of cotton and the production of cotton, but not between the price of farm labor and the production of cotton. It’s reasonable to think that if cotton prices change, farmers won’t be able to react immediately because it takes a while for cotton to be planted and to grow. The meaning of the regression coefficient of a lagged variable is not the same as the meaning of the coefficient of an unlagged variable. The estimated coefficient of a lagged X measures the change in this year’s Y attributed to a one-unit increase in last year’s X (holding constant the other Xs in the equa- tion). Thus in Equation 17 measures the extra number of units of cotton that would be produced this year as a result of a one-unit increase in last year’s price of cotton, holding this year’s price of farm labor constant. If the lag structure is hypothesized to take place over more than one time period, or if a lagged dependent variable is included on the right-hand side of an equation, the question becomes significantly more complex. Such cases are called distributed lags. Using Dummy Variables We introduce the concept of a dummy variable, which we define as one that takes on the values of 0 or 1, depending on a qualitative attribute such as gender. We can use a dummy variable as an intercept dummy, a dummy vari- able that changes the constant or intercept term, depending on whether the qualitative condition is met. These take the general form: (18) where Di 5 e 1 if the ith observation meets a particular condition 0 otherwise Yi 5 �0 1 �1Xi 1 �2Di 1 �i 4 �1 t 2 1PCt21 Ct 5 f( P 1 Ct21, P F 2 t) 1 �t 5 �0 1 �1PCt21 1 �2PFt 1 �t SPECIFICATION: CHOOSING A FUNCTIONAL FORM 235 SPECIFICATION: CHOOSING A FUNCTIONAL FORM As can be seen in Figure 6, the intercept dummy does indeed change the intercept depending on the value of D, but the slopes remain constant no matter what value D takes. This is true even if we define the dummy variable “backwards” and have D � 0 if the particular condition is met and D � 1 otherwise. The slopes still remain constant. Note that in this example only one dummy variable is used even though there were two conditions. This is because one fewer dummy variable is constructed than conditions. The event not explicitly represented by a dummy variable, the omitted condition, forms the basis against which the included conditions are compared. Thus, for dual situations only one dummy variable is entered as an independent variable; the coefficient is interpreted as the effect of the included condition relative to the omitted condition. What happens if you use two dummy variables to describe the two condi- tions? For example, suppose you decide to include gender in an equation by specifying that X1 � 1 if a person is male and X2 � 1 if a person is female. In such a situation, X1 plus X2 would always add up to 1—do you see why? Y 0 X Di = 0 �2 �0 �0 + �2 (�2 > 0)
Di = 1
Both Slopes = �1
Yi = �0 + �1Xi + �2Di
Figure 6 An Intercept Dummy
If an intercept dummy is added to an equation, a graph of the equation will
have different intercepts for the two qualitative conditions specified by the dummy vari-
able. The difference between the two intercepts is The slopes are constant with re-
spect to the qualitative condition.
�2.
(�2Di)
236

Thus X1 would be perfectly, linearly correlated with X2, and the equation
would violate Classical Assumption VI! If you were to make this mistake,
sometimes called a dummy variable trap, you’d have perfect multicollinearity
and OLS almost surely would fail to estimate the equation.
For an example of the meaning of the coefficient of a dummy variable, let’s
look at a study of the relationship between fraternity/sorority membership and
grade point average (GPA). Most noneconometricians would approach this re-
search problem by calculating the mean grades of fraternity/sorority (so-called
Greek) members and comparing them to the mean grades of nonmembers.
However, such a technique ignores the relationship that grades have to charac-
teristics other than Greek membership.
Instead, we’d want to build a regression model that explains college GPA.
Independent variables would include not only Greek membership but also
other predictors of academic performance such as SAT scores and high school
grades. Being a member of a social organization is a qualitative variable, how-
ever, so we’d have to create a dummy variable to represent fraternity or sorority
membership quantitatively in a regression equation:
If we collect data from all the students in our class and estimate the equa-
tion implied in this example, we obtain:
(19)
where: CGi � the cumulative college GPA (4-point scale) of the ith student
HGi � the cumulative high school GPA (4-point scale) of the ith
student
Si � the sum of the highest verbal and mathematics SAT scores
earned by the ith student
The meaning of the estimated coefficient of Gi in Equation 19 is very specific.
Stop for a second and figure it out for yourself. What is it? The estimate that
means that, for this sample, the GPA of fraternity/sorority mem-
bers is 0.38 lower than for nonmembers, holding SATs and high school GPA
constant. Thus, Greek members are doing about a third of a grade worse than
otherwise might be expected. To understand this example better, try using
Equation 19 to predict your own GPA; how close does it come?
�̂G 5 20.38
R2 5 .45 N 5 25
CGi 5 0.37 1 0.81HGi 1 0.00001Si 2 0.38Gi
Gi 5 •
1 if the ith student is an active member of
a fraternity or sorority
0 otherwise
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
237

Before you rush out and quit whatever social organization you’re in, how-
ever, note that this sample is quite small and that we’ve surely omitted some
important determinants of academic success from the equation. As a result,
we shouldn’t be too quick to conclude that Greeks are dummies.
To this point, we’ve used dummy variables to represent just those qualita-
tive variables that have exactly two possibilities (such as gender). What about
situations where a qualitative variable has three or more alternatives? For ex-
ample, what if you’re trying to measure the impact of education on salaries in
business and you want to distinguish high school graduates from holders of
B.A.s and M.B.A.s? The answer certainly isn’t to have MBA � 2, BA � 1, and 0
otherwise, because we have no reason to think that the impact of having an
M.B.A. is exactly twice that of having a B.A. If not that, then what?
The answer is to create one less dummy variable than there are alternatives
and to use each dummy to represent just one of the possible conditions. In
the salary case, for example, you’d create two variables, the first equal to 1 if
the employee had an M.B.A. (0 otherwise) and the second equal to 1 if the
employee’s highest degree was a B.A. (and 0 otherwise). As before, the omit-
ted condition is represented by having both dummies equal to 0. This way
you can measure the impact of each degree independently without having to
link the impacts of having an M.B.A. and a B.A.
A dummy variable that has only a single observation with a value of 1
while the rest of the observations are 0 (or vice versa) is to be avoided unless
the variable is required by theory. Such a “one-time dummy” acts merely to
eliminate that observation from the data set, improving the fit artificially by
setting the dummy’s coefficient equal to the residual for that observation.
One would obtain exactly the same estimates of the other coefficients if that
observation were deleted, but the deletion of an observation is rarely, if ever,
appropriate. Finally, dummy variables can be used as dependent variables.
Slope Dummy Variables
Until now, every independent variable in this text has been multiplied by ex-
actly one other item: the slope coefficient. To see this, take another look at
Equation 18:
(18)
In this equation X is multiplied only by and D is multiplied only by
and there are no other factors involved.
�2,�1,
Yi 5 �0 1 �1Xi 1 �2Di 1 �i
5
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
238

This restriction does not apply to a new kind of variable called an interac-
tion term. An interaction term is an independent variable in a regression equa-
tion that is the multiple of two or more other independent variables. Each inter-
action term has its own regression coefficient, so the end result is that the
interaction term has three or more components, as in Such interaction
terms are used when the change in Y with respect to one independent variable
(in this case X) depends on the level of another independent variable (in this
case D). For an example of the use of interaction terms, see Exercise 14.
Interaction terms can involve two quantitative variables (B3X1X2) or two
dummy variables (B3D1D2), but the most frequent application of interaction
terms involves one quantitative variable and one dummy variable (B3X1D1), a
combination that is typically called a slope dummy. Slope dummy variables
allow the slope of the relationship between the dependent variable and an
independent variable to be different depending on whether the condition
specified by a dummy variable is met. This is in contrast to an intercept
dummy variable, which changes the intercept, but does not change the slope,
when a particular condition is met.
In general, a slope dummy is introduced by adding to the equation a vari-
able that is the multiple of the independent variable that has a slope you
want to change and the dummy variable that you want to cause the changed
slope. The general form of a slope dummy equation is:
(20)
Note the difference between Equations 18 and 20. Equation 20 is the same as
Equation 18, except that we have added an interaction term in which the
dummy variable is multiplied by an independent variable Let’s
check to make sure that the slope of Y with respect to X does indeed change if
D changes:
In essence, the coefficient of X changes when the condition specified by D is
met. To see this, substitute D � 0 and D � 1, respectively, into Equation 20
and factor out X.
Note that Equation 20 includes both a slope dummy and an intercept
dummy. It turns out that whenever a slope dummy is used, it’s vital to also
have and in the equation to avoid bias in the estimate of the coeffi-
cient of the slope dummy term. If there are other Xs in an equation, they
should not be multiplied by D unless you hypothesize that their slopes
change with respect to D as well.
�2D�1Xi
When D 5 1, �Y>�X 5 (�1 1 �3)
When D 5 0, �Y>�X 5 �1
(�3XiDi).
Yi 5 �0 1 �1Xi 1 �2Di 1 �3XiDi 1 �i
�3XiDi.
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
239

Take a look at Figure 7, which has both a slope dummy and an intercept
dummy. In Figure 7 the intercept will be when D � 0 and when
D � 1. In addition, the slope of Y with respect to X will be when D � 0
and when D � 1. As a result, there really are two equations:
In practice, slope dummies have many realistic uses. For example, consider
the question of earnings differentials between men and women. Although there
is little argument that these differentials exist, there is quite a bit of controversy
over the extent to which these differentials are caused by sexual discrimination
(as opposed to other factors). Suppose you decide to build a model of earnings
to get a better view of this controversy. If you hypothesized that men earn more
than women on average, then you would want to use an intercept dummy vari-
able for gender in an earnings equation that included measures of experience,
special skills, education, and so on, as independent variables:
(21)ln(Earningsi) 5 �0 1 �1Di 1 �2EXPi 1
c1 �i
Yi 5 (�0 1 �2) 1 (�1 1 �3)Xi 1 �i fwhen D 5 1g
Yi 5 �0 1 �1Xi 1 �i fwhen D 5 0g
�1 1 �3
�1
�0 1 �2�0
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
Y
0 X
Di = 0
�2
�0
�0 + �2
(�2 > 0)
Di = 1
Slope = �1
Slope = �1 + �3
(�3 > 0)
Yi = �0 + �1Xi + �2Di + �3XiDi
Figure 7 Slope and Intercept Dummies
If slope dummy terms are added to an equation,
a graph of the equation will have different intercepts and different slopes depending on
the value of the qualitative condition specified by the dummy variable. The difference
between the two intercepts is whereas the difference between the two slopes is �3.�2,
(�3XiDi) and intercept dummy (�2Di)
240

where: Di � 1 if the ith worker is male and 0 otherwise
EXPi � the years experience of the ith worker
� a classical error term
In Equation 21, would be an estimate of the average difference between
males and females, holding constant their experience and the other factors in
the equation. Equation 21 also forces the impact of increases in experience
(and the other factors in the equation) to have the same effect for females as
for males because the slopes are the same for both genders.
If you hypothesized that men also increase their earnings more per year of
experience than women, then you would include a slope dummy as well as
an intercept dummy in such a model:
(22)
In Equation 22, would be an estimate of the differential impact of an
extra year of experience on earnings between men and women. We could test
the possibility of a positive true by running a one-tailed t-test on . If
were significantly different from zero in a positive direction, then we could
reject the null hypothesis of no difference due to gender in the impact of ex-
perience on earnings, holding constant the other variables in the equation.
Problems with Incorrect Functional Forms
Once in a while a circumstance will arise in which the model is logically non-
linear in the variables, but the exact form of this nonlinearity is hard to spec-
ify. In such a case, the linear form is not correct, and yet a choice between the
various nonlinear forms cannot be made on the basis of economic theory.
Even in these cases, however, it still pays (in terms of understanding the true
relationships) to avoid choosing a functional form on the basis of fit alone.
If functional forms are similar, and if theory does not specify exactly which
form to use, why should we try to avoid using goodness of fit over the sample
to determine which equation to use? This section will highlight two answers to
this question:
1. are difficult to compare if the dependent variable is transformed.
2. An incorrect functional form may provide a reasonable fit within the
sample but have the potential to make large forecast errors when used
outside the range of the sample.
Are Difficult to Compare When Y Is Transformed
When the dependent variable is transformed from its linear version, the overall
measure of fit, the cannot be used for comparing the fit of the nonlinearR2,
R 2s
R2s
6
�̂3�̂3�3
�̂3
ln(Earningsi) 5 �0 1 �1Di 1 �2EXPi 1 �3DiEXPi 1
c1 �i
�̂1
�i
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
241

equation with the original linear one. This problem is not especially important
in most cases because the emphasis in applied regression analysis is usually on
the coefficient estimates. However, if are ever used to compare the fit of two
different functional forms, then it becomes crucial that this lack of comparabil-
ity be remembered. For example, suppose you were trying to compare a linear
equation:
(23)
with a semilog version of the same equation (using the version of a semilog
function that takes the log of the dependent variable):
(24)
Notice that the only difference between Equations 23 and 24 is the func-
tional form of the dependent variable. The reason that the of the respec-
tive equations cannot be used to compare overall fits of the two equations is
that the total sum of squares (TSS) of the dependent variable around its
mean is different in the two formulations. That is, the are not comparable
because the dependent variables are different. There is no reason to expect
that different dependent variables will have the identical (or easily compara-
ble) degrees of dispersion around their means.
Incorrect Functional Forms Outside the Range of the Sample
If an incorrect functional form is used, then the probability of mistaken infer-
ences about the true population parameters will increase. Using an incorrect
functional form is a kind of specification error that is similar to the omitted
variable bias. Even if an incorrect functional form provides good statistics
within a sample, large residuals almost surely will arise when the misspecified
equation is used on data that were not part of the sample used to estimate the
coefficients.
In general, the extrapolation of a regression equation to data that are out-
side the range over which the equation was estimated runs increased risks of
large forecasting errors and incorrect conclusions about population values.
This risk is heightened if the regression uses a functional form that is inappro-
priate for the particular variables being studied.
Two functional forms that behave similarly over the range of the sample
may behave quite differently outside that range. If the functional form is cho-
sen on the basis of theory, then the researcher can take into account how the
equation would act over any range of values, even if some of those values are
R2s
R2s
lnY 5 �0 1 �1X1 1 �2X2 1 �
Y 5 �0 1 �1X1 1 �2X2 1 �
R2s
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
242

outside the range of the sample. If functional forms are chosen on the basis of
fit, then extrapolating outside the sample becomes tenuous.
Figure 8 contains a number of hypothetical examples. As can be seen, some
functional forms have the potential to fit quite poorly outside the sample range.
Such graphs are meant as examples of what could happen, not as statements of
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
Y
0 X
(a) Double-Log (� < 0) Sample Y 0 X Out of Sample (b) Polynomial Sample Y 0 X Out of Sample Out of Sample (c) Semilog Right Sample Out of Sample Y 0 X (d) Linear Sample Figure 8 Incorrect Functional Forms Outside the Sample Range If an incorrect form is applied to data outside the range of the sample on which it was estimated, the probability of large mistakes increases. In particular, note how the poly- nomial functional form can change slope rapidly outside the sample range (panel b) and that even a linear form can cause mistakes if the true functional form is nonlinear (panel d). 243 what necessarily will happen, when incorrect functional forms are pushed outside the range of the sample over which they were estimated. Do not con- clude from these diagrams that nonlinear functions should be avoided com- pletely. If the true relationship is nonlinear, then the linear functional form will make large forecasting errors outside the sample. Instead, the researcher must take the time to think through how the equation will act for values both inside and outside the sample before choosing a functional form to use to estimate the equation. If the theoretically appropriate nonlinear equation appears to work well over the relevant range of possible values, then it should be used without concern over this issue. Summary 1. Do not suppress the constant term even if it appears to be theoreti- cally likely to equal zero. On the other hand, don’t rely on estimates of the constant term for inference even if it appears to be statistically significant. 2. The choice of a functional form should be based on the underlying eco- nomic theory to the extent that theory suggests a shape similar to that provided by a particular functional form. A form that is linear in the variables should be used unless a specific hypothesis suggests otherwise. 3. Functional forms that are nonlinear in the variables include the double-log form, the semilog form, the polynomial form, and the in- verse form. The double-log form is especially useful if the elasticities involved are expected to be constant. The semilog and inverse forms have the advantage of allowing the effect of an independent variable to tail off as that variable increases. The polynomial form is useful if the slopes are expected to change sign, depending on the level of an independent variable. 4. A slope dummy is a dummy variable that is multiplied by an inde- pendent variable to allow the slope of the relationship between the dependent variable and the particular independent variable to change, depending on whether a particular condition is met. 5. The use of nonlinear functional forms has a number of potential prob- lems. In particular, the are difficult to compare if Y has been trans- formed, and the residuals are potentially large if an incorrect functional form is used for forecasting outside the range of the sample. R2s 7 SPECIFICATION: CHOOSING A FUNCTIONAL FORM 244 SPECIFICATION: CHOOSING A FUNCTIONAL FORM EXERCISES (The answer to Exercise 2 is at the end of the chapter.) 1. Write out the meaning of each of the following terms without refer- ring to the book (or your notes), and compare your definition with the version in the text for each: a. elasticity b. double-log functional form c. semilog functional form d. polynomial functional form e. inverse functional form f. slope dummy g. natural log h. omitted condition i. interaction term j. linear in the variables k. linear in the coefficients 2. For each of the following pairs of dependent (Y) and independent (X) variables, pick the functional form that you think is likely to be appropriate, and then explain your reasoning (assume that all other relevant independent variables are included in the equation): a. Y � sales of shoes X � disposable income b. Y � the attendance at the Hollywood Bowl outdoor symphony concerts on a given night X � whether the orchestra’s most famous conductor was scheduled to conduct that night c. Y � aggregate consumption of goods and services in the United States X � aggregate disposable income in the United States d. Y � the money supply in the United States X � the interest rate on Treasury Bills (in a demand function) e. Y � the average production cost of a box of pasta X � the number of boxes of pasta produced 3. Look over the following equations and decide whether they are linear in the variables, linear in the coefficients, both, or neither: a. b. Yi 5 �0 1 �1ln Xi 1 �i Yi 5 �0 1 �1X˛ 3 i 1 �i 245 SPECIFICATION: CHOOSING A FUNCTIONAL FORM c. d. e. 4. Consider the following estimated semilog equation (standard errors in parentheses): where: lnSALi � the log of the salary of the ith worker EDi � the years of education of the ith worker EXPi � the years of experience of the ith worker a. Make appropriate hypotheses for signs, calculate t-scores, and test your hypotheses. b. What is the economic meaning of the constant in this equation? c. Why do you think a left-side semilog functional form is used in this model? (Hint: What are the slopes of salary with respect to educa- tion and experience?) d. Suppose you ran the linear version of this equation and obtained an of .46. What can you conclude from this result? 5. In 2003, Ray Fair6 analyzed the relationship between stock prices and risk aversion by looking at the 1996–2000 performance of the 65 companies that had been a part of Standard and Poor’s famous index (the S&P 500) since its inception in 1957. Fair focused on the P/E ratio (the ratio of a company’s stock price to its earnings per share) and its relationship to the � coefficient (a measure of a company’s riskiness—a high � implies high risk). Hypothesizing that the stock price would be a positive function of earnings growth and dividend growth, he estimated the following equation: LNPEi � 2.74 � 0.22BETAi � 0.83EARNi � 2.81DIVi (0.11) (0.57) (0.84) t � �1.99 1.45 3.33 N � 65 R2 � .232 � .194 R2 R2 R2 5 .48 N 5 28 (0.025) (0.050) lnSALi 5 2̨8.10 1 0.100EDi 1 0.110EXPi Y˛ �0 i 5 �1 1 �2X˛ 2 i 1 �i Yi 5 �0 1 �1X˛ �2 i 1 �i ln Yi 5 �0 1 �1ln Xi 1 �i 6. Ray C. Fair, “Risk Aversion and Stock Prices,” Cowles Foundation Discussion Papers 1382, Cowles Foundation: Yale University, revised February 2003. Most of the article is well beyond the scope of this text, but Fair generously included the data (including proprietary data that he generated) necessary to replicate his regression results. 246 SPECIFICATION: CHOOSING A FUNCTIONAL FORM where: LNPEi � the log of the median P/E ratio of the ith com- pany from 1996 to 2000 BETAi � the mean � of the ith company from 1958 to 1994 EARNi � the median percentage earnings growth rate for the ith company from 1996 to 2000 DIVi � the median percentage dividend growth rate for the ith company from 1996 to 2000 a. Create and test appropriate hypotheses about the slope coefficients of this equation at the 5-percent level. b. One of these variables is lagged and yet this is a cross-sectional equa- tion. Explain which variable is lagged and why you think Fair lagged it. c. Is one of Fair’s variables potentially irrelevant? Which one? Use EViews, Stata, or your own regression program on the data in Table 2 to estimate Fair’s equation without your potentially irrelevant variable and then use our four specification criteria to determine whether the variable is indeed irrelevant. d. What functional form does Fair use? Does this form seem appropriate on the basis of theory? (Hint: A review of the literature would certainly help you answer this question, but before you start that review, think through the meaning of slope coefficients in this functional form.) e. (optional) Suppose that your review of the literature makes you concerned that Fair should have used a double-log functional form for his equation. Use the data in Table 2 to estimate that functional form on Fair’s data. What is your estimated result? Does it support your concern? Explain. 6. In an effort to explain regional wage differentials, you collect wage data from 7,338 unskilled workers, divide the country into four regions (Northeast, South, Midwest, and West), and estimate the following equa- tion (standard errors in parentheses): where: Yi � the hourly wage (in dollars) of the ith unskilled worker Ei � a dummy variable equal to 1 if the ith worker lives in the Northeast, 0 otherwise Si � a dummy variable equal to 1 if the ith worker lives in the South, 0 otherwise Wi � a dummy variable equal to 1 if the ith worker lives in the West, 0 otherwise R2 5 .49 N 5 7,338 (0.019) (0.010) (0.012) Ŷi 5 4.78 2 0.038Ei 2 0.041Si 2 0.048Wi 247 SPECIFICATION: CHOOSING A FUNCTIONAL FORM Table 2 Data for the Stock Price Example COMPANY PE BETA EARN DIV 1 Alcan 12.64 0.466 0.169 –0.013 2 TXU Corp. 10.80 0.545 0.016 0.014 3 Procter & Gamble 19.90 0.597 0.066 0.050 4 PG&E 11.30 0.651 0.021 0.014 5 Phillips Petroleum 13.27 0.678 0.071 0.006 6 AT&T 13.71 0.697 –0.004 –0.008 7 Minnesota Mining & Mfg. 17.61 0.781 0.054 0.051 8 Alcoa 15.97 0.795 0.120 –0.015 9 American Electric Power 10.68 0.836 –0.001 –0.021 10 Public Service Entrp 9.63 0.845 –0.018 –0.011 11 Hercules 16.07 0.851 0.077 –0.008 12 Air Products & Chemicals 16.20 0.865 0.051 0.074 13 Bristol Myers Squibb 17.01 0.866 0.068 0.110 14 Kimberly-Clark 13.42 0.869 0.063 0.018 15 Aetna 8.98 0.894 –0.137 0.007 16 Wrigley 14.49 0.898 0.062 0.044 17 Halliburton 17.84 0.906 0.120 –0.011 18 Deere & Co. 12.15 0.916 –0.010 0.004 19 Kroger 11.82 0.931 0.010 0.000 20 Intl Business Machines 16.08 0.944 0.081 0.045 21 Caterpillar 16.95 0.952 –0.043 –0.005 22 Goodrich 12.06 0.958 0.028 –0.015 23 General Mills 17.16 0.965 0.060 0.048 24 Winn-Dixie Stores 16.10 0.973 0.045 0.047 25 Heinz (H J) 13.49 0.979 0.079 0.079 26 Eastman Kodak 28.28 0.983 0.023 0.009 27 Campbell Soup 16.33 0.986 0.028 0.025 28 Philip Morris 12.25 0.993 0.129 0.130 29 Southern Co. 11.26 0.995 0.034 0.000 30 Du Pont 14.16 0.996 0.099 0.001 31 Phelps Dodge 11.47 1.008 0.186 –0.011 32 Pfizer Inc. 17.63 1.019 0.052 0.062 33 Hershey Foods 14.66 1.022 0.025 0.058 34 Ingersoll-Rand 14.24 1.024 0.045 –0.018 35 FPL Group 11.86 1.048 0.038 0.019 36 Pitney Bowes 16.11 1.064 0.049 0.086 37 Archer-Daniels-Midland 14.43 1.073 0.073 –0.011 (continued) 248 SPECIFICATION: CHOOSING A FUNCTIONAL FORM a. What is the omitted condition in this equation? b. If you add a dummy variable for the omitted condition to the equation without dropping Ei, Si, or Wi, what will happen? c. If you add a dummy variable for the omitted condition to the equation and drop Ei, what will the sign of the new variable’s esti- mated coefficient be? Table 2 (continued) COMPANY PE BETA EARN DIV 38 Rockwell 9.42 1.075 0.062 0.020 39 Dow Chemical 15.25 1.081 0.042 0.026 40 General Electric 15.16 1.091 0.051 0.015 41 Abbott Laboratories 17.58 1.097 0.114 0.098 42 Merck & Co. 23.29 1.122 0.066 0.072 43 J C Penney 13.14 1.133 0.094 –0.003 44 Union Pacific Corp. 12.99 1.136 0.010 0.021 45 Schering-Plough 18.18 1.137 0.112 0.060 46 Pepsico 18.94 1.147 0.082 0.046 47 McGraw-Hill 16.93 1.150 0.051 0.052 48 Household International 8.36 1.184 0.019 0.008 49 Emerson Electric 17.52 1.196 0.047 0.044 50 General Motors 11.21 1.206 0.052 –0.023 51 Colgate-Palmolive 16.60 1.213 0.067 0.025 52 Eaton Corp. 10.64 1.216 0.137 0.001 53 Dana Corp. 10.26 1.222 0.069 –0.011 54 Sears Roebuck 12.41 1.256 0.030 –0.014 55 Corning Inc. 19.33 1.258 0.052 –0.013 56 General Dynamics 9.06 1.285 0.056 –0.023 57 Coca-Cola 21.68 1.290 0.085 0.055 58 Boeing 11.93 1.306 0.169 0.017 59 Ford 8.62 1.308 0.016 0.026 60 Peoples Energy 9.58 1.454 0.000 0.005 61 Goodyear 12.02 1.464 0.022 0.012 62 May Co. 11.32 1.525 0.050 0.006 63 ITT Industries 9.92 1.630 0.038 0.018 64 Raytheon 11.75 1.821 0.112 0.050 65 Cooper Industries 12.41 1.857 0.108 0.037 Source: Ray C. Fair, “Risk Aversion and Stock Prices,” Cowles Foundation Discussion Papers 1382, Cowles Foundation: Yale University, revised February 2003. Datafile � STOCK7 249 SPECIFICATION: CHOOSING A FUNCTIONAL FORM d. Which of the following three statements is most correct? Least cor- rect? Explain your answer. i. The equation explains 49 percent of the variation of Y around its mean with regional variables alone, so there must be quite a bit of wage variation by region. ii. The coefficients of the regional variables are virtually identical, so there must not be much wage variation by region. iii. The coefficients of the regional variables are quite small com- pared with the average wage, so there must not be much wage variation by region. e. If you were going to add one variable to this model, what would it be? Justify your choice. 7. V. N. Murti and V. K. Sastri7 investigated the production characteris- tics of various Indian industries, including cotton and sugar. They specified Cobb–Douglas production functions for output (Q) as a double-log function of labor (L) and capital (K): and obtained the following estimates (standard errors in parentheses): Industry Cotton 0.97 0.92 0.12 .98 (0.03) (0.04) Sugar 2.70 0.59 0.33 .80 (0.14) (0.17) a. What are the elasticities of output with respect to labor and capital for each industry? b. What economic significance does the sum have? c. Murti and Sastri expected positive slope coefficients. Test their hy- potheses at the 5-percent level of significance. (Hint: This is much harder than it looks!) 8. Suppose you are studying the rate of growth of income in a country as a function of the rate of growth of capital in that country and of the per capita income of that country. You’re using a cross-sectional data set that includes both developed and developing countries. Suppose further that the underlying theory suggests that income growth rates (�̂1 1 �̂2) R2�̂2�̂1�̂0 lnQi 5 �0 1 �1lnLi 1 �2lnKi 1 �i 7. V. N. Murti and V. K. Sastri, “Production Functions for Indian Industry,” Econometrica, Vol. 25, No. 2, pp. 205–221. 250 SPECIFICATION: CHOOSING A FUNCTIONAL FORM will increase as per capita income increases and then start decreasing past a particular point. Describe how you would model this relation- ship with each of the following functional forms: a. a quadratic function b. a semilog function c. a slope dummy equation 9. A study of hotel investments in Waikiki estimated this revenue pro- duction function: lnR � �0 � �1 lnL � �2 lnK � where: R � the annual net revenue of the hotel (in thousands of dollars) L � land input (site area in square feet) K � capital input (construction cost in thousands of dollars) a. Create specific null and alternative hypotheses for this equation. b. Calculate the appropriate t-values and run t-tests given the follow- ing regression result (standard errors in parentheses): c. If you were going to build a Waikiki hotel, what input would you most want to use? Is there an additional piece of information you would need to know before you could answer? 10. William Comanor and Thomas Wilson8 specified the following re- gression in their study of advertising’s effect on the profit rates of 41 consumer goods firms: where: PRi � the profit rate of the ith firm ADVi � the advertising expenditures in the ith firm (in dollars) SALESi � the total gross sales of the ith firm (in dollars) CAPi � the capital needed to enter the ith firm’s market at an efficient size PRi 5 �0 1 �1ADVi>SALESi 1 �2 lnCAPi 1 �3 lnESi 1 �4 lnDGi 1 �i
N 5 25
(0.125) (0.135)
lnR 5 2 0.91750 1 0.273lnL 1 0.733lnK
�
8. William S. Comanor and Thomas A. Wilson, “Advertising, Market Structure and Performance,”
Review of Economics and Statistics, Vol. 49, p. 432.
251

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
ESi � the degree to which economies of scale exist in
the ith firm’s industry
DGi � percent growth in sales (demand) of the ith firm
over the last 10 years
ln � natural logarithm
� a classical error term
a. Hypothesize expected signs for each of the slope coefficients.
b. Note that there are two different kinds of nonlinear (in the vari-
ables) relationships in this equation. For each independent vari-
able, determine the shape that the chosen functional form implies,
and state whether you agree or disagree with this shape. Explain
your reasoning in each case.
c. Comanor and Wilson state that the simple correlation coefficient
between and each of the other independent variables
is positive. If one of these other variables were omitted, in which
direction would likely be biased?
11. Suggest the appropriate functional forms for the relationships be-
tween the following variables. Be sure to explain your reasoning:
a. The age of the ith house in a cross-sectional equation for the sales
price of houses in Cooperstown, New York. (Hint: Cooperstown is
known as a lovely town with a number of elegant historic homes.)
b. The price of natural gas in year t in a demand-side time-series equa-
tion for the consumption of natural gas in the United States.
c. The income of the ith individual in a cross-sectional equation for
the number of suits owned by individuals.
d. A dummy variable for being a student (1 � yes) in the equation
specified in part c.
e. The number of long-distance telephone calls handled per year in a
cross-sectional equation for the marginal cost of a telephone call
faced by various competing long-distance telephone carriers.
12. Suppose you’ve been hired by a union that wants to convince workers in
local dry cleaning establishments that joining the union will improve
their well-being. As your first assignment, your boss asks you to build a
model of wages for dry cleaning workers that measures the impact of
union membership on those wages. Your first equation (standard errors
in parentheses) is:
N 5 34 R2 5 .14
(0.10) (0.002) (0.20) (1.00)
Ŵi 5 2 11.40 1 0.30Ai 2 0.003A
2
i 1 1.00Si 1 1.20Ui
�̂1
ADVi>SALESi
�i
252

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
where: Wi � the hourly wage (in dollars) of the ith worker
Ai � the age of the ith worker
Si � the number of years of education of the ith worker
Ui � a dummy variable � 1 if the ith worker is a union
member, 0 otherwise
a. Evaluate the equation. How do and the signs and significance of
the coefficients compare with your expectations?
b. What is the meaning of the A2 term? What relationship between A
and W does it imply? Why doesn’t the inclusion of A and A2 violate
Classical Assumption VI of no perfect collinearity between two in-
dependent variables?
c. Do you think you should have used the log of W as your depen-
dent variable? Why or why not? (Hint: Compare this equation to
the one in Exercise 4.)
d. Even though we’ve been told not to analyze the value of the inter-
cept, isn’t �$11.40 too low to ignore? What should be done to cor-
rect this problem?
e. On the basis of your regression, should the workers be convinced
that joining the union will improve their well-being? Why or why
not?
13. Your boss manages to use the regression results in Exercise 12 to con-
vince the dry cleaning workers to join your union. About a year later,
they go on strike, a strike that turns violent. Now your union is being
sued by all the local dry cleaning establishments for some of the rev-
enues lost during the strike. Their claim is that the violence has intim-
idated replacement workers, thus decreasing production. Your boss
doesn’t believe that the violence has had a significant impact on pro-
duction efficiency and asks you to test his hypothesis with a regres-
sion. Your results (standard errors in parentheses) are:
where: LEt � the natural log of the efficiency rate (defined as the
ratio of actual total output to the goal output in
week t)
LQt � the natural log of actual total output in week t
At � the absentee rate (%) during week t
Vt � the number of incidents of violence during week t
N 5 24 R2 5 .855
(0.04) (0.010) (0.0008)
LEt 5 3.08 1 0.16LQt 2 0.020At 2 0.0001Vt
R2
253

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
a. Hypothesize signs and develop and test the appropriate hypotheses
for the individual estimated coefficients (5-percent level).
b. If the functional form is correct, what does its use suggest about the
theoretical elasticity of E with respect to Q compared with the elas-
ticities of E with respect to A and V?
c. On the basis of this result, do you think the court will conclude
that the violence had a significant impact on the efficiency rate?
Why or why not?
d. What problems appear to exist in this equation? (Hint: The prob-
lems may be theoretical as well as econometric.) If you could make
one change in the specification of this equation, what would it be?
14. Richard Fowles and Peter Loeb studied the interactive effect of drink-
ing and altitude on traffic deaths.9 The authors hypothesized that
drunk driving fatalities are more likely at high altitude than at low
altitude because higher elevations diminish the oxygen intake of the
brain, increasing the impact of a given amount of alcohol. To test this
hypothesis, they used an interaction variable between altitude and
beer consumption. They estimated the following cross-sectional
model (by state for the continental United States) of the motor vehi-
cle fatality rate (t-scores in parentheses):
(25)
where: Fi � traffic fatalities per motor vehicle mile driven in the
ith state
Bt � per capita consumption of beer (malt beverages) in
state i
Si � average highway driving speed in state i
Di � a dummy variable equal to 1 if the ith state had a
vehicle safety inspection program, 0 otherwise
Ai � the average altitude of metropolitan areas in state i
(in thousands)
N 5 48 R2 5 .499
(2 0.08) (1.85) (2 1.29) (4.05)
F̂i 5 2 3.36 2 0.002Bi 1 0.17Si 2 0.31Di 1 0.011Bi Ai
9. Richard Fowles and Peter D. Loeb, “The Interactive Effect of Alcohol and Altitude on Traffic
Fatalities,” Southern Economic Journal, Vol. 59, pp. 108–111. To focus the analysis, we have omit-
ted the coefficients of three other variables (the minimum legal drinking age, the percent of the
population between 18 and 24, and the variability of highway driving speeds) that were in-
significant in Equations 25 and 26.
254

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
a. Carefully state and test appropriate hypotheses about the coeffi-
cients of B, S, and D at the 5-percent level. Do these results give any
indication of econometric problems in the equation? Explain.
b. Think through the interaction variable. What is it measuring? Care-
fully state the meaning of the coefficient of
c. Create and test appropriate hypotheses about the coefficient of the
interaction variable at the 5-percent level.
d. Note that Ai is included in the equation in the interaction variable
but not as an independent variable on its own. If an equation in-
cludes an interaction variable, should both components of the in-
teraction be independent variables in the equation as a matter of
course? Why or why not? (Hint: Recall that with slope dummies,
we emphasized that both the intercept dummy term and the slope
dummy variable term should be in the equation.)
e. When the authors included Ai in their model, the results were as in
Equation 26. Which equation do you prefer? Explain.
(26)
15. Walter Primeaux used slope dummies to help test his hypothesis that
monopolies tend to advertise less intensively than do duopolies in the
electric utility industry.10 His estimated equation (which also included
a number of geographic dummies and a time variable) was (t-scores in
parentheses):
where: Yi � advertising and promotional expense (in dollars) per
1,000 residential kilowatt hours (KWH) of the ith
electric utility
R2 5 .456 N 5 350
(2 5.0) (2.3)
2 20.0Si ? Di 1 0.49Gi ? Di
(4.5) (0.4) (2.9)
Ŷi 5 0.15 1 5.0Si 1 0.015Gi 1 0.35Di
N 5 48 R2 5 .501
(2 0.80) (1.53) (2 0.96) (2 1.07) (1.97)
F̂i 5 2 2.33 2 0.024Bi 1 0.14Si 2 0.24Di 2 0.35Ai 1 0.023Bi Ai
B*A.
10. Walter J. Primeaux, Jr., “An Assessment of the Effects of Competition on Advertising Inten-
sity,” Economic Inquiry, Vol. 19, No. 4, pp. 613–625.
255

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
Si � number of residential customers of the ith utility
(hundreds of thousands)
Gi � annual percentage growth in residential KWH of the
ith utility
Di � a dummy variable equal to 1 if the ith utility is a du-
opoly, 0 if a monopoly
a. Carefully explain the economic meaning of each of the five slope co-
efficients. Note that both independent variables have slope dummies.
b. Hypothesize and test the relevant null hypotheses with the t-test at
the 5-percent level of significance. (Hint: Primeaux expected posi-
tive coefficients for all five.)
c. Assuming that Primeaux’s equation is correct, graph the relation-
ship between advertising (Yi) and size (Si) for monopolies and for
duopolies.
d. Assuming that Primeaux’s equation is correct, graph the relation-
ship between advertising and growth (Gi) for monopolies and for
duopolies.
16. What attributes make a car accelerate well? If you’re like most people,
you’d answer that the fastest accelerators are high-powered, light cars
with aerodynamic shapes. To test this, we used the data in Table 3 for
2009 model vehicles to estimate the following equation (standard er-
rors in parentheses):
TIMEi � 7.43 � 1.90TOPi � 0.0007WEIGHTi � 0.005HPi (27)
(0.29) (0.0003) (0.00060)
t � � 6.49 2.23 � 7.74
N � 30 � .877
where: TIMEi � the time (in seconds) it takes the ith car to ac-
celerate from 0 to 60 miles per hour
TOPi � a dummy equal to 1 if the ith car has a hard
top, 0 if it has a soft top (convertible)
WEIGHTi � the curb weight (in pounds) of the ith car
HPi � the base horsepower of the ith car
a. Create and test appropriate hypotheses about the slope coefficients
of the equation at the 1-percent level.
b. What possible econometric problems, out of omitted variables, ir-
relevant variables, or incorrect functional form, does Equation 27
appear to have? Explain.
R2
256

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
Table 3 Acceleration Times for 2009 Model Vehicles
MAKE MODEL TIME SPEED TOP WEIGHT HP
1 Audi TT Roadster 8.9 133 0 1335 150
2 Mini Cooper S 7.4 134 0 1240 168
3 Volvo C70 T5 Sport 7.4 150 0 1711 220
4 Saab Nine-Three 7.9 149 0 1680 247
5 Mercedes-
Benz SL350 6.6 155 0 1825 268
6 Jaguar XK8 6.7 154 0 1703 290
7 Bugatti Veyron 16.4 2.4 253 1 1950 1000
8 Lotus Exige 4.9 147 1 875 189
9 BMW M3 (E30) 6.7 144 1 1257 220
10 BMW 330i Sport 5.9 155 1 1510 231
11 Porsche Cayman S 5.3 171 1 1350 291
12 Nissan Skyline GT-R
(R34) 4.7 165 1 1560 276
13 Porsche 911 RS 4.7 172 1 1270 300
14 Ford Shelby GT 5 150 1 1584 319
15 Mitsubishi Evo VII RS
Sprint 4.4 150 1 1260 320
16 Aston Martin V8 Vantage 5.2 175 1 1630 380
17 Mercedes-
Benz SLK55 AMG 4.8 155 1 1540 355
18 Maserati Quattroporte
Sport GT 5.1 171 1 1930 394
19 Spyker C8 4.5 187 1 1275 400
20 Ferrari 288GTO 4.9 189 1 1161 400
21 Mosler MT900 3.9 190 1 1130 435
22 Lamborghini Countach QV 4.9 180 1 1447 455
23 Chrysler Viper GTS-R 4 190 1 1290 460
24 Bentley Arnage T 5.2 179 1 2585 500
25 Ferrari 430 Scuderia 3.5 198 1 1350 503
26 Saleen S7 3.3 240 1 1247 550
27 Lamborghini Murcielago 4 205 1 1650 570
28 Pagani Zonda F 3.6 214 1 1230 602
29 McLaren F1 3.2 240 1 1140 627
30 Koenigsegg CCR 3.2 242 1 1180 806
Source: StrikeEngine. “Performance Car Specs: 0–60, 0–100, Power to Weight Ratio, Top Speed.”
StrikeEngine.com. 2009.
257

SPECIFICATION: CHOOSING A FUNCTIONAL FORM
c. Suppose that your next-door neighbor is a physics major who
tells you that horsepower can be expressed in terms of the follow-
ing equation: where , ,
and TIME and HP are as defined previously. Does
this change your answer to part b? How? Why?
d. On the basis of your answer to part c, you decide to change the func-
tional form of the relationship between TIME and HP to an inverse
because that’s the appropriate theoretical relationship between the
two variables. What would the expected sign of the coefficient of
be? Explain.
e. Equation 28 shows what happens if you switch your horsepower
functional form to an inverse. Which equation do you prefer? Why?
If Equation 28 had a higher and higher t-scores, would that
change your answer? Why or why not?
TIMEi � 2.26 � 1.26TOPi � 0.001WEIGHTi � 765.44(1/HPi) (28)
(0.33) (0.0003) (99.61)
t � � 3.74 3.06 7.68
N � 30 � .875
f. Since the two equations have different functional forms, can be
used to compare the overall fit of the equations? Why or why not?
g. (optional) Note that Table 3 also includes data on SPEEDi, defined
as the top speed of the ith vehicle. Use EViews, Stata, or your com-
puter’s regression program to estimate Equations 27 and 28 with
SPEED as the dependent variable instead of TIME, and then answer
parts a–f of this exercise for the new dependent variable.
R2
R2
R2
1>HP
A 5 acceleration,
D 5 distanceM 5 massHP 5 MDA>TIME
258

Answers
Exercise 2
a. Semilog right [where Y � f(lnX)]; as income increases, the sales of
shoes will increase, but at a declining rate.
b. Linear (intercept dummy); there is little justification for any other
form.
c. Semilog right [where Y � f(lnX)] or linear are both justifiable.
d. Inverse function [where Y � f(1/X)]; as the interest rate gets higher,
the quantity of money demanded will decrease, but even at very
high interest rates, there still will be some money held to allow
for transactions.
e. Quadratic function [where Y � f(X,X2)]; as output levels are in-
creased, we will encounter diminishing returns to scale.
SPECIFICATION: CHOOSING A FUNCTIONAL FORM
259

260

1 Perfect versus Imperfect Multicollinearity
2 The Consequences of Multicollinearity
3 The Detection of Multicollinearity
4 Remedies for Multicollinearity
5 An Example of Why Multicollinearity Often Is Best Left Unadjusted
6 Summary and Exercises
7 Appendix: The SAT Interactive Regression Learning Exercise
Multicollinearity
This chapter addresses multicollinearity; a violation of the Classical Assump-
tions, and remedies. We will attempt to answer the following questions:
1. What is the nature of the problem?
2. What are the consequences of the problem?
3. How is the problem diagnosed?
4. What remedies for the problem are available?
Strictly speaking, perfect multicollinearity is the violation of Classical As-
sumption VI—that no independent variable is a perfect linear function of
one or more other independent variables. Perfect multicollinearity is rare, but
severe imperfect multicollinearity, although not violating Classical Assump-
tion VI, still causes substantial problems.
Recall that the coefficient can be thought of as the impact on the de-
pendent variable of a one-unit increase in the independent variable Xk,
holding constant the other independent variables in the equation. But if two
explanatory variables are significantly related, then the OLS computer pro-
gram will find it difficult to distinguish the effects of one variable from the
effects of the other.
�k
From Chapter 8 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
261

MULTICOLLINEARITY
1. The word collinearity describes a linear correlation between two independent variables, and
multicollinearity indicates that more than two independent variables are involved. In common
usage, multicollinearity is used to apply to both cases, and so we’ll typically use that term in
this text even though many of the examples and techniques discussed relate, strictly speaking,
to collinearity.
In essence, the more highly correlated two (or more) independent vari-
ables are, the more difficult it becomes to accurately estimate the coefficients
of the true model. If two variables move identically, then there is no hope of
distinguishing between the impacts of the two; but if the variables are only
roughly correlated, then we still might be able to estimate the two effects ac-
curately enough for most purposes.
Perfect versus Imperfect Multicollinearity
Perfect Multicollinearity
Perfect multicollinearity1 violates Classical Assumption VI, which specifies
that no explanatory variable is a perfect linear function of any other explana-
tory variables. The word perfect in this context implies that the variation in
one explanatory variable can be completely explained by movements in an-
other explanatory variable. Such a perfect linear function between two inde-
pendent variables would be:
(1)
where the are constants and the Xs are independent variables in:
(2)
Notice that there is no error term in Equation 1. This implies that X1 can be
exactly calculated given X2 and the equation. Examples of such perfect linear
relationships would be:
(3)
(4)
Figure 1 shows a graph of explanatory variables that are perfectly corre-
lated. As can be seen in Figure 1, a perfect linear function has all data points
on the same straight line. There is none of the variation that accompanies the
data from a typical regression.
X1i 5 2 1 4X2i
X1i 5 3X2i
Yi 5 �0 1 �1X1i 1 �2X2i 1 �i
�s
X1i 5 �0 1 �1X2i
1
262

MULTICOLLINEARITY
X1
0 X2
Figure 1 Perfect Multicollinearity
With perfect multicollinearity, an independent variable can be completely explained by
the movements of one or more other independent variables. Perfect multicollinearity
can usually be avoided by careful screening of the independent variables before a re-
gression is run.
What happens to the estimation of an econometric equation where there
is perfect multicollinearity? OLS is incapable of generating estimates of the
regression coefficients, and most OLS computer programs will print out an
error message in such a situation. Using Equation 2 as an example, we theoret-
ically would obtain the following estimated coefficients and standard errors:
(5)
(6)
Perfect multicollinearity ruins our ability to estimate the coefficients because
the two variables cannot be distinguished. You cannot “hold all the other in-
dependent variables in the equation constant” if every time one variable
changes, another changes in an identical manner.
Fortunately, instances in which one explanatory variable is a perfect linear
function of another are rare. More important, perfect multicollinearity should
be fairly easy to discover before a regression is run. You can detect perfect mul-
ticollinearity by asking whether one variable equals a multiple of another or if
one variable can be derived by adding a constant to another or if a variable
equals the sum of two other variables. If so, then one of the variables should
be dropped because there is no essential difference between the two.
�̂2 5 indeterminate SE(�̂2) 5 `
�̂1 5 indeterminate SE(�̂1) 5 `
263

MULTICOLLINEARITY
A special case related to perfect multicollinearity occurs when a variable that
is definitionally related to the dependent variable is included as an independent
variable in a regression equation. Such a dominant variable is by definition so
highly correlated with the dependent variable that it completely masks the ef-
fects of all other independent variables in the equation. In a sense, this is a case
of perfect collinearity between the dependent and an independent variable.
For example, if you include a variable measuring the amount of raw mate-
rials used by the shoe industry in a production function for that industry, the
raw materials variable would have an extremely high t-score, but otherwise
important variables like labor and capital would have quite insignificant
t-scores. Why? In essence, if you knew how much leather was used by a shoe
factory, you could predict the number of pairs of shoes produced without
knowing anything about labor or capital. The relationship is definitional, and
the dominant variable should be dropped from the equation to get reason-
able estimates of the coefficients of the other variables.
Be careful, though! Dominant variables shouldn’t be confused with highly
significant or important explanatory variables. Instead, they should be recog-
nized as being virtually identical to the dependent variable. While the fit be-
tween the two is superb, knowledge of that fit could have been obtained from
the definitions of the variables without any econometric estimation.
Imperfect Multicollinearity
Since perfect multicollinearity is fairly easy to avoid, econometricians almost
never talk about it. Instead, when we use the word multicollinearity, we
really are talking about severe imperfect multicollinearity. Imperfect multi-
collinearity can be defined as a linear functional relationship between two
or more independent variables that is so strong that it can significantly affect
the estimation of the coefficients of the variables.
In other words, imperfect multicollinearity occurs when two (or more) ex-
planatory variables are imperfectly linearly related, as in:
(7)
Compare Equation 7 to Equation 1; notice that Equation 7 includes ui, a sto-
chastic error term. This implies that although the relationship between X1 and
X2 might be fairly strong, it is not strong enough to allow X1 to be completely
explained by X2; some unexplained variation still remains. Figure 2 shows the
graph of two explanatory variables that might be considered imperfectly mul-
ticollinear. Notice that although all the observations in the sample are fairly
close to the straight line, there is still some variation in X1 that cannot be
explained by X2.
X1i 5 �0 1 �1X2i 1 ui
264

MULTICOLLINEARITY
Imperfect multicollinearity is a strong linear relationship between the ex-
planatory variables. The stronger the relationship between the two (or more)
explanatory variables, the more likely it is that they’ll be considered signifi-
cantly multicollinear. Two variables that might be only slightly related in one
sample might be so strongly related in another that they could be considered
to be imperfectly multicollinear. In this sense, it is fair to say that multi-
collinearity is a sample phenomenon as well as a theoretical one. This con-
trasts with perfect multicollinearity because two variables that are perfectly
related probably can be detected on a logical basis. The detection of multi-
collinearity will be discussed in more detail in Section 3.
The Consequences of Multicollinearity
If the multicollinearity in a particular sample is severe, what will happen to
estimates calculated from that sample? The purpose of this section is to ex-
plain the consequences of multicollinearity and then to explore some exam-
ples of such consequences.
Recall the properties of OLS estimators that might be affected by this or some
other econometric problem. We stated that the OLS estimators are
2
X1
0 X2
Figure 2 Imperfect Multicollinearity
With imperfect multicollinearity, an independent variable is a strong but not perfect
linear function of one or more other independent variables. Imperfect multicollinearity
varies in degree from sample to sample.
265

MULTICOLLINEARITY
BLUE (or MvLUE) if the Classical Assumptions hold. This means that OLS esti-
mates can be thought of as being unbiased and having the minimum variance
possible for unbiased linear estimators.
What Are the Consequences of Multicollinearity?
The major consequences of multicollinearity are:
1. Estimates will remain unbiased. Even if an equation has significant multi-
collinearity, the estimates of the still will be centered around the
true population if all the Classical Assumptions are met for a cor-
rectly specified equation.
2. The variances and standard errors of the estimates will increase. This is the
principal consequence of multicollinearity. Since two or more of the
explanatory variables are significantly related, it becomes difficult to
precisely identify the separate effects of the multicollinear variables.
When it becomes hard to distinguish the effect of one variable from the
effect of another, then we’re much more likely to make large errors in
estimating the than we were before we encountered multicollinear-
ity. As a result, the estimated coefficients, although still unbiased, now
come from distributions with much larger variances and, therefore,
larger standard errors.2
Figure 3 compares a distribution of from a sample with severe
multicollinearity to one with virtually no correlation between any of
the independent variables. Notice that the two distributions have the
same mean, indicating that multicollinearity does not cause bias. Also
note how much wider the distribution of becomes when multi-
collinearity is severe; this is the result of the increase in the standard
error of that is caused by multicollinearity.
Because of this larger variance, multicollinearity increases the likeli-
hood of obtaining an unexpected sign3 for a coefficient even though, as
mentioned earlier, multicollinearity causes no bias.
�̂
�̂
�̂s
�s
�s
�s
2. Even though the variances and standard errors are larger with multicollinearity than they are
without it, OLS is still BLUE when multicollinearity exists. That is, no other linear unbiased esti-
mation technique can get lower variances than OLS even in the presence of multicollinearity. Thus,
although the effect of multicollinearity is to increase the variance of the estimated coefficients, OLS
still has the property of minimum variance. These “minimum variances” are just fairly large.
3. These unexpected signs generally occur because the distribution of the with multi-
collinearity is wider than without it, increasing the chance that a particular observed will be
on the other side of zero from the true (have an unexpected sign).�
�̂
�̂s
266

MULTICOLLINEARITY
3. The computed t-scores will fall. Multicollinearity tends to decrease the
t-scores of the estimated coefficients mainly because of the formula for
the t-statistic:
(8)
Notice that this equation is divided by the standard error of the esti-
mated coefficient. Multicollinearity increases the standard error of the
estimated coefficient, and if the standard error increases, then the
t-score must fall, as can be seen from Equation 8. Not surprisingly, it’s
quite common to observe low t-scores in equations with severe
multicollinearity.
4. Estimates will become very sensitive to changes in specification. The addition
or deletion of an explanatory variable or of a few observations will
tk 5
(�̂
k
2 �̂
H
0
)
SE(�̂k)
� �
With Severe
Multicollinearity
Without Severe
Multicollinearity
Figure 3 Severe Multicollinearity Increases the Variances of the
Severe multicollinearity produces a distribution of the that is centered around the
true but that has a much wider variance. Thus, the distribution of with multi-
collinearity is much wider than otherwise.
�̂s�
�̂s
�̂s
267

MULTICOLLINEARITY
often cause major changes in the values of the when significant
multicollinearity exists. If you drop a variable, even one that appears to
be statistically insignificant, the coefficients of the remaining variables
in the equation sometimes will change dramatically.
These large changes occur because OLS estimation is sometimes
forced to emphasize small differences between variables in order to
distinguish the effect of one multicollinear variable from another. If
two variables are virtually identical throughout most of the sample, the
estimation procedure relies on the observations in which the variables
move differently in order to distinguish between them. As a result, a
specification change that drops a variable that has an unusual value for
one of these crucial observations can cause the estimated coefficients of
the multicollinear variables to change dramatically.
5. The overall fit of the equation and the estimation of the coefficients of non-
multicollinear variables will be largely unaffected. Even though the individ-
ual t-scores are often quite low in a multicollinear equation, the overall
fit of the equation, as measured by , will not fall much, if at all, in
the face of significant multicollinearity. Given this, one of the first indi-
cations of severe multicollinearity is the combination of a high with
no statistically significant individual regression coefficients. Similarly,
if an explanatory variable in an equation is not multicollinear with the
other variables, then the estimation of its coefficient and standard error
usually will not be affected.
Because multicollinearity has little effect on the overall fit of the
equation, it will also have little effect on the use of that equation for
prediction or forecasting, as long as the independent variables main-
tain the same pattern of multicollinearity in the forecast period that
they demonstrated in the sample.
Two Examples of the Consequences of Multicollinearity
To see what severe multicollinearity does to an estimated equation, let’s look
at a hypothetical example. Suppose you decide to estimate a “student
consumption function.” After the appropriate preliminary work, you come
up with the following hypothesized equation:
(9)
where: COi � the annual consumption expenditures of the ith student
on items other than tuition and room and board
COi 5 f(Yd
1
i, LA
1
i) 1 �i 5 �0 1 �1Ydi 1 �2LAi 1 �i
R2
R2
�̂s
268

MULTICOLLINEARITY
Ydi � the annual disposable income (including gifts) of that
student
LAi � the liquid assets (savings, etc.) of the ith student
� a stochastic error term
You then collect a small amount of data from people who are sitting near you
in class:
Student COi Ydi LAi
Mary $2000 $2500 $25000
Robby 2300 3000 31000
Jim 2800 3500 33000
Lesley 3800 4000 39000
Sita 3500 4500 48000
Jerry 5000 5000 54000
Harwood 4500 5500 55000
Datafile � CONS8
If you run an OLS regression on your data set for Equation 9, you obtain:
(10)
On the other hand, if you had consumption as a function of disposable in-
come alone, then you would have obtained:
(11)
Notice from Equations 10 and 11 that the t-score for disposable income
increases more than tenfold when the liquid assets variable is dropped
from the equation. Why does this happen? First of all, the simple correla-
tion coefficient between Yd and LA is quite high: rYd,LA � .986. This high
degree of correlation causes the standard errors of the estimated coeffi-
cients to be very high when both variables are included. In the case of
the standard error goes from 0.157 to 1.03 with the inclusion of LA!�̂Yd,
R2 5 .861
t 5 6.187
(0.157)
COi 5 2 471.43 1 0.9714Ydi
R2 5 .835
t 5 0.496 0.453
(1.0307) (0.0942)
COi 5 2 367.83 1 0.5113Ydi 1 0.0427LAi
�i
269

MULTICOLLINEARITY
In addition, the coefficient estimate itself changes somewhat. Further, note
that the of the two equations are quite similar despite the large differ-
ences in the significance of the explanatory variables in the two equations.
It’s quite common for to stay virtually unchanged when multicollinear
variables are dropped. All of these results are typical of equations with
multicollinearity.
Which equation is better? If the liquid assets variable theoretically belongs
in the equation, then to drop it will run the risk of omitted variable bias, but
to include the variable will mean certain multicollinearity. There is no auto-
matic answer when dealing with multicollinearity. We’ll discuss this issue in
more detail in Sections 4 and 5.
A second example of the consequences of multicollinearity is based on ac-
tual rather than hypothetical data. Suppose you’ve decided to build a cross-
sectional model of the demand for gasoline by state:
(12)
where: PCONi � petroleum consumption in the ith state (trillions of
BTUs)
UHMi � urban highway miles within the ith state
TAXi � the gasoline tax rate in the ith state (cents per gallon)
REGi � motor vehicle registrations in the ith state (thousands)
Given the definitions, let’s move on to the estimation of Equation 12
using a linear functional form (assuming a classical error term):
(13)
What’s wrong with this equation? The motor vehicle registrations variable
has an insignificant coefficient with an unexpected sign, but it’s hard to
believe that the variable is irrelevant. Is an omitted variable causing bias?
It’s possible, but adding a variable is unlikely to fix things. Does it help
to know that the simple correlation coefficient between REG and UHM is
0.98? Given that, it seems fair to say that one of the two variables is
redundant; both variables are really measuring the size of the state, so we have
multicollinearity.
N 5 50 R2 5 .919
t 5 5.92 2 2.77 2 1.43
(10.3) (13.2) (0.043)
PCONi 5 389.6 1 60.8UHMi 2 36.5TAXi 2 0.061REGi
PCONi 5 f(UHM
1
i, TAX
2
i, REG
1
i) 1 �i
R2
R2s
270

MULTICOLLINEARITY
Notice the impact of the multicollinearity on the equation. The coefficient
of a variable such as motor vehicle registrations, which has a very strong the-
oretical relationship to petroleum consumption, is insignificant and has a
sign contrary to our expectations. This is mainly because the multicolline-
arity has increased the variance of the distribution of the estimated
What would happen if we were to drop one of the multicollinear variables?
(14)
Dropping UHM has made REG extremely significant. Why did this occur?
The answer is that the standard error of the coefficient of REG has fallen
substantially (from 0.043 to 0.012) now that the multicollinearity has
been removed from the equation. Also note that the sign of the estimated
coefficient has now become positive as hypothesized. The reason is that
REG and UHM are virtually indistinguishable from an empirical point of
view, and so the OLS program latched onto minor differences between the
variables to explain the movements of PCON. Once the multicollinearity
was removed, the direct positive relationship between REG and PCON was
obvious.
Either UHM or REG could have been dropped with similar results because
the two variables are, in a quantitative sense, virtually identical. In this case,
REG was judged to be theoretically superior to UHM. Even though fell
when UHM was dropped, Equation 14 should be considered superior to
Equation 13. This is an example of the point that the fit of the equation is
not the most important criterion to be used in determining its overall quality.
The Detection of Multicollinearity
How do we decide whether an equation has a severe multicollinearity prob-
lem? A first step is to recognize that some multicollinearity exists in every
equation. It’s virtually impossible in a real-world example to find a set of ex-
planatory variables that are totally uncorrelated with each other (except for
designed experiments). Our main purpose in this section will be to learn to
determine how much multicollinearity exists in an equation, not whether any
multicollinearity exists.
3
R2
N 5 50 R2 5 .861
t 5 2 3.18 15.88
(16.9) (0.012)
PCONi 5 551.7 2 53.6TAXi 1 0.186REGi
�̂s.
271

MULTICOLLINEARITY
A second key point is that the severity of multicollinearity in a given equa-
tion can change from sample to sample depending on the characteristics of the
sample. As a result, the theoretical underpinnings of the equation are not quite
as important in the detection of multicollinearity as they are in the detection of
an omitted variable or an incorrect functional form. Instead, we tend to rely
more on data-oriented techniques to determine the severity of the multi-
collinearity in a given sample. Of course, we can never ignore the theory be-
hind an equation. The trick is to find variables that are theoretically relevant
(for meaningful interpretation) and that are also statistically nonmulti-
collinear (for meaningful inference).
Because multicollinearity is a sample phenomenon, and the level of
damage of its impact is a matter of degree, many of the methods used to
detect it are informal tests without critical values or levels of significance.
Indeed, there are no generally accepted, true statistical tests for multi-
collinearity. Most researchers develop a general feeling for the severity of
multicollinearity in an estimated equation by looking at a number of the
characteristics of that equation. Let’s examine two of the most-used of
those characteristics.
High Simple Correlation Coefficients
One way to detect severe multicollinearity is to examine the simple correla-
tion coefficients between the explanatory variables. If an r is high in ab-
solute value, then we know that these two particular Xs are quite correlated
and that multicollinearity is a potential problem. For example, in Equation
10, the simple correlation coefficient between disposable income and liquid
assets is 0.986. A simple correlation coefficient this high, especially in an
equation with only two independent variables, is a certain indication of se-
vere multicollinearity.
How high is high? Some researchers pick an arbitrary number, such as
0.80, and become concerned about multicollinearity any time the absolute
value of a simple correlation coefficient exceeds 0.80. A better answer might
be that r is high if it causes unacceptably large variances in the coefficient es-
timates in which we’re interested.
Be careful; the use of simple correlation coefficients as an indication of the
extent of multicollinearity involves a major limitation if there are more than
two explanatory variables. It is quite possible for groups of independent vari-
ables, acting together, to cause multicollinearity without any single simple
correlation coefficient being high enough to indicate that multicollinearity is
in fact severe. As a result, simple correlation coefficients must be considered
to be sufficient but not necessary tests for multicollinearity. Although a high r
272

MULTICOLLINEARITY
does indeed indicate the probability of severe multicollinearity, a low r by no
means proves otherwise.4
High Variance Inflation Factors (VIFs)
The use of tests to give an indication of the severity of multicollinearity in a
particular sample is controversial. Some econometricians reject even the sim-
ple indicator described previously, mainly because of the limitations cited.
Others tend to use a variety of more formal tests.5
One measure of the severity of multicollinearity that is easy to use and that
is gaining in popularity is the variance inflation factor. The variance infla-
tion factor (VIF) is a method of detecting the severity of multicollinearity by
looking at the extent to which a given explanatory variable can be explained
by all the other explanatory variables in the equation. There is a VIF for each
explanatory variable in an equation. The VIF is an index of how much multi-
collinearity has increased the variance of an estimated coefficient. A high VIF
indicates that multicollinearity has increased the estimated variance of the es-
timated coefficient by quite a bit, yielding a decreased t-score.
Suppose you want to use the VIF to attempt to detect multicollinearity in
an original equation with K independent variables:
Doing so requires calculating K different VIFs, one for each Xi. Calculating
the VIF for a given Xi involves two steps:
1. Run an OLS regression that has Xi as a function of all the other explanatory
variables in the equation. For i � 1, this equation would be:
(15)
where v is a classical stochastic error term. Note that X1 is not included
on the right-hand side of Equation 15, which is referred to as an
X1 5 �1 1 �2X2 1 �3X3 1
c1 �KXK 1 v
Y 5 �0 1 �1X1 1 �2X2 1
c1 �KXK 1 �
4. Most authors criticize the use of simple correlation coefficients to detect multicollinearity in
equations with large numbers of explanatory variables, but many researchers continue to do so
because a scan of the simple correlation coefficients is a “quick and dirty” way to get a feel for
the degree of multicollinearity in an equation.
5. Perhaps the best of these is the Condition number. For more on the Condition number,
which is a single index of the degree of multicollinearity in the overall equation, see D. A. Belsley,
Conditioning Diagnostics (New York: Wiley, 1991).
273

MULTICOLLINEARITY
(16)VIF(�̂i) 5
1
(1 2 R
2
i )
auxiliary or secondary regression. Thus there are K auxiliary regressions,
one for each independent variable in the original equation.
2. Calculate the variance inflation factor for :�̂i
where is the coefficient of determination (the unadjusted R2) of the
auxiliary regression in step one. Since there is a separate auxiliary regres-
sion for each independent variable in the original equation, there also is
an The higher the VIF, the more severe the
effects of multicollinearity.
How high is high? An of 1, indicating perfect multicollinearity, pro-
duces a VIF of infinity, whereas an of 0, indicating no multicollinearity at
all, produces a VIF of 1. While there is no table of formal critical VIF values, a
common rule of thumb is that if the multicollinearity is severe.
As the number of independent variables increases, it makes sense to increase
this number slightly.
For example, let’s return to Equation 10 and calculate the VIFs for both inde-
pendent variables. Both VIFs equal 36, confirming the quite severe multi-
collinearity we already know exists. It’s no coincidence that the VIFs for the two
variables are equal. In an equation with exactly two independent variables, the
two auxiliary equations will have identical leading to equal VIFs.6
Some authors and statistical software programs replace the VIF with its
reciprocal, called tolerance, or TOL. Whether we calculate VIF or
TOL is a matter of personal preference, but either way, the general approach
is the most comprehensive multicollinearity detection technique we’ve dis-
cussed in this text.
Unfortunately, there are a couple of problems with using VIFs. First, as
mentioned, there is no hard-and-fast VIF decision rule. Second, it’s possible
to have multicollinear effects in an equation that has no large VIFs. For in-
stance, if the simple correlation coefficient between X1 and X2 is 0.88, multi-
collinear effects are quite likely, and yet the VIF for the equation (assuming
no other Xs) is only 4.4.
(1 2 R2i ),
R2i s,
VIF(�i) . 5,
R2i
R2i
R2i and a VIF(�̂i) for each Xi.
R2i
6. Another use for the R2s of these auxiliary equations is to compare them with the overall
equation’s R2. If an auxiliary equation’s R2 is higher, it’s yet another sign of multicollinearity.
274

MULTICOLLINEARITY
In essence, then, the VIF is a sufficient but not necessary test for multi-
collinearity, just like the other test described in this section. Indeed, as is
probably obvious to the reader by now, there is no test that allows a re-
searcher to reject the possibility of multicollinearity with any real certainty.
Remedies for Multicollinearity
What can be done to minimize the consequences of severe multicollinearity?
There is no automatic answer to this question because multicollinearity is a
phenomenon that could change from sample to sample even for the same
specification of a regression equation. The purpose of this section is to out-
line a number of alternative remedies for multicollinearity that might be ap-
propriate under certain circumstances.
Do Nothing
The first step to take once severe multicollinearity has been diagnosed is to
decide whether anything should be done at all. As we’ll see, it turns out that
every remedy for multicollinearity has a drawback of some sort, and so it
often happens that doing nothing is the correct course of action.
One reason for doing nothing is that multicollinearity in an equation will
not always reduce the t-scores enough to make them insignificant or change
the enough to make them differ from expectations. In other words, the mere
existence of multicollinearity does not necessarily mean anything. A remedy
for multicollinearity should be considered only if the consequences cause in-
significant t-scores or unreliable estimated coefficients. For example, it’s possi-
ble to observe a simple correlation coefficient of .97 between two explanatory
variables and yet have each individual t-score be significant. It makes no sense
to consider remedial action in such a case, because any remedy for multi-
collinearity would probably cause other problems for the equation. In a sense,
multicollinearity is similar to a non-life-threatening human disease that re-
quires general anesthesia to operate on the patient: The risk of the operation
should be undertaken only if the disease is causing a significant problem.
A second reason for doing nothing is that the deletion of a multicollinear vari-
able that belongs in an equation will cause specification bias. If we drop such a
variable, then we are purposely creating bias. Given all the effort typically spent
avoiding omitted variables, it seems foolhardy to consider running that risk on
purpose. As a result, experienced econometricians often will leave multicollinear
variables in equations despite low t-scores.
�̂s
4
275

MULTICOLLINEARITY
The final reason for considering doing nothing to offset multicollinearity
is that every time a regression is rerun, we risk encountering a specification
that fits because it accidentally works for the particular data set involved, not
because it is the truth. The larger the number of experiments, the greater the
chances of finding the accidental result. To make things worse, when there is
significant multicollinearity in the sample, the odds of strange results in-
crease rapidly because of the sensitivity of the coefficient estimates to slight
specification changes.
To sum, it is often best to leave an equation unadjusted in the face of all
but extreme multicollinearity. Such advice might be difficult for beginning
researchers to take, however, if they think that it’s embarrassing to report that
their final regression is one with insignificant t-scores. Compared to the alter-
natives of possible omitted variable bias or accidentally significant regression
results, the low t-scores seem like a minor problem. For an example of “doing
nothing” in the face of severe multicollinearity, see Section 5.
Drop a Redundant Variable
On occasion, the simple solution of dropping one of the multicollinear vari-
ables is a good one. For example, some inexperienced researchers include too
many variables in their regressions, not wanting to face omitted variable bias.
As a result, they often have two or more variables in their equations that are
measuring essentially the same thing. In such a case the multicollinear vari-
ables are not irrelevant, since any one of them is quite probably theoretically
and statistically sound. Instead, the variables might be called redundant;
only one of them is needed to represent the effect on the dependent variable
that all of them currently represent. For example, in an aggregate demand
function, it would not make sense to include disposable income and GDP
because both are measuring the same thing: income. A bit more subtle is the
inference that population and disposable income should not both be in-
cluded in the same aggregate demand function because, once again, they
really are measuring the same thing: the size of the aggregate market. As pop-
ulation rises, so too will income. Dropping these kinds of redundant multi-
collinear variables is doing nothing more than making up for a specification
error; the variables should never have been included in the first place.
To see how this solution would work, let’s return to the student consump-
tion function example of Equation 10:
(10)
t 5 0.496 0.453 R2 5 .835
(1.0307) (0.0942)
COi 5 2367.83 1 0.5113Ydi 1 0.0427LAi
276

MULTICOLLINEARITY
where CO � consumption, Yd � disposable income, and LA � liquid assets.
When we first discussed this example, we compared this result to the same
equation without the liquid assets variable:
(11)
If we had instead dropped the disposable income variable, we would have
obtained:
(17)
Note that dropping one of the multicollinear variables has eliminated both the
multicollinearity between the two explanatory variables and also the low t-score
of the coefficient of the remaining variable. By dropping Yd, we were able to in-
crease tLA from 0.453 to 6.153. Since dropping a variable changes the meaning
of the remaining coefficient (because the dropped variable is no longer being
held constant), such dramatic changes are not unusual. The coefficient of the re-
maining included variable also now measures almost all of the joint impact on
the dependent variable of the multicollinear explanatory variables.
Assuming you want to drop a variable, how do you decide which variable
to drop? In cases of severe multicollinearity, it makes no statistical difference
which variable is dropped. As a result, it doesn’t make sense to pick the vari-
able to be dropped on the basis of which one gives superior fit or which one
is more significant (or has the expected sign) in the original equation. In-
stead, the theoretical underpinnings of the model should be the basis for
such a decision. In the example of the student consumption function, there is
more theoretical support for the hypothesis that disposable income deter-
mines consumption than there is for the liquid assets hypothesis. Therefore,
Equation 11 should be preferred to Equation 17.
Increase the Size of the Sample
Another way to deal with multicollinearity is to attempt to increase the size of
the sample to reduce the degree of multicollinearity. Although such an increase
may be impossible, it’s a useful alternative to be considered when feasible.
The idea behind increasing the size of the sample is that a larger data set
(often requiring new data collection) will allow more accurate estimates than
t 5 6.153 R2 5 .860
(0.01443)
COi 5 2199.44 1 0.08876LAi
t 5 6.187 R2 5 .861
(0.157)
COi 5 2471.43 1 0.9714Ydi
277

MULTICOLLINEARITY
a small one, since the larger sample normally will reduce the variance of the
estimated coefficients, diminishing the impact of the multicollinearity.
For most time series data sets, however, this solution isn’t feasible. After
all, samples typically are drawn by getting all the available data that seem
similar. As a result, new data are generally impossible or quite expensive to
find. Going out and generating new data is much easier with a cross-sectional
or experimental data set than it is when the observations must be generated
by the passage of time.
An Example of Why Multicollinearity Often
Is Best Left Unadjusted
Let’s look at an example of the idea that multicollinearity often should be left
unadjusted. Suppose you work in the marketing department of a hypotheti-
cal soft drink company and you build a model of the impact on sales of your
firm’s advertising:
(18)
where: St � sales of the soft drink in year t
Pt � average relative price of the drink in year t
At � advertising expenditures for the company in year t
Bt � advertising expenditures for the company’s main competitor
in year t
Assume that there are no omitted variables. All variables are measured in real
dollars; that is, the nominal values are divided, or deflated, by a price index.
On the face of it, this is a reasonable-looking result. Estimated coefficients
are significant in the directions implied by the underlying theory, and both
the overall fit and the size of the coefficients seem acceptable. Suppose you
now were told that advertising in the soft drink industry is cut-throat in na-
ture and that firms tend to match their main competitor’s advertising expen-
ditures. This would lead you to suspect that significant multicollinearity was
possible. Further suppose that the simple correlation coefficient between the
two advertising variables is .974 and that their respective VIFs are well over 5.
Such a correlation coefficient is evidence that there is severe multi-
collinearity in the equation, but there is no reason even to consider doing
R2 5 .825 N 5 28
t 5 2 3.00 3.99 2 2.04
(25,000) (1.06) (0.51)
Ŝt 5 3080 2 75,000Pt 1 4.23At 2 1.04Bt
5
278

MULTICOLLINEARITY
anything about it, because the coefficients are so powerful that their t-scores
remain significant, even in the face of severe multicollinearity. Unless multi-
collinearity causes problems in the equation, it should be left unadjusted. To
change the specification might give us better-looking results, but the adjust-
ment would decrease our chances of obtaining the best possible estimates of
the true coefficients. Although it’s certainly lucky that there were no major
problems due to multicollinearity in this example, that luck is no reason to
try to fix something that isn’t broken.
When a variable is dropped from an equation, its effect will be absorbed
by the other explanatory variables to the extent that they are correlated with
the newly omitted variable. It’s likely that the remaining multicollinear vari-
able(s) will absorb virtually all the bias, since the variables are highly corre-
lated. This bias may destroy whatever usefulness the estimates had before the
variable was dropped.
For example, if a variable, say B, is dropped from the equation to fix the
multicollinearity, then the following might occur:
(19)
What’s going on here? The company’s advertising coefficient becomes less in-
stead of more significant when one of the multicollinear variables is
dropped. To see why, first note that the expected bias on is negative be-
cause the product of the expected sign of the coefficient of B and of the corre-
lation between A and B is negative:
(20)
Second, this negative bias is strong enough to decrease the estimated coeffi-
cient of A until it is insignificant. Although this problem could have been
avoided by using a relative advertising variable (A divided by B, for instance),
that formulation would have forced identical absolute coefficients on A and
1/B. Such identical coefficients will sometimes be theoretically expected or
empirically reasonable but, in most cases, these kinds of constraints will
force bias onto an equation that previously had none.
This example is simplistic, but its results are typical in cases in which
equations are adjusted for multicollinearity by dropping a variable with-
out regard to the effect that the deletion is going to have. The point here
Bias 5 �B ? f(rA,B) 5 (2) ? (1) 5 2
�̂A
R2 5 .531 N 5 28
t 5 2 3.25 0.12
(24,000) (4.32)
Ŝt 5 2586 2 78,000Pt 1 0.52A t
279

MULTICOLLINEARITY
is that it’s quite often theoretically or operationally unwise to drop a vari-
able from an equation and that multicollinearity in such cases is best left
unadjusted.
Summary
1. Perfect multicollinearity is the violation of the assumption that no ex-
planatory variable is a perfect linear function of other explanatory
variable(s). Perfect multicollinearity results in indeterminate esti-
mates of the regression coefficients and infinite standard errors of
those estimates.
2. Imperfect multicollinearity, which is what is typically meant when the
word “multicollinearity” is used, is a linear relationship between two
or more independent variables that is strong enough to significantly
affect the estimation of that equation. Multicollinearity is a sample
phenomenon as well as a theoretical one. Different samples can ex-
hibit different degrees of multicollinearity.
3. The major consequence of severe multicollinearity is to increase the
variances of the estimated regression coefficients and therefore de-
crease the calculated t-scores of those coefficients. Multicollinearity
causes no bias in the estimated coefficients, and it has little effect on
the overall significance of the regression or on the estimates of the co-
efficients of any nonmulticollinear explanatory variables.
4. Since multicollinearity exists, to one degree or another, in virtually
every data set, the question to be asked in detection is how severe the
multicollinearity in a particular sample is.
5. Two useful methods for the detection of severe multicollinearity are:
a. Are the simple correlation coefficients between the explanatory
variables high?
b. Are the variance inflation factors high?
If either of these answers is yes, then multicollinearity certainly exists,
but multicollinearity can also exist even if the answers are no.
6. The three most common remedies for multicollinearity are:
a. Do nothing (and thus avoid specification bias).
b. Drop a redundant variable.
c. Increase the size of the sample.
6
280

MULTICOLLINEARITY
7. Quite often, doing nothing is the best remedy for multicollinearity. If
the multicollinearity has not decreased t-scores to the point of in-
significance, then no remedy should even be considered. Even if the
t-scores are insignificant, remedies should be undertaken cautiously,
because all impose costs on the estimation that may be greater than
the potential benefit of ridding the equation of multicollinearity.
EXERCISES
(The answer to Exercise 2 appears of the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or your notes), and then compare your definition with the
version in the text for each:
a. perfect multicollinearity
b. severe imperfect multicollinearity
c. dominant variable
d. auxiliary (or secondary) equation
e. variance inflation factor
f. redundant variable
2. A recent study of the salaries of elementary school teachers in a small
school district in Northern California came up with the following es-
timated equation (note: t-scores in parentheses!):
(21)
where: SALi � the salary of the ith teacher (in dollars)
EMPi � the years that the ith teacher has worked in this
school district
UNITSi � the units of graduate work completed by the ith
teacher
LANGi � a dummy variable equal to 1 if the ith teacher
speaks two languages
EXPi � the total years of teaching experience of the ith
teacher
R
—

2
5 .866 N 5 25
(20.98) (2.39) (2.08) (4.97)
lnSALi 5 10.5 2 0.006EMPi 1 0.002UNITSi 1 0.079LANGi 1 0.020EXPi
281

MULTICOLLINEARITY
a. Make up and test appropriate hypotheses for the coefficients of this
equation at the 5-percent level.
b. What is the functional form of this equation? Does it seem appro-
priate? Explain.
c. What econometric problems (out of irrelevant variables, omitted
variables, and multicollinearity) does this equation appear to have?
Explain.
d. Suppose that you now are told that the simple correlation coeffi-
cient between EMP and EXP is .89 and that the VIFs for EMP and
EXP are both just barely over 5. Does this change your answer to
part c above? How?
e. What remedy for the problem you identify in part d do you recom-
mend? Explain.
f. If you drop EMP from the equation, the estimated equation be-
comes Equation 22. Use our four specification criteria to decide
whether you prefer Equation 21 or Equation 22. Which do you like
better? Why?
(22)
3. A researcher once attempted to estimate an asset demand equation
that included the following three explanatory variables: current
wealth Wt, wealth in the previous quarter and the change in
wealth What problem did this researcher en-
counter? What should have been done to solve this problem?
4. In each of the following situations, determine whether the variable in-
volved is a dominant variable:
a. games lost in year t in an equation for the number of games won in
year t by a baseball team that plays the same number of games each
year
b. number of Woody’s restaurants in a model of the total sales of the
entire Woody’s chain of restaurants
c. disposable income in an equation for aggregate consumption ex-
penditures
d. number of tires purchased in an annual model of the number of
automobiles produced by an automaker that does not make its
own tires
e. number of acres planted in an agricultural supply function
�Wt 5 Wt 2 Wt21.
Wt21,
R
—

2
5 .871 N 5 25
(2.47) (2.09) (8.65)
lnSALi 5 10.5 1 0.002UNITSi 1 0.081LANGi 1 0.015EXPi
282

MULTICOLLINEARITY
5. Beginning researchers quite often believe that they have multi-
collinearity when they’ve accidentally included in their equation two
or more explanatory variables that basically serve the same purpose or
are, in essence, measuring the same thing. Which of the following
pairs of variables are likely to include such a redundant variable?
a. GDP and NDP in a macroeconomic equation of some sort
b. the price of refrigerators and the price of washing machines in a
durable goods demand function
c. the number of acres harvested and the amount of seed used in an
agricultural supply function
d. long-term interest rates and the money supply in an investment
function
6. You’ve been hired by the Dean of Students Office to help reduce dam-
age done to dorms by rowdy students, and your first step is to build a
cross-sectional model of last term’s damage to each dorm as a func-
tion of the attributes of that dorm (standard errors in parentheses):
where: Di � the amount of damage (in dollars) done to the ith
dorm last term
Fi � the percentage of the ith dorm residents who are
frosh
Si � the number of students who live in the ith dorm
Ai � the number of incidents involving alcohol that were
reported to the Dean of Students Office from the ith
dorm last term (incidents involving alcohol may or
may not involve damage to the dorm)
a. Hypothesize signs, calculate t-scores, and test hypotheses for this
result (5-percent level).
b. What problems (omitted variables, irrelevant variables, or multi-
collinearity) appear to exist in this equation? Why?
c. Suppose you were now told that the simple correlation coefficient
between Si and Ai was .94; would that change your answer? How?
d. Is it possible that the unexpected sign of could have been caused
by multicollinearity? Why?
7. Suppose that your friend was modeling the impact of income on con-
sumption in a quarterly model and discovered that income’s impact
�̂s
N 5 33 R2 5 .84
(253) (0.752) (12.4)
D̂i 5 210 1 733Fi 2 0.805Si 1 74.0Ai
283

MULTICOLLINEARITY
on consumption lasts at least a year. As a result, your friend estimated
the following model:
a. Would this equation be subject to perfect multicollinearity?
b. Would this equation be subject to imperfect multicollinearity?
c. What, if anything, could be done to rid this equation of any mul-
ticollinearity it might have? (One answer to this question, the
autoregressive approach to distributed lags, will be covered in
Chapter 12.)
8. In 1998, Mark McGwire hit 70 homers to break Roger Maris’s old
record of 61, and yet McGwire wasn’t voted the Most Valuable Player
(MVP) in his league. To try to understand how this happened, you
collect the following data on MVP votes, batting average (BA), home
runs (HR), and runs batted in (RBI) from the 1998 National League:
Name Votes (V) BA HR RBI
Sosa 438 .308 66 158
McGwire 272 .299 70 147
Alou 215 .312 38 124
Vaughn 185 .272 50 119
Biggio 163 .325 20 88
Galarraga 147 .305 44 121
Bonds 66 .303 37 122
Jones 56 .313 34 107
Datafile � MVP8
Just as you are about to run the regression, your friend (trying to get
back at you for your comments on Exercise 7) warns you that you
probably have multicollinearity.
a. What should you do about your friend’s warning before running
the regression?
b. Run the regression implied in this example:
on the data given. What signs of multicollinearity are there?
c. What suggestions would you make for another run of this equa-
tion? In particular, what would you do about multicollinearity?
9. A full-scale regression model for the total annual gross sales in thou-
sands of dollars of J. C. Quarter’s durable goods for the last 26 years
V 5 f(BA
1
, HR
1
, RBI
1
) 1 �
Ct 5 �0 1 �1Ydt 1 �2Ydt21 1 �3Ydt22 1 �4Ydt23 1 �t
284

MULTICOLLINEARITY
produces the following result (all measurements are in real dollars—
or billions of real dollars; standard errors in parentheses):
where: SQt � sales of durable goods at J. C. Quarter’s in year t
PCt � average price of durables in year t at J. C. Quarter’s
main competition
PQt � the average price of durables at J. C. Quarter’s in
year t
Yt � U.S. gross domestic product in year t
Ct � U.S. aggregate consumption in year t
Nt � the number of J. C. Quarter’s stores open in year t
a. Hypothesize signs, calculate t-scores, and test hypotheses for this
result (5-percent level).
b. What problems (out of omitted variables, irrelevant variables, and
multicollinearity) appear to exist in this equation? Explain.
c. Suppose you were now told that the was .821, that rY,C was .993,
and that rPC,PQ was .813. Would this change your answer to the
previous question? How?
d. What recommendation would you make for a rerun of this equa-
tion with different explanatory variables? Why?
10. A cross-sectional regression was run on a sample of 44 states in an ef-
fort to understand federal defense spending by state (standard errors
in parentheses):
where: Si � annual spending (millions of dollars) on defense in
the ith state
Ci � contracts (millions of dollars) awarded in the ith state
(contracts are often for many years of service) per year
Pi � annual payroll (millions of dollars) for workers in
defense-oriented industries in the ith state
Ei � the number of civilians employed in defense-oriented
industries in the ith state
(0.027) (0.1664) (0.0092)
Ŝi 5 2 148.0 1 0.841Ci 2 0.0115Pi 2 0.0078Ei
R2
(10.6) (103.8)
2 15.8Ct 1 201.1Nt
(250.1) (125.6) (40.1)
SQt 5 2 7.2 1 200.3PCt 2 150.6PQt 1 20.6Yt
285

MULTICOLLINEARITY
a. Hypothesize signs, calculate t-scores, and test hypotheses for this
result (5-percent level).
b. The VIFs for this equation are all above 20, and those for P and C are
above 30. What conclusion does this information allow you to draw?
c. What recommendation would you make for a rerun of this equa-
tion with a different specification? Explain your answer.
11. Consider the following regression result paraphrased from a study
conducted by the admissions office at the Stanford Business School
(standard errors in parentheses):
where: Gi � the Stanford Business School GPA of the ith student
(4 � high)
Mi � the score on the graduate management admission
test of the ith student (800 � high)
Bi � the number of years of business experience of the ith
student
Ai � the age of the ith student
Si � dummy equal to 1 if the ith student was an econom-
ics major, 0 otherwise
a. Theorize the expected signs of all the coefficients (try not to look at
the results) and test these expectations with appropriate hypothe-
ses (including choosing a significance level).
b. Do any problems appear to exist in this equation? Explain your
answer.
c. How would you react if someone suggested a polynomial func-
tional form for A? Why?
d. What suggestions (if any) would you have for another run of this
equation?
12. Calculating VIFs typically involves running sets of auxiliary regres-
sions, one regression for each independent variable in an equation.
To get practice with this procedure, calculate the following:
a. the VIFs for N, P, and I from the Woody’s data in Table 1 from
Chapter 3
b. the VIFs for PB, PC, and YD from the chicken demand data in
Table 2 from Chapter 6 (using Equation 8 from Chapter 6)
R2 5 0.20 N 5 1000
(0.001) (0.20) (0.10) (0.10)
Ĝi 5 1.00 1 0.005Mi 1 0.20Bi 2 0.10Ai 1 0.25Si
286

MULTICOLLINEARITY
c. the VIF for X1 in an equation where X1 and X2 are the only inde-
pendent variables, given that the VIF for X2 is 3.8 and N � 28
d. the VIF for X1 in an equation where X1 and X2 are the only inde-
pendent variables, given that the simple correlation coefficient be-
tween X1 and X2 is 0.80 and N � 15
13. Let’s take a look at a classic example, a model of the demand for fish
in the United States from 1946 to 1970. This time period is interest-
ing because it includes the Pope’s 1966 decision to allow Catholics to
eat meat on non-Lent Fridays. Before the Pope’s decision, many
Catholics ate fish on Fridays (when they weren’t allowed to eat meat),
and the purpose of the research is to determine whether the Pope’s
decision decreased the demand for fish or simply changed the days of
the week when fish was eaten.
If you use the data in Table 1, you can estimate the following equa-
tion:
(23)
where: Ft � average pounds of fish consumed per capita in year t
PFt � price index for fish in year t
PBt � price index for beef in year t
Ydt � real per capita disposable income in year t (in billions of
dollars)
Pt � a dummy variable equal to 1 after the Pope’s 1966 decision
and 0 otherwise
a. Create and test appropriate hypotheses about the slope coefficients
of Equation 23 at the 5-percent level.
b. What’s going on here? How is it possible to have a reasonably high
and have t-scores of less than 1 for all the slope coefficients?
c. One possibility is an omitted variable, and a friend suggests adding
a variable (N) that measures the number of Catholics in the United
States in year t. Do you agree with this suggestion? Explain your
reasoning.
d. A second possibility is an irrelevant variable, and another friend
suggests dropping P. Do you agree with this suggestion? Explain
your reasoning.
R2
R2 5 .667 N 5 25
t 5 0.98 0.24 0.31 2 0.48
(0.03) (0.019) (1.15) (0.26)
F̂t 5 7.96 1 0.03PFt 1 0.0047PBt 1 0.36 ln Ydt 2 0.12 Pt
287

MULTICOLLINEARITY
e. A third possibility is multicollinearity, and the simple correlation
coefficient of .958 between PF and PB certainly is high! Are the two
price variables redundant? Should you drop one? If so, which one?
Explain your reasoning.
f. (optional) Using the data in Table 1, calculate the VIFs for Equa-
tion 23. Do they support the possibility of multicollinearity?
Explain.
g. You decide to replace the individual price variables with a relative
price variable:
RPt � PFt/PBt
Table 1 Data for the Fish/Pope Example
Year F PF PB N Yd
1946 12.8 56.0 50.1 24402 1606
1947 12.3 64.3 71.3 25268 1513
1948 13.1 74.1 81.0 26076 1567
1949 12.9 74.5 76.2 26718 1547
1950 13.8 73.1 80.3 27766 1646
1951 13.2 83.4 91.0 28635 1657
1952 13.3 81.3 90.2 29408 1678
1953 13.6 78.2 84.2 30425 1726
1954 13.5 78.7 83.7 31648 1714
1955 12.9 77.1 77.1 32576 1795
1956 12.9 77.0 74.5 33574 1839
1957 12.8 78.0 82.8 34564 1844
1958 13.3 83.4 92.2 36024 1831
1959 13.7 84.9 88.8 39505 1881
1960 13.2 85.0 87.2 40871 1883
1961 13.7 86.9 88.3 42105 1909
1962 13.6 90.5 90.1 42882 1969
1963 13.7 90.3 88.7 43847 2015
1964 13.5 88.2 87.3 44874 2126
1965 13.9 90.8 93.9 45640 2239
1966 13.9 96.7 102.6 46246 2335
1967 13.6 100.0 100.0 46864 2403
1968 14.0 101.6 102.3 47468 2486
1969 14.2 107.2 111.4 47873 2534
1970 14.8 118.0 117.6 47872 2610
Source: Historical Statistics of the U.S., Colonial Times to 1970 (Washington, D.C.: U.S. Bureau
of the Census, 1975).
Datafile � FISH8
288

MULTICOLLINEARITY
Such a variable would make sense if theory calls for keeping both
prices in the equation and if the two price coefficients are expected
to be close in absolute value with opposite signs. (Opposite ex-
pected signs are required because an increase in PF will increase RP
while an increase in PB will decrease it.) What is the expected sign
of the coefficient of RP?
h. You replace PF and PB with RP and estimate:
(24)
Which equation do you prefer, Equation 23 or Equation 24? Ex-
plain your reasoning.
i. What’s your conclusion? Did the Pope’s decision reduce the overall
demand for fish?
14. Let’s assume that you were hired by the Department of Agriculture to
do a cross-sectional study of weekly expenditures for food consumed
at home by the ith household (Fi) and that you estimated the follow-
ing equation (standard errors in parentheses):
where: Yi � the weekly disposable income of the ith household
Hi � the number of people in the ith household
Ai � the number of children (under 19) in the ith household
a. Create and test appropriate hypotheses at the 10-percent level.
b. Isn’t the estimated coefficient for Y impossible? (There’s no way that
people can spend twice their income on food.) Explain your answer.
c. Which econometric problems (omitted variables, irrelevant vari-
ables, or multicollinearity) appear to exist in this equation? Ex-
plain your answer.
d. Suppose that you were now told that the VIFs for A and H were both
between 5 and 10. How does this change your answer to part c?
e. Would you suggest changing this specification for one final run of
this equation? How? Why? What are the possible econometric costs
of estimating another specification?
R2 5 .46 N 5 235
(0.7) (.05) (2.0) (2.0)
F̂i 5 2 10.50 1 2.1Yi 2 .04Y
2
i 1 13.0Hi 2 2.0Ai
R2 5 .588 N 5 25
t 5 2 1.35 4.13 0.019
(1.43) (0.66) (0.2801)
F̂t 5 2 5.17 2 1.93RPt 1 2.71 ln Ydt 1 0.0052Pt
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15. Suppose you hear that because of the asymmetry of the human heart,
the heartbeat of any individual is a function of the difference between
the lengths of that individual’s legs rather than of the length of either
leg. You decide to collect data and build a regression model to test
this hypothesis, but you can’t decide which of the following two mod-
els to estimate7:
where: Hi � the heartbeat of the ith cardiac patient
Ri � the length of the ith patient’s right leg
Li � the length of the ith patient’s left leg
a. Model A seems more likely to encounter multicollinearity than
does Model B, at least as measured by the simple correlation coef-
ficient. Why? What remedy for this multicollinearity would you
recommend?
b. Suppose you estimate a set of coefficients for Model A. Can you
calculate estimates of the coefficients of Model B from this infor-
mation? If so, how? If not, why?
c. What does your answer to part b tell you about which of the two
models is more vulnerable to multicollinearity?
d. Suppose you had dropped Li from Model A because of the high
simple correlation coefficient between Li and Ri. What would this
deletion have done to your answers to parts b and c?
Appendix: The SAT Interactive Regression
Learning Exercise
Econometrics is difficult to learn by reading examples, no matter how good
they are. Most econometricians, the author included, had trouble under-
standing how to use econometrics, particularly in the area of specification
choice, until they ran their own regression projects. This is because there’s an
element of econometric understanding that is better learned by doing than by
reading about what someone else is doing.
Unfortunately, mastering the art of econometrics by running your own re-
gression projects without any feedback is also difficult because it takes quite a
7
Model B: Hi 5 �0 1 �1Ri 1 �2(Li 2 Ri) 1 �i
Model A: Hi 5 �0 1 �1Ri 1 �2Li 1 �i
7. Potluri Rao and Roger Miller, Applied Econometrics (Belmont, CA: Wadsworth, 1971), p. 48.
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while to learn to avoid some fairly simple mistakes. Probably the best way to
learn is to work on your own regression project, analyzing your own prob-
lems and making your own decisions, but with a more experienced econo-
metrician nearby to give you one-on-one feedback on exactly which of your
decisions were inspired and which were flawed (and why).
This section is an attempt to give you an opportunity to make indepen-
dent specification decisions and to then get feedback on the advantages or
disadvantages of those decisions. Using the interactive learning exercise of
this section requires neither a computer nor a tutor, although either would
certainly be useful. Instead, we have designed an exercise that can be used on
its own to help to bridge the gap between the typical econometrics examples
(which require no decision making) and the typical econometrics projects
(which give little feedback).
STOP!
To get the most out of the exercise, it’s important to follow the instructions
carefully. Reading the pages in order as with any other example will waste
your time, because once you have seen even a few of the results, the benefits
to you of making specification decisions will diminish. In addition, you
shouldn’t look at any of the regression results until you have specified your
first equation.
Building a Model of Scholastic Aptitude Test Scores
The dependent variable for this interactive learning exercise is the combined
“two-test” SAT score, math plus verbal, earned by students in the senior class
at Arcadia High School. Arcadia is an upper-middle-class suburban commu-
nity located near Los Angeles, California. Out of a graduating class of about
640, a total of 65 students who had taken the SATs were randomly selected
for inclusion in the data set. In cases in which a student had taken the test
more than once, the highest score was recorded.
A review of the literature on the SAT shows many more psychological stud-
ies and popular press articles than econometric regressions. Many articles
have been authored by critics of the SAT, who maintain (among other things)
that it is biased against women and minorities. In support of this argument,
these critics have pointed to national average scores for women and some
minorities, which in recent years have been significantly lower than the na-
tional averages for white males. Any reader interested in reviewing a portion
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of the applicable literature should do so now before continuing on with the
section.8
If you were going to build a single-equation linear model of SAT scores,
what factors would you consider? First, you’d want to include some measures
of a student’s academic ability. Three such variables are cumulative high
school grade point average (GPA) and participation in advanced placement
math and English courses (APMATH and APENG). Advanced placement (AP)
classes are academically rigorous courses that may help a student do well on
the SAT. More important, students are invited to be in AP classes on the basis
of academic potential, and students who choose to take AP classes are reveal-
ing their interest in academic subjects, both of which bode well for SAT
scores. GPAs at Arcadia High School are weighted GPAs; each semester that a
student takes an AP class adds one extra point to his or her total grade points.
(For example, a semester grade of “A” in an AP math class counts for five
grade points as opposed to the conventional four points.)
A second set of important considerations includes qualitative factors that
may affect performance on the SAT. Available dummy variables in this cate-
gory include measures of a student’s gender (GEND), ethnicity (RACE), and
native language (ESL). All of the students in the sample are either Asian or
Caucasian, and RACE is assigned a value of one if a student is Asian. Asian
students are a substantial proportion of the student body at Arcadia High.
The ESL dummy is given a value of one if English is a student’s second lan-
guage. In addition, studying for the test may be relevant, so a dummy vari-
able indicating whether or not a student has attended an SAT preparation
class (PREP) is also included in the data.
To sum, the explanatory variables available for you to choose for your
model are:
GPAi � the weighted GPA of the ith student
APMATHi � a dummy variable equal to 1 if the ith student has taken AP
math, 0 otherwise
APENGi � a dummy variable equal to 1 if the ith student has taken AP
English, 0 otherwise
APi � a dummy variable equal to 1 if the ith student has taken AP
math and/or AP English, 0 if the ith student has taken neither
ESLi � a dummy variable equal to 1 if English is not the ith student’s
first language, 0 otherwise
8. See, for example, James Fallows, “The Tests and the ‘Brightest’: How Fair Are the College
Boards?” The Atlantic, Vol. 245, No. 2, pp. 37–48. We are grateful to former Occidental student
Bob Sego for his help in preparing this interactive exercise.
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RACEi � a dummy variable equal to 1 if the ith student is Asian, 0 if the
student is Caucasian
GENDi � a dummy variable equal to 1 if the ith student is male, 0 if the
student is female
PREPi � a dummy variable equal to 1 if the ith student has attended a
SAT preparation course, 0 otherwise
The data for these variables are presented in Table 2.
Table 2 Data for the SAT Interactive Learning Exercise
SAT GPA APMATH APENG AP ESL GEND PREP RACE
1060 3.74 0 1 1 0 0 0 0
740 2.71 0 0 0 0 0 1 0
1070 3.92 0 1 1 0 0 1 0
1070 3.43 0 1 1 0 0 1 0
1330 4.35 1 1 1 0 0 1 0
1220 3.02 0 1 1 0 1 1 0
1130 3.98 1 1 1 1 0 1 0
770 2.94 0 0 0 0 0 1 0
1050 3.49 0 1 1 0 0 1 0
1250 3.87 1 1 1 0 1 1 0
1000 3.49 0 0 0 0 0 1 0
1010 3.24 0 1 1 0 0 1 0
1320 4.22 1 1 1 1 1 0 1
1230 3.61 1 1 1 1 1 1 1
840 2.48 1 0 1 1 1 0 1
940 2.26 1 0 1 1 0 0 1
910 2.32 0 0 0 1 1 1 1
1240 3.89 1 1 1 0 1 1 0
1020 3.67 0 0 0 0 1 0 0
630 2.54 0 0 0 0 0 1 0
850 3.16 0 0 0 0 0 1 0
1300 4.16 1 1 1 1 1 1 0
950 2.94 0 0 0 0 1 1 0
1350 3.79 1 1 1 0 1 1 0
1070 2.56 0 0 0 0 1 0 0
1000 3.00 0 0 0 0 1 1 0
770 2.79 0 0 0 0 0 1 0
1280 3.70 1 0 1 1 0 1 1
590 3.23 0 0 0 1 0 1 1
1060 3.98 1 1 1 1 1 0 1
1050 2.64 1 0 1 0 0 0 0
1220 4.15 1 1 1 1 1 1 1
(continued )
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Table 2 (continued)
SAT GPA APMATH APENG AP ESL GEND PREP RACE
930 2.73 0 0 0 0 1 1 0
940 3.10 1 1 1 1 0 0 1
980 2.70 0 0 0 1 1 1 1
1280 3.73 1 1 1 0 1 1 0
700 1.64 0 0 0 1 0 1 1
1040 4.03 1 1 1 1 0 1 1
1070 3.24 0 1 1 0 1 1 0
900 3.42 0 0 0 0 1 1 0
1430 4.29 1 1 1 0 1 0 0
1290 3.33 0 0 0 0 1 0 0
1070 3.61 1 0 1 1 0 1 1
1100 3.58 1 1 1 0 0 1 0
1030 3.52 0 1 1 0 0 1 0
1070 2.94 0 0 0 0 1 1 0
1170 3.98 1 1 1 1 1 1 0
1300 3.89 1 1 1 0 1 0 0
1410 4.34 1 1 1 1 0 1 1
1160 3.43 1 1 1 0 1 1 0
1170 3.56 1 1 1 0 0 0 0
1280 4.11 1 1 1 0 0 1 0
1060 3.58 1 1 1 1 0 1 0
1250 3.47 1 1 1 0 1 1 0
1020 2.92 1 0 1 1 1 1 1
1000 4.05 0 1 1 1 0 0 1
1090 3.24 1 1 1 1 1 1 1
1430 4.38 1 1 1 1 0 0 1
860 2.62 1 0 1 1 0 0 1
1050 2.37 0 0 0 0 1 0 0
920 2.77 0 0 0 0 0 1 0
1100 2.54 0 0 0 0 1 1 0
1160 3.55 1 0 1 1 1 1 1
1360 2.98 0 1 1 1 0 1 0
970 3.64 1 1 1 0 0 1 0
Datafile � SAT8
Now:
1. Hypothesize expected signs for the coefficients of each of these variables
in an equation for the SAT score of the ith student. Examine each vari-
able carefully; what is the theoretical content of your hypothesis?
2. Choose carefully the best set of explanatory variables. Start off by in-
cluding GPA, APMATH, and APENG; what other variables do you think
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should be specified? Don’t simply include all the variables, intending
to drop the insignificant ones. Instead, think through the problem
carefully and find the best possible equation.
Once you’ve specified your equation, you’re ready to move on. Keep follow-
ing the instructions in the exercise until you have specified your equation com-
pletely. You may take some time to think over the questions or take a break,
but when you return to the interactive exercise make sure to go back to the
exact point from which you left rather than starting all over again. To the extent
you can do it, try to avoid looking at the hints until after you’ve completed the
entire project. The hints are there to help you if you get stuck, not to allow you
to check every decision you make.
One final bit of advice: each regression result is accompanied by a series of
questions. Take the time to answer all these questions, in writing if possible.
Rushing through this interactive exercise will lessen its effectiveness.
The SAT Score Interactive Regression Exercise
To start, choose the specification you’d like to estimate, find the regression
run number9 of that specification in the following list, and then turn to that
regression. Note that the simple correlation coefficient matrix for this data set
is in Table 3 just before the results begin.
All the equations include SAT as the dependent variable and GPA,
APMATH, and APENG as explanatory variables. Find the combination of ex-
planatory variables (from ESL, GEND, PREP, and RACE) that you wish to in-
clude and go to the indicated regression:
None of them, go to regression run 1
ESL only, go to regression run 2
GEND only, go to regression run 3
PREP only, go to regression run 4
RACE only, go to regression run 5
ESL and GEND, go to regression run 6
ESL and PREP, go to regression run 7
ESL and RACE, go to regression run 8
GEND and PREP, go to regression run 9
9. All the regression results appear exactly as they are produced by the EViews regression
package. Instructors who would prefer to use results produced by the Stata regression program
can find these results in the Instructor’s Manual on the book’s website at www.pearsonhighered
.com/studenmund.
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GEND and RACE, go to regression run 10
PREP and RACE, go to regression run 11
ESL, GEND, and PREP, go to regression run 12
ESL, GEND, and RACE, go to regression run 13
ESL, PREP, and RACE, go to regression run 14
GEND, PREP, and RACE, go to regression run 15
All four, go to regression run 16
Table 3 Means, Standard Deviations, and Simple Correlation Coefficients
for the SAT Interactive Regression Learning Exercise
Means, Standard Deviations, and Correlations
Sample Range: 1–65
Variable Mean Standard Deviation
SAT 1075.538 191.3605
GPA 3.362308 0.612739
APMATH 0.523077 0.503354
APENG 0.553846 0.500961
AP 0.676923 0.471291
ESL 0.400000 0.493710
GEND 0.492308 0.503831
PREP 0.738462 0.442893
RACE 0.323077 0.471291
Correlation Coeff Correlation Coeff
APMATH,GPA 0.497 GPA,SAT 0.678
APENG,SAT 0.608 APMATH,SAT 0.512
APENG,APMATH 0.444 APENG,GPA 0.709
AP,SAT 0.579 AP,GPA 0.585
AP,APMATH 0.723 AP,APENG 0.769
ESL,GPA 0.071 ESL,SAT 0.024
ESL,APENG 0.037 ESL,APMATH 0.402
GEND,GPA –0.008 ESL,AP 0.295
GEND,APENG –0.044 GEND,SAT 0.293
GEND,ESL –0.050 GEND,APMATH 0.077
PREP,SAT –0.100 GEND,AP –0.109
PREP,APMATH –0.147 PREP,GPA 0.001
PREP,AP –0.111 PREP,APENG 0.029
PREP,GEND –0.044 PREP,ESL –0.085
RACE,SAT –0.085 RACE,GPA –0.025
RACE,APMATH 0.330 RACE,APENG –0.107
RACE,AP 0.195 RACE,ESL 0.846
RACE,GEND –0.022 RACE,PREP –0.187
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Regression Run 1
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 2 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to add ESL to the equation (go to run 2).
iii. I would like to add GEND to the equation (go to run 3).
iv. I would like to add PREP to the equation (go to run 4).
v. I would like to add RACE to the equation (go to run 5).
If you need feedback on your answer, see hint 6 in the material at the end of
this chapter.
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Regression Run 2
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 3 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 1).
iii. I would like to add GEND to the equation (go to run 6).
iv. I would like to add RACE to the equation (go to run 8).
v. I would like to add PREP to the equation (go to run 7).
If you need feedback on your answer, see hint 6 in the material at the end of
this chapter.
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Regression Run 3
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 5 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to add ESL to the equation (go to run 6).
iii. I would like to add PREP to the equation (go to run 9).
iv. I would like to add RACE to the equation (go to run 10).
If you need feedback on your answer, see hint 19 in the material at the end of
this chapter.
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Regression Run 4
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 8 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop PREP from the equation (go to run 1).
iii. I would like to add ESL to the equation (go to run 7).
iv. I would like to add GEND to the equation (go to run 9).
v. I would like to replace APMATH and APENG with AP, a linear
combination of the two variables (go to run 17).
If you need feedback on your answer, see hint 12 in the material at the end of
this chapter.
300

MULTICOLLINEARITY
Regression Run 5
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 3 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop RACE from the equation (go to run 1).
iii. I would like to add ESL to the equation (go to run 8).
iv. I would like to add GEND to the equation (go to run 10).
v. I would like to add PREP to the equation (go to run 11).
If you need feedback on your answer, see hint 14 in the material at the end of
this chapter.
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Regression Run 6
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 7 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 3).
iii. I would like to add PREP to the equation (go to run 12).
iv. I would like to add RACE to the equation (go to run 13).
If you need feedback on your answer, see hint 4 in the material at the end of
this chapter.
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Regression Run 7
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 8 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 4).
iii. I would like to drop PREP from the equation (go to run 2).
iv. I would like to add GEND to the equation (go to run 12).
v. I would like to add RACE to the equation (go to run 14).
If you need feedback on your answer, see hint 18 in the material at the end of
this chapter.
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Regression Run 8
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 9 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 5).
iii. I would like to drop RACE from the equation (go to run 2).
iv. I would like to add GEND to the equation (go to run 13).
v. I would like to add PREP to the equation (go to run 14).
If you need feedback on your answer, see hint 15 in the material at the end of
this chapter.
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Regression Run 9
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 8 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop PREP from the equation (go to run 3).
iii. I would like to add ESL to the equation (go to run 12).
iv. I would like to add RACE to the equation (go to run 15).
If you need feedback on your answer, see hint 17 in the material at the end of
this chapter.
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Regression Run 10
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 10 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop RACE from the equation (go to run 3).
iii. I would like to add ESL to the equation (go to run 13).
iv. I would like to add PREP to the equation (go to run 15).
If you need feedback on your answer, see hint 4 in the material at the end
of this chapter.
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Regression Run 11
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 8 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop PREP from the equation (go to run 5).
iii. I would like to drop RACE from the equation (go to run 4).
iv. I would like to add GEND to the equation (go to run 15).
v. I would like to replace APMATH and APENG with AP, a linear
combination of the two variables (go to run 18).
If you need feedback on your answer, see hint 18 in the material at the end of
this chapter.
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Regression Run 12
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 8 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 9).
iii. I would like to drop PREP from the equation (go to run 6).
iv. I would like to add RACE to the equation (go to run 16).
If you need feedback on your answer, see hint 17 in the material at the end of
this chapter.
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Regression Run 13
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 9 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 10).
iii. I would like to drop RACE from the equation (go to run 6).
iv. I would like to add PREP to the equation (go to run 16).
If you need feedback on your answer, see hint 15 in the material at the end of
this chapter.
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Regression Run 14
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 9 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 11).
iii. I would like to drop PREP from the equation (go to run 8).
iv. I would like to add GEND to the equation (go to run 16).
v. I would like to replace APMATH and APENG with AP, a linear
combination of the two variables (go to run 19).
If you need feedback on your answer, see hint 15 in the material at the end of
this chapter.
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Regression Run 15
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 8 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop PREP from the equation (go to run 10).
iii. I would like to drop RACE from the equation (go to run 9).
iv. I would like to add ESL to the equation (go to run 16).
If you need feedback on your answer, see hint 17 in the material at the end of
this chapter.
311

MULTICOLLINEARITY
Regression Run 16
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 9 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 15).
iii. I would like to drop PREP from the equation (go to run 13).
iv. I would like to drop RACE from the equation (go to run 12).
If you need feedback on your answer, see hint 15 in the material at the end of
this chapter.
312

MULTICOLLINEARITY
Regression Run 17
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 11 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop PREP from the equation (go to run 20).
iii. I would like to add RACE to the equation (go to run 18).
iv. I would like to replace the AP combination variable with
APMATH and APENG (go to run 4).
If you need feedback on your answer, see hint 16 in the material at the end of
this chapter.
313

MULTICOLLINEARITY
Regression Run 18
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 11 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop RACE from the equation (go to run 17).
iii. I would like to add ESL to the equation (go to run 19).
iv. I would like to replace the AP combination variable with
APMATH and APENG (go to run 11).
If you need feedback on your answer, see hint 16 in the material at the end of
this chapter.
314

MULTICOLLINEARITY
Regression Run 19
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 11 in the material at the
end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to drop ESL from the equation (go to run 18).
iii. I would like to replace the AP combination variable with
APMATH and APENG (go to run 14).
If you need feedback on your answer, see hint 16 in the material at the end of
this chapter.
315

MULTICOLLINEARITY
Regression Run 20
Answer each of the following questions for this regression run.
a. Evaluate this result with respect to its economic meaning, overall
fit, and the signs and significance of the individual coefficients.
b. What econometric problems (out of omitted variables, irrelevant
variables, or multicollinearity) does this regression have? Why? If
you need feedback on your answer, see hint 13 in the material at
the end of this chapter.
c. Which of the following statements comes closest to your recom-
mendation for further action to be taken in the estimation of this
equation?
i. No further specification changes are advisable (see the end of
the chapter).
ii. I would like to add PREP to the equation (go to run 17).
iii. I would like to replace the AP combination variable with
APMATH and APENG (go to run 1).
If you need feedback on your answer, see hint 13 in the material at the end of
this chapter.
316

MULTICOLLINEARITY
Evaluating the Results from Your Interactive Exercise
Congratulations! If you’ve reached this section, you must have found a speci-
fication that met your theoretical and econometric goals. Which one did you
pick? Our experience is that most beginning econometricians end up with ei-
ther regression run 3, 6, or 10, but only after looking at three or more regres-
sion results (or a hint or two) before settling on that choice.
In contrast, we’ve found that most experienced econometricians gravitate
to regression run 6, usually after inspecting, at most, one other specification.
What lessons can we learn from this difference?
1. Learn that a variable isn’t irrelevant simply because its t-score is low. In our
opinion, ESL belongs in the equation for strong theoretical reasons,
and a slightly insignificant t-score in the expected direction isn’t
enough evidence to get us to rethink the underlying theory.
2. Learn to spot redundant (multicollinear) variables. ESL and RACE wouldn’t
normally be redundant, but in this high school, with its particular eth-
nic diversity, they are. Once one is included in the equation, the other
shouldn’t even be considered.
3. Learn to spot false variables. At first glance, PREP is a tempting variable to
include because prep courses almost surely improve the SAT scores of the
students who choose to take them. The problem is that a student’s deci-
sion to take a prep course isn’t independent of his or her previous SAT
scores (or expected scores). We trust the judgment of students who feel a
need for a prep course, and we think that all the course will do is bring
them up to the level of their peers who didn’t feel they needed a course.
As a result, we wouldn’t expect a significant effect in either direction.
Answers
Exercise 2
a. EMPi UNITS LANGi EXPi
H0 �1 � 0 �2 � 0 �3 � 0 �4 � 0
HA �1 � 0 �2 � 0 �3 � 0 �4 � 0
tEM � �.098 tU � 2.39 tL � 2.08 tEX � 4.97
tc � 1.725 tc � 1.725 tc � 1.725 tc � 1.725
317

For the first last three coefficients, we can reject H0, because the
absolute value of tk is greater than tc and the sign of tk is that
specified in HA. For EMP, however, we cannot reject H0, because
the sign of the coefficient is unexpected and because the absolute
value of tEM is less than 1.725.
b. The functional form is semilog left (or semilog lnY). Semilog left
is an appropriate functional form for an equation with salary as the
dependent variable, because salaries often increase in percentage
terms when an independent variable (like experience) increases
by one unit.
c. There’s a chance that an omitted variable is pulling down the coeffi-
cient of EMP, but it’s more likely that EMP and EXP are redundant
(because in essence they measure the same thing) and are causing
multicollinearity.
d. This lends support to our opinion that EMPi and EXPi are redundant.
e. If we knew that this particular school district didn’t give credit for
teaching experience elsewhere, then it would make sense to drop
EXP. Without that specific knowledge, however, we’d drop EMP
because EXP includes EMP.
f. Theory: EMP clearly has a theoretically strong impact on salary,
but EMP and EXP are redundant, so we should keep only one.
t-Test: The variable’s estimated coefficient is insignificant in the
unexpected direction.
: The overall fit of the equation (adjusted for degrees of
freedom) improves when the variable is dropped from the
equation.
Bias: The exercise gives t-scores only, but if you work backward,
you can calculate the SE( )s. If you do this, you’ll find that
the coefficient of EXP does indeed change by more than a stan-
dard error when EMP is dropped from the equation. This is ex-
actly what you’d expect to happen when a redundant variable is
dropped from an equation; the coefficient of the remaining
redundant variable will adjust to pick up the effect of both
variables.
Thus even though it might appear that two of the specifica-
tion criteria support keeping EMP in the equation, in actuality all
four support the conclusion that they’re redundant and that EMP
should be removed. As a result, we have a strong preference for
Equation 22 over Equation 21.
�̂
R2
MULTICOLLINEARITY
318

Hints for the SAT Interactive Regression Learning Exercise
1. Severe multicollinearity between APMATH and APENG is the
only possible problem in this regression. You should switch to
the AP linear combination immediately.
2. An omitted variable is a distinct possibility, but be sure to choose
the one to add on the basis of theory.
3. Either an omitted or irrelevant variable is a possibility. In this
case, theory seems more important than any mild statistical
insignificance.
4. On balance, this is a reasonable regression. We see no reason to
worry about theoretically sound variables that have slightly in-
significant coefficients with expected signs. We’re concerned that
the coefficient of GEND seems larger in absolute size than those
reported in the literature, but none of the specification alterna-
tives seems remotely likely to remedy this problem.
5. An omitted variable is a possibility, but there are no signs of bias
and this is a fairly reasonable equation already.
6. We’d prefer not to add PREP (since many students take prep
courses because they did poorly on their first shots at the SAT) or
RACE (because of its redundancy with ESL and the lack of real di-
versity at Arcadia High). If you make a specification change, be
sure to evaluate the change with our four specification criteria.
7. Either an omitted or irrelevant variable is a possibility, although
GEND seems theoretically and statistically strong.
8. The unexpected sign makes us concerned with the possibility
that an omitted variable is causing bias or that PREP is irrelevant.
If PREP is relevant, what omission could have caused this result?
How strong is the theory behind PREP?
9. This is a case of imperfect multicollinearity. Even though the VIFs
are only between 3.8 and 4.0, the definitions of ESL and RACE
(and the high simple correlation coefficient between them) make
them seem like redundant variables. Remember to use theory
(and not statistical fit) to decide which one to drop.
10. An omitted variable or irrelevant variable is a possibility, but
there are no signs of bias and this is a fairly reasonable equation
already.
11. Despite the switch to the AP linear combination, we still have an
unexpected sign, so we’re still concerned with the possibility that
an omitted variable is causing bias or that PREP is irrelevant. If
PREP is relevant, what omission could have caused this result?
How strong is the theory behind PREP?
MULTICOLLINEARITY
319

12. All of the choices would improve this equation except switching
to the AP linear combination. If you make a specification change,
be sure to evaluate the change with our four specification criteria.
13. To get to this result, you had to have made at least three suspect
specification decisions, and you’re running the risk of bias due to a
sequential specification search. Our advice is to stop, take a break,
and then try this interactive exercise again.
14. We’d prefer not to add PREP (since many students take prep
courses because they did poorly on their first shots at the SAT) or
ESL (because of its redundancy with RACE and the lack of real di-
versity at Arcadia High). If you make a specification change, be
sure to evaluate the change with our four specification criteria.
15. Unless you drop one of the redundant variables, you’re going to
continue to have severe multicollinearity.
16. From theory and from the results, it seems as if the decision to
switch to the AP linear combination was a waste of a regression
run. Even if there were severe collinearity between APMATH and
APENG (which there isn’t), the original coefficients are significant
enough in the expected direction to suggest taking no action to
offset any multicollinearity.
17. On reflection, PREP probably should not have been chosen in the
first place. Many students take prep courses only because they
did poorly on their first shots at the SAT or because they antici-
pate doing poorly. Thus, even if the PREP courses improve SAT
scores, which they probably do, the students who think they
need to take them were otherwise going to score worse than their
colleagues (holding the other variables in the equation con-
stant). The two effects seem likely to offset each other, making
PREP an irrelevant variable. If you make a specification change,
be sure to evaluate the change with our four specification criteria.
18. Either adding GEND or dropping PREP would be a good choice,
and it’s hard to choose between the two. If you make a specifica-
tion change, be sure to evaluate the change with our four specifi-
cation criteria.
19. On balance, this is a reasonable regression. We’d prefer not to add
PREP (since many students take prep courses because they did
poorly on their first shots at the SAT), but the theoretical case for
ESL (or RACE) seems strong. We’re concerned that the coefficient of
GEND seems larger in absolute size than those reported in the liter-
ature, but none of the specification alternatives seems remotely
likely to remedy this problem. If you make a specification change,
be sure to evaluate the change with our four specification criteria.
MULTICOLLINEARITY
320

1 Pure versus Impure Serial Correlation
2 The Consequences of Serial Correlation
3 The Durbin–Watson d Test
4 Remedies for Serial Correlation
5 Summary and Exercises
Serial Correlation
We’ll investigate the final component of the specification of a regression
equation—choosing the correct form of the stochastic error term. Our first
topic, serial correlation, is the violation of Classical Assumption IV that dif-
ferent observations of the error term are uncorrelated with each other. Serial
correlation, also called autocorrelation, can exist in any research study in
which the order of the observations has some meaning. It therefore occurs
most frequently in time-series data sets. In essence, serial correlation implies
that the value of the error term from one time period depends in some sys-
tematic way on the value of the error term in other time periods. Since time-
series data are used in many applications of econometrics, it’s important to
understand serial correlation and its consequences for OLS estimators.
The approach of this chapter to the problem of serial correlation will be
presented here. We’ll attempt to answer the four questions:
1. What is the nature of the problem?
2. What are the consequences of the problem?
3. How is the problem diagnosed?
4. What remedies for the problem are available?
322

Pure versus Impure Serial Correlation
Pure Serial Correlation
Pure serial correlation occurs when Classical Assumption IV, which assumes
uncorrelated observations of the error term, is violated in a correctly specified
equation. Assumption IV implies that:
If the expected value of the simple correlation coefficient between any two
observations of the error term is not equal to zero, then the error term is said
to be serially correlated. When econometricians use the term serial correla-
tion without any modifier, they are referring to pure serial correlation.
The most commonly assumed kind of serial correlation is first-order serial
correlation, in which the current value of the error term is a function of the
previous value of the error term:
E(r�i�j
) 5 0 (i 2 j)
1
SERIAL CORRELATION
(1)�t 5 ��t21 1 ut
where: � the error term of the equation in question
� the first-order autocorrelation coefficient
u � a classical (not serially correlated) error term
The functional form in Equation 1 is called a first-order Markov scheme. The
new symbol, (rho, pronounced “row”), called the first-order auto-
correlation coefficient, measures the functional relationship between the
value of an observation of the error term and the value of the previous obser-
vation of the error term.
The magnitude of indicates the strength of the serial correlation in an
equation. If is zero, then there is no serial correlation (because would
equal u, a classical error term). As approaches one in absolute value, the
value of the previous observation of the error term becomes more important
in determining the current value of and a high degree of serial correlation
exists. For to be greater than one in absolute value is unreasonable because
it implies that the error term has a tendency to continually increase in ab-
solute value over time (“explode”). As a result of this, we can state that:
�
�t,
�
��
�
�
�
�
(2)2 1 , � , 1 1
323

SERIAL CORRELATION
The sign of indicates the nature of the serial correlation in an equation. A
positive value for implies that the error term tends to have the same sign
from one time period to the next; this is called positive serial correlation.
Such a tendency means that if happens by chance to take on a large value
in one time period, subsequent observations would tend to retain a portion
of this original large value and would have the same sign as the original. For
example, in time-series models, a large external shock to an economy (like an
earthquake) in one period may linger on for several time periods. The error
term will tend to be positive for a number of observations, then negative for
several more, and then back again.
Figure 1 shows two different examples of positive serial correlation. The
error term observations plotted in Figure 1 are arranged in chronological
order, with the first observation being the first period for which data are
available, the second being the second, and so on. To see the difference be-
tween error terms with and without positive serial correlation, compare
the patterns in Figure 1 with the depiction of no serial correlation in
Figure 2.
A negative value of implies that the error term has a tendency to switch
signs from negative to positive and back again in consecutive observations;
this is called negative serial correlation. It implies that there is some sort
of cycle (like a pendulum) behind the drawing of stochastic disturbances.
Figure 3 shows two different examples of negative serial correlation. For in-
stance, negative serial correlation might exist in the error term of an equa-
tion that is in first differences because changes in a variable often follow a
cyclical pattern. In most time-series applications, however, negative pure se-
rial correlation is much less likely than positive pure serial correlation. As a
result, most econometricians analyzing pure serial correlation concern
themselves primarily with positive serial correlation.
Serial correlation can take on many forms other than first-order serial cor-
relation. For example, in a quarterly model, the current quarter’s error term
observation may be functionally related to the observation of the error term
from the same quarter in the previous year. This is called seasonally based se-
rial correlation:
Similarly, it is possible that the error term in an equation might be a function
of more than one previous observation of the error term:
Such a formulation is called second-order serial correlation.
�t 5 �1�t21 1 �2�t22 1 ut
�t 5 ��t24 1 ut
�
(� 5 0)
�t
�
�
324

SERIAL CORRELATION
Impure Serial Correlation
By impure serial correlation we mean serial correlation that is caused by a
specification error such as an omitted variable or an incorrect functional
form. While pure serial correlation is caused by the underlying distribution
of the error term of the true specification of an equation (which cannot be
�
�
0 Time
�
�
�
0 Time
�
Figure 1 Positive Serial Correlation
With positive first-order serial correlation, the current observation of the error term
tends to have the same sign as the previous observation of the error term. An example
of positive serial correlation would be external shocks to an economy that take more
than one time period to completely work through the system.
325

SERIAL CORRELATION
�
�
0 Time
�
Figure 2 No Serial Correlation
With no serial correlation, different observations of the error term are completely
uncorrelated with each other. Such error terms would conform to Classical
Assumption IV.
changed by the researcher), impure serial correlation is caused by a specifica-
tion error that often can be corrected.
How is it possible for a specification error to cause serial correlation? Recall
that the error term can be thought of as the effect of omitted variables, non-
linearities, measurement errors, and pure stochastic disturbances on the de-
pendent variable. This means, for example, that if we omit a relevant variable
or use the wrong functional form, then the portion of that omitted effect that
cannot be represented by the included explanatory variables must be ab-
sorbed by the error term. The error term for an incorrectly specified equation
thus includes a portion of the effect of any omitted variables and/or a portion
of the effect of the difference between the proper functional form and the one
chosen by the researcher. This new error term might be serially correlated even
if the true one is not. If this is the case, the serial correlation has been caused
by the researcher’s choice of a specification and not by the pure error term as-
sociated with the correct specification.
As you’ll see in Section 4, the proper remedy for serial correlation
depends on whether the serial correlation is likely to be pure or impure. Not
surprisingly, the best remedy for impure serial correlation is to attempt to find
the omitted variable (or at least a good proxy) or the correct functional form
for the equation. Both the bias and the impure serial correlation will disap-
pear if the specification error is corrected. As a result, most econometricians
326

SERIAL CORRELATION
try to make sure they have the best specification possible before they spend
too much time worrying about pure serial correlation.
To see how an omitted variable can cause the error term to be serially cor-
related, suppose that the true equation is:
(3)Yt 5 �0 1 �1X1t 1 �2X2t 1 �t
�
�
0 Time
�
�
�
0 Time
�
Figure 3 Negative Serial Correlation
With negative first-order serial correlation, the current observation of the error term
tends to have the opposite sign from the previous observation of the error term. In
most time-series applications, negative serial correlation is much less likely than posi-
tive serial correlation.
327

SERIAL CORRELATION
where is a classical error term. If X2 is accidentally omitted from the equa-
tion (or if data for X2 are unavailable), then:
(4)
Thus, the error term in the omitted variable case is not the classical error
term Instead, it’s also a function of one of the independent variables, X2.
As a result, the new error term, , can be serially correlated even if the true
error term is not. In particular, the new error term will tend to be serially
correlated when:
1. X2 itself is serially correlated (this is quite likely in a time series) and
2. the size of is small compared to the size1 of
These tendencies hold even if there are a number of included and/or omitted
variables.
For example, suppose that X2 in Equation 3 is disposable income (Yd).
What would happen to this equation if Yd were omitted?
The most obvious effect would be that the estimated coefficient of X2
would be biased, depending on the correlation of X2 with Yd. A secondary ef-
fect would be that the error term would now include a large portion of the
omitted effect of disposable income. That is, would be a function of
It’s reasonable to expect that disposable income might follow a
fairly serially correlated pattern:
(5)
Why is this likely? Observe Figure 4, which plots U.S. disposable income over
time. Note that the continual rise of disposable income over time makes it
act in a serially correlated or autoregressive manner. But if disposable income
is serially correlated (and if its impact is not small relative to ), then is
likely to also be serially correlated, which can be expressed as:
(6)�t* 5 ��t*21 1 ut
�*�
Ydt 5 f(Ydt21) 1 ut
�t 1 �2 Ydt.
�t*
�2X2.�
�*�,
�*
�.
Yt 5 �0 1 �1X1t 1 �t* where �t* 5 �2X2t 1 �t
�t
1. If typical values of are significantly larger in absolute size than then even a serially
correlated omitted variable (X2) will not change very much. In addition, recall that the omit-
ted variable, X2, will cause bias in the estimate of depending on the correlation between the
two Xs. If is biased because of the omission of X2, then a portion of the effect must
have been absorbed by and will not end up in the residuals. As a result, tests for serial corre-
lation based on those residuals may give incorrect readings. Just as important, such residuals
may leave misleading clues as to possible specification errors. This is only one of many reasons
why an analysis of the residuals should not be the only procedure used to determine the nature
of possible specification errors.
�̂1
�2X2�̂1
�1,
�*
�2X2,�
328

SERIAL CORRELATION
where is the autocorrelation coefficient and u is a classical error term. This
example has shown that it is indeed possible for an omitted variable to intro-
duce “impure” serial correlation into an equation.
Another common kind of impure serial correlation is that caused by an in-
correct functional form. Here, the choice of the wrong functional form can
cause the error term to be serially correlated. Let’s suppose that the true equa-
tion is polynomial in nature:
(7)
but that instead a linear regression is run:
(8)
The new error term is now a function of the true error term and of the dif-
ferences between the linear and the polynomial functional forms. As can be
seen in Figure 5, these differences often follow fairly autoregressive patterns.
That is, positive differences tend to be followed by positive differences, and
negative differences tend to be followed by negative differences. As a result,
using a linear functional form when a nonlinear one is appropriate will
usually result in positive impure serial correlation.
��*
Yt 5 �0 1 �1X1t 1 �t*
Yt 5 �0 1 �X1t 1 �2X
2
1t 1 �t
�
0
Yd
Time
Figure 4 U.S. Disposable Income as a Function of Time
U.S. disposable income (and most other national aggregates) tends to increase steadily
over time. As a result, such variables are serially correlated (or autocorrelated), and the
omission of such a variable from an equation could potentially introduce impure serial
correlation into the error term of that equation.
329

SERIAL CORRELATION
The Consequences of Serial Correlation
The consequences of serial correlation are quite different in nature from the
consequences of the problems discussed so far in this text. Omitted variables,
irrelevant variables, and multicollinearity all have fairly recognizable external
symptoms. Each problem changes the estimated coefficients and standard
2
0
Y
X1
Y = �0 + �1X1
0�
�
�
X1
Figure 5 Incorrect Functional Form as a Source of Impure
Serial Correlation
The use of an incorrect functional form tends to group positive and negative residuals
together, causing positive impure serial correlation.
330

SERIAL CORRELATION
errors in a particular way, and an examination of these changes (and the un-
derlying theory) often provides enough information for the problem to be
detected. As we shall see, serial correlation is more likely to have internal
symptoms; it affects the estimated equation in a way that is not easily observ-
able from an examination of just the results themselves.
The existence of serial correlation in the error term of an equation violates
Classical Assumption IV, and the estimation of the equation with OLS has at
least three consequences:2
2. If the regression includes a lagged dependent variable as an independent variable, then the
problems worsen significantly.
1. Pure serial correlation does not cause bias in the coefficient
estimates.
2. Serial correlation causes OLS to no longer be the minimum variance
estimator (of all the linear unbiased estimators).
3. Serial correlation causes the OLS estimates of the SE to be
biased, leading to unreliable hypothesis testing.
(�̂)s
1. Pure serial correlation does not cause bias in the coefficient estimates. If the
error term is serially correlated, one of the assumptions of the Gauss–
Markov Theorem is violated, but this violation does not cause the coeffi-
cient estimates to be biased. If the serial correlation is impure, however,
bias may be introduced by the use of an incorrect specification.
This lack of bias does not necessarily mean that the OLS estimates
of the coefficients of a serially correlated equation will be close to the
true coefficient values; the single estimate observed in practice can
come from a wide range of possible values. In addition, the standard
errors of these estimates will typically be increased by the serial correla-
tion. This increase will raise the probability that a will differ signifi-
cantly from the true value. What unbiased means in this case is that
the distribution of the is still centered around the true
2. Serial correlation causes OLS to no longer be the minimum variance estimator
(of all the linear unbiased estimators). Although the violation of Classical
Assumption IV causes no bias, it does affect the other main conclusion of
the Gauss–Markov Theorem, that of minimum variance. In particular, we
�.�̂s
�
�̂
331

SERIAL CORRELATION
cannot prove that the distribution of the OLS is minimum variance
(among the linear unbiased estimators) when Assumption IV is violated.
The serially correlated error term causes the dependent variable to
fluctuate in a way that the OLS estimation procedure sometimes attrib-
utes to the independent variables. Thus, OLS is more likely to misesti-
mate the true in the face of serial correlation. On balance, the are
still unbiased because overestimates are just as likely as underestimates,
but these errors increase the variance of the distribution of the esti-
mates, increasing the amount that any given estimate is likely to differ
from the true
3. Serial correlation causes the OLS estimates of the SE( )s to be biased, leading
to unreliable hypothesis testing. With serial correlation, the OLS formula
for the standard error produces biased estimates of the SE( )s. Because
the SE( ) is a prime component in the t-statistic, these biased SE( )s
cause biased t-scores and unreliable hypothesis testing in general. In
essence, serial correlation causes OLS to produce incorrect SE( )s and
t-scores! Not surprisingly, most econometricians therefore are very hesi-
tant to put much faith in hypothesis tests that were conducted in the
face of pure serial correlation.3
What sort of bias does serial correlation tend to cause? Typically, the
bias in the estimate of SE( ) is negative, meaning that OLS underesti-
mates the size of the standard errors of the coefficients. This comes
about because serial correlation usually results in a pattern of observa-
tions that allows a better fit than the actual (not serially correlated)
observations would otherwise justify. This tendency of OLS to underes-
timate the SE( ) means that OLS typically overestimates the t-scores of
the estimated coefficients, since:
(9)
Thus the t-scores printed out by a typical software regression package in
the face of serial correlation are likely to be too high.
What will happen to hypothesis testing if OLS underestimates the
SE( )s and therefore overestimates the t-scores? Well, the “too low” SE( ) �̂�̂
t 5
A�̂ 2 �H0
B
SEA�̂B
�̂
�̂
�̂
�̂�̂
�̂
�̂
�.
�̂s�
�̂s
3. While our discussion here involves the t-test, the same conclusion of unreliability in the face
of serial correlation applies to all other test statistics.
332

SERIAL CORRELATION
will cause a “too high” t-score for a particular coefficient, and this will make
it more likely that we will reject a null hypothesis (for example H0: )
when it is in fact true. This increased chance of rejecting H0 means that we’re
more likely to make a Type I Error, and we’re more likely to make the mistake
of keeping an irrelevant variable in an equation because its coefficient’s t-
score has been overestimated. In other words, hypothesis testing becomes
both biased and unreliable when we have pure serial correlation.
The Durbin–Watson d Test
How can we detect serial correlation? While the first step in detecting serial
correlation often is observing a pattern in the residuals like that in Figure 1,
the test for serial correlation that is most widely used is the Durbin–Watson
d test.
The Durbin–Watson d Statistic
The Durbin–Watson d statistic is used to determine if there is first-order
serial correlation in the error term of an equation by examining the
residuals of a particular estimation of that equation.4 It’s important to use
the Durbin–Watson d statistic only when the assumptions that underlie its
derivation are met:
1. The regression model includes an intercept term.
2. The serial correlation is first-order in nature:
where is the autocorrelation coefficient and u is a classical (normally
distributed) error term.
3. The regression model does not include a lagged dependent variable
as an independent variable.5
�
�t 5 ��t21 1 ut
3
� # 0
4. J. Durbin and G. S. Watson, “Testing for Serial Correlation in Least-Squared Regression,”
Biometrika, 1951, pp. 159–177. The second most-used test, the Lagrange Multiplier test, is
presented.
5. In such a circumstance, the Durbin–Watson d is biased toward 2, but other tests can be used
instead.
333

SERIAL CORRELATION
The equation for the Durbin–Watson d statistic for T observations is:
(10)
where the ets are the OLS residuals. Note that the numerator has one fewer
observation than the denominator because an observation must be used to
calculate The Durbin–Watson d statistic equals 0 if there is extreme pos-
itive serial correlation, 2 if there is no serial correlation, and 4 if there is ex-
treme negative serial correlation. To see this, let’s put appropriate residual
values into Equation 10 for these three cases:
1. Extreme Positive Serial Correlation: d � 0
In this case,
2. Extreme Negative Serial Correlation:
In this case, Substituting into
Equation 10, we obtain and
3. No Serial Correlation:
When there is no serial correlation, the mean of the distribution of d
is equal to 2.6 That is, if there is no serial correlation,
Using the Durbin–Watson d Test
The Durbin–Watson d test is unusual in two respects. First, econometricians
almost never test the one-sided null hypothesis that there is negative serial
correlation in the residuals because negative serial correlation, as mentioned
previously, is quite difficult to explain theoretically in economic or business
analysis. Its existence usually means that impure serial correlation has been
caused by some error of specification.
Second, the Durbin–Watson test is sometimes inconclusive. Whereas pre-
viously explained decision rules always have had only “acceptance” regions
and rejection regions, the Durbin–Watson test has a third possibility, called
d < 2. d < 2 d < 4.d 5 g (2et) 2>g (et)
2
et 5 2et21, and (et 2 et21) 5 (2et).
d < 4 et 5 et21, so (et 2 et21) 5 0 and d 5 0. et21. d 5 g T 2 (et 2 et21) 2^g T 1 e2t 6. To see this, multiply out the numerator of Equation 10, obtaining (11) If there is no serial correlation, then and are not related, and, on average, g (etet21) 5 0.et21et d 5 cg T 2 e2t 2 2g T 2 (etet21) 1 g T 2 e2t21d ^g T 1 e2t < cg T 2 e2t 1 g T 2 e2t21d ^g T 1 e2t < 2 334 SERIAL CORRELATION the inconclusive region.7 For reasons outlined in Section 4, we do not recom- mend the application of a remedy for serial correlation if the Durbin– Watson test is inconclusive. With these exceptions, the use of the Durbin–Watson d test is quite similar to the use of the t-test. To test for positive serial correlation, the following steps are required: 1. Obtain the OLS residuals from the equation to be tested and calculate the d statistic by using Equation 10. 2. Determine the sample size and the number of explanatory variables and then consult Statistical Tables B-4, B-5, or B-6 in Appendix B to find the upper critical d value, dU, and the lower critical d value, dL, respectively. Instructions for the use of these tables are also in that appendix. 3. Given the null hypothesis of no positive serial correlation and a one- sided alternative hypothesis: (12) the appropriate decision rule is: In rare circumstances, perhaps first differenced equations, a two-sided d test might be appropriate. In such a case, steps 1 and 2 are still used, but step 3 is now: Given the null hypothesis of no serial correlation and a two-sided alter- native hypothesis: (13) HA : � 2 0 (serial correlation) H0 : � 5 0 (no serial correlation) if dL # d # dU Inconclusive if d . dU Do not reject H0 if d , dL Reject H0 HA : � . 0 (positive serial correlation) H0 : � # 0 (no positive serial correlation) 7. This inconclusive region is troubling, but the development of exact Durbin–Watson tests may eliminate this problem in the near future. Some computer programs allow the user the op- tion of calculating an exact Durbin–Watson probability (of first-order serial correlation). Alter- natively, it’s worth noting that there is a growing trend toward the use of dU as a sole critical value. This trend runs counter to our view that if the Durbin–Watson test is inconclusive, then no remedial action should be taken except to search for a possible cause of impure serial correlation. 335 SERIAL CORRELATION the appropriate decision rule is: Examples of the Use of the Durbin–Watson d Statistic Let’s work through some applications of the Durbin–Watson test. First, turn to Statistical Tables B-4, B-5, and B-6. Note that the upper and lower critical d values (dU and dL) depend on the number of explanatory variables (do not count the constant term), the sample size, and the level of significance of the test. Now let’s set up a one-sided 5-percent test for a regression with three ex- planatory variables and 25 observations. As can be seen from the 5-percent table (B-4), the critical d values are dL � 1.12 and dU � 1.66. As a result, if the hypotheses are: (14) the appropriate decision rule is: A computed d statistic of 1.78, for example, would indicate that there is no evidence of positive serial correlation, a value of 1.28 would be inconclusive, and a value of 0.60 would imply positive serial correlation. Figure 6 provides a graph of the “acceptance,” rejection, and inconclusive regions for this example. For a more familiar example, we return to the chicken demand model of Equation 6.8. As can be confirmed with the data provided in Table 6.2, the Durbin–Watson statistic from Equation 6.8 is 0.99. Is that cause to be con- cerned about serial correlation? What would be the result of a one-sided 5-percent test of the null hypothesis of no positive serial correlation? Our first step would be to consult Statistical Table B-4. In that table, with K (the num- ber of explanatory variables) equal to 3 and N (the number of observations) equal to 29, we would find the critical d values dL � 1.20 and dU � 1.65. if 1.12 # d # 1.66 Inconclusive if d . 1.66 Do not reject H0 if d , 1.12 Reject H0 HA: � . 0 (positive serial correlation) H0: � # 0 (no positive serial correlation) otherwise Inconclusive if 4 2 dU . d . dU Do not reject H0 if d . 4 2 dL Reject H0 if d , dL Reject H0 336 SERIAL CORRELATION The decision rule would thus be: Since 0.99 is less than the critical lower limit of the d statistic, we would reject the null hypothesis of no positive serial correlation, and we would have to decide how to cope with that serial correlation. Remedies for Serial Correlation Suppose that the Durbin–Watson d statistic detects serial correlation in the residuals of your equation. Is there a remedy? Some students suggest reordering the observations of Y and the Xs to avoid serial correlation. They think that if this time’s error term appears to be affected by last time’s error term, why not reorder the data randomly to get rid of the problem? The answer is that the reordering of the data does not get rid of the serial correlation; it just makes the problem harder to detect. If and we reorder the data, then the error term observations are still related to each other, but they now no longer follow each other, and it becomes almost impossible to discover the serial correlation. �2 5 f(�1) 4 if 1.20 # d # 1.65 Inconclusive if d . 1.65 Do not reject H0 if d , 1.20 Reject H0 0 20.60 = 1.66 = 1.12 dU dL 1.28 1.78 4 Inconclusive Region dL < d < dU Rejection Region d < dL “Acceptance” Region dU < d Positive Serial Correlation No Positive Serial Correlation Figure 6 An Example of a One-Sided Durbin–Watson d Test In a one-sided Durbin–Watson test for positive serial correlation, only values of d sig- nificantly below 2 cause the null hypothesis of no positive serial correlation to be re- jected. In this example, a d of 1.78 would indicate no positive serial correlation, a d of 0.60 would indicate positive serial correlation, and a d of 1.28 would be inconclusive. 337 SERIAL CORRELATION Interestingly, reordering the data changes the Durbin–Watson d statistic but does not change the estimates of the coefficients or their standard errors at all.8 The place to start in correcting a serial correlation problem is to look carefully at the specification of the equation for possible errors that might be causing impure serial correlation. Is the functional form cor- rect? Are you sure that there are no omitted variables? Only after the specification of the equation has been reviewed carefully should the possibility of an adjustment for pure serial correlation be considered. It’s worth noting that if an omitted variable increases or decreases over time, as is often the case, or if the data set is logically reordered (say, accord- ing to the magnitude of one of the variables), then the Durbin–Watson statis- tic can help detect impure serial correlation. A significant Durbin–Watson statistic can easily be caused by an omitted variable or an incorrect functional form. In such circumstances, the Durbin–Watson test does not distinguish between pure and impure serial correlation, but the detection of negative serial correlation is often a strong hint that the serial correlation is impure. If you conclude that you have pure serial correlation, then the appropriate response is to consider the application of Generalized Least Squares or Newey–West standard errors, as described in the following sections. Generalized Least Squares Generalized least squares (GLS) is a method of ridding an equation of pure first-order serial correlation and in the process restoring the minimum vari- ance property to its estimation. GLS starts with an equation that does not meet the Classical Assumptions (due in this case to the pure serial correlation in the error term) and transforms it into one (Equation 19) that does meet those assumptions. At this point, you could skip directly to Equation 19, but it’s easier to un- derstand the GLS estimator if you examine the transformation from which it comes. Start with an equation that has first-order serial correlation: (15)Yt 5 �0 1 �1X1t 1 �t 8. This can be proven mathematically, but it is usually more instructive to estimate a regression yourself, change the order of the observations, and then reestimate the regression. See Exercise 3 at the end of the chapter. 338 SERIAL CORRELATION which, if (due to pure serial correlation), also equals: (16) where is the serially correlated error term, is the autocorrelation coeffi- cient, and u is a classical (not serially correlated) error term. If we could get the term out of Equation 16, the serial correlation would be gone, because the remaining portion of the error term (ut) has no serial correlation in it. To rid from Equation 16, multiply Equation 15 by and then lag the new equation by one time period, obtaining (17) Notice that we now have an equation with a term in it. If we now sub- tract Equation 17 from Equation 16, the equivalent equation that remains no longer contains the serially correlated component of the error term: (18) Equation 18 can be rewritten as: (19) where: (20) Equation 19 is called a Generalized Least Squares (or “quasi-differenced”) version of Equation 16. Notice that: 1. The error term is not serially correlated. As a result, OLS estimation of Equation 19 will be minimum variance. (This is true if we know or if we accurately estimate ) 2. The slope coefficient is the same as the slope coefficient of the orig- inal serially correlated equation, Equation 16. Thus coefficients esti- mated with GLS have the same meaning as those estimated with OLS. 3. The dependent variable has changed compared to that in Equation 16. This means that the GLS is not directly comparable to the OLS 4. To forecast with GLS, adjustments like those discussed in Section 2 from Chapter 15 are required. R2.R2 �1 �. � �0* 5 �0 2 ��0 X1*t 5 X1t 2 �X1t21 Yt* 5 Yt 2 �Yt21 Yt* 5 �0* 1 �1X1*t 1 ut Yt 2 �Yt21 5 �0(1 2 �) 1 �1(X1t 2 �X1t21) 1 ut ��t21 �Yt21 5 ��0 1 ��1X1t21 1 ��t21 � ��t21 ��t21 �� Yt 5 �0 1 �1X1t 1 ��t21 1 ut �t 5 ��t21 1 ut 339 SERIAL CORRELATION Unfortunately we can’t use OLS to estimate a Generalized Least Squares model because GLS equations are inherently nonlinear in the coefficients. To see why, take a look at Equation 18. We need to estimate values not only for and but also for and is multiplied by and (which you can see if you multiply out the right-hand side of the equation). Since OLS requires that the equation be linear in the coefficients, we need a different estimation procedure. Luckily, there are a number of techniques that can be used to estimate GLS equations. Perhaps the best known of these is the Cochrane–Orcutt method, a two-step iterative technique9 that first produces an estimate of and then estimates the GLS equation using that . The two steps are: 1. Estimate by running a regression based on the residuals of the equa- tion suspected of having serial correlation: (21) where the ets are the OLS residuals from the equation suspected of having pure serial correlation and ut is a classical error term. 2. Use this to estimate the GLS equation by substituting into Equation 18 and using OLS to estimate Equation 18 with the adjusted data. These two steps are repeated (iterated) until further iteration results in little change in . Once has converged (usually in just a few iterations), the last estimate of step 2 is used as a final estimate of Equation 18. As popular as Cochrane–Orcutt is, we suggest a different method, the AR(1) method, for GLS models. The AR(1) method estimates a GLS equa- tion like Equation 18 by estimating and simultaneously with itera- tive nonlinear regression techniques that are well beyond the scope of this chapter.10 The AR(1) method tends to produce the same coefficient estimates as Cochrane–Orcutt but with superior estimates of the standard errors, so we recommend the AR(1) approach as long as your software can support such nonlinear regression. Let’s apply Generalized Least Squares, using the AR(1) estimation method, to the chicken demand example that was found to have positive serial correlation ��0, �1, �̂�̂ �̂�̂ et 5 �et 2 1 1 ut � �̂ � �1�0��,�1 �0 9. D. Cochrane and G. H. Orcutt, “Application of Least Squares Regression to Relationships Containing Autocorrelated Error Terms,” Journal of the American Statistical Association, 1949, pp. 32–61. 10. To run GLS with EViews, simply add “AR(1)” to the equation as if it were an independent variable. The resulting equation is a GLS estimate where will appear as the estimated coeffi- cient of the variable AR(1). To run GLS with Stata, click on “linear regression with AR(1) distur- bance” in the appropriate drop-down window. �̂ 340 SERIAL CORRELATION in the previous section. Recall that we estimated the per capita demand for chicken as a function of the price of chicken, the price of beef, and disposable income: (6.8) (0.03) (0.02) (0.01) t � �3.38 � 1.86 � 15.7 DW d � 0.99 Note that we have added the Durbin–Watson d statistic to the documenta- tion with the notation DW. All future time-series results will include the DW statistic, but cross-sectional documentation of the DW is not required unless the observations are ordered in some meaningful manner (like smallest to largest or youngest to oldest). If we reestimate Equation 6.8 with the AR(1) approach to GLS, we obtain: (22) Let’s compare Equations 6.8 and 22. Note that the used in Equation 22 is 0.56. This means that Y was actually run as PC as etc. Second, replaces DW in the documentation of GLS estimates in part because the DW of Equation 22 isn’t strictly compa- rable to non-GLS DWs (it is biased toward 2). Finally, the sample size of the GLS regression is 28 because the first observation has to be used to create the lagged values for the calculation of the quasi-differenced variables in Equation 20. Generalized Least Squares estimates, no matter how produced, have at least two problems. First, even though serial correlation causes no bias in the estimates of the the GLS estimates usually are different from the OLS ones. For example, note that all three slope coefficients change as we move from OLS in Equation 6.8 to GLS in Equation 22. This isn’t surpris- ing, since two different estimates can have different values even though their expected values are the same. The second problem is more important, however. It turns out that GLS works well if is close to the actual , but the GLS is biased in small samples. If is biased, then the biased intro- duces bias into the GLS estimates of the Luckily, there is a remedy for serial correlation that avoids both of these problems: Newey–West stan- dard errors. �̂s. �̂�̂�̂ ��̂ �̂s, �̂PC* 5 PCt 2 0.56PCt 2 1, Y* 5 Yt 2 0.56Yt 2 1, �̂ N 5 28 �̂ 5 0.56R2 5 .9921 t 5 2 1.70 1 0.76 1 12.06 (0.05) (0.02) (0.02) Ŷt 5 27.7 2 0.08PCt 1 0.02PBt 1 0.24YDt R2 5 .9904 N 5 29 Ŷt 5 27.7 2 0.11PCt 1 0.03PBt 1 0.23YDt 341 SERIAL CORRELATION Newey–West Standard Errors Not all corrections for pure serial correlation involve Generalized Least Squares. Newey–West standard errors are SE( )s that take account of serial correlation without changing the themselves in any way.11 The logic behind Newey–West standard errors is powerful. If serial correlation does not cause bias in the but does impact the standard errors, then it makes sense to adjust the estimated equation in a way that changes the SE( )s but not the Thus Newey–West standard errors have been calculated specifically to avoid the consequences of pure first-order serial correlation. The Newey–West proce- dure yields an estimator of the standard errors that, while they are biased, is gen- erally more accurate than uncorrected standard errors for large samples in the face of serial correlation. As a result, Newey–West standard errors can be used for t-tests and other hypothesis tests in most samples without the errors of inference potentially caused by serial correlation. Typically, Newey–West SE( )s are larger than OLS SE( )s, thus producing lower t-scores and decreasing the probability that a given estimated coefficient will be significantly different from zero. To see how Newey–West standard errors work, let’s apply them to the same serially correlated chicken demand equation to which we applied GLS in Equation 22. If we use Newey–West standard errors in the estimation of Equation 8 from Chapter 6, we get: (23) (0.03) (0.02) (0.01) Let’s compare Equations 8 from Chapter 6 and 23. First of all, the are identical in Equations 8 from Chapter 6 and 23. This is because Newey–West standard errors do not change the OLS Second, while we can’t observe the change because of rounding, the Newey–West standard errors must be different from the OLS standard errors because the t-scores have changed even though the estimated coefficients are identical. However, the Newey- West SE( )s are slightly lower than the OLS SE( )s, which is a surprise even in a small sample like this one. Such a result indicates that there may well be an omitted variable or nonstationarity in this equation. �̂�̂ �̂s. �̂s N 5 29 R2 5 .9904 t 5 2 3.51 1 1.92 1 19.4 Ŷt 5 27.7 2 0.11PCt 1 0.03PBt 1 0.23YDt �̂ �̂ �̂s.�̂ �̂s �̂s �̂ 11. W. K. Newey and K. D. West, “A Simple, Positive Semi-Definite Heteroskedasticity and Auto- correlation Consistent Covariance Matrix,” Econometrica, 1987, pp. 703–708. Newey–West stan- dard errors are similar to HC standard errors (or White standard errors), discussed in Section 10.4. 342 SERIAL CORRELATION Summary 1. Serial correlation, or autocorrelation, is the violation of Classical As- sumption IV that the observations of the error term are uncorrelated with each other. Usually, econometricians focus on first-order serial correlation, in which the current observation of the error term is as- sumed to be a function of the previous observation of the error term and a not serially correlated error term (u): where is “rho,” the autocorrelation coefficient. 2. Pure serial correlation is serial correlation that is a function of the error term of the correctly specified regression equation. Impure se- rial correlation is caused by specification errors such as an omitted variable or an incorrect functional form. While impure serial correla- tion can be positive pure se- rial correlation in economics or business situations is almost always positive. 3. The major consequence of serial correlation is bias in the OLS SE ( )s, causing unreliable hypothesis testing. Pure serial correlation does not cause bias in the estimates of the 4. The most commonly used method of detecting first-order serial corre- lation is the Durbin–Watson d test, which uses the residuals of an esti- mated regression to test the possibility of serial correlation in the error term. A d value of 0 indicates extreme positive serial correlation, a d value of 2 indicates no serial correlation, and a d value of 4 indi- cates extreme negative serial correlation. 5. The first step in ridding an equation of serial correlation is to check for possible specification errors. Only once the possibility of impure serial correlation has been reduced to a minimum should remedies for pure serial correlation be considered. 6. Generalized Least Squares (GLS) is a method of transforming an equation to rid it of pure first-order serial correlation. The use of GLS requires the estimation of 7. Newey–West standard errors are an alternative remedy for serial corre- lation that adjusts the OLS estimates of the SE( )s to take account of the serial correlation without changing the .�̂s �̂ �. �s. �̂ (0 , � , 1) or negative (21 , � , 0), � �t 5 ��t21 1 ut 21 , � , 1 5 343 SERIAL CORRELATION EXERCISES (The answer to Exercise 2 is at the end of the chapter.) 1. Write the meaning of each of the following terms without referring to the book (or your notes), and compare your definition with the ver- sion in the text for each: a. impure serial correlation b. first-order serial correlation c. first-order autocorrelation coefficient d. Durbin–Watson d statistic e. Generalized Least Squares f. positive serial correlation g. Newey–West standard errors 2. Consider the following equation for U.S. per capita consumption of beef: (24) (7.4) (0.13) (0.12) (4.1) t � 6.6 �2.6 2.7 �3.7 where: Bt � the annual per capita pounds of beef consumed in the United States in year t � the log of real per capita disposable real income in the U.S. in year t PBt � average annualized real wholesale price of beef in year t (in cents per pound) PRPt � average annualized real wholesale price of pork in year t (in cents per pound) Dt � a dummy variable equal to 1 for years in which there was a “health scare” about the dangers of red meat, 0 otherwise a. Develop and test your own hypotheses with respect to the individ- ual estimated slope coefficients. b. Test for serial correlation in Equation 24 using the Durbin–Watson d test at the 5-percent level. c. What econometric problem(s) (if any) does Equation 24 appear to have? What remedy would you suggest? ln Yt R2 5 .700 N 5 28 DW 5 0.94 B̂ t 5 2330.3 1 49.1ln Yt 2 0.34PBt 1 0.33PRPt 2 15.4Dt 344 SERIAL CORRELATION d. You take your own advice, and apply GLS to Equation 24, obtaining: (25) Compare Equations 24 and 25. Which do you prefer? Why? 3. Recall from Section 4 that switching the order of a data set will not change its coefficient estimates. A revised order will change the Durbin–Watson statistic, however. To see both these points, run re- gressions and compare the coefficient esti- mates and DW d statistics for this data set: Year Housing Starts Population 1 9090 2200 2 8942 2222 3 9755 2244 4 10327 2289 5 10513 2290 in the following three orders (in terms of year): a. 1, 2, 3, 4, 5 b. 5, 4, 3, 2, 1 c. 2, 4, 3, 5, 1 4. Use Statistical Tables B-4, B-5, and B-6 to test for serial correlation given the following Durbin–Watson d statistics for serial correlation. a. 5-percent, one-sided positive test b. 1-percent, one-sided positive test c. 2.5-percent, one-sided positive test d. 5-percent, two-sided test e. 5-percent, one-sided positive test f. 2-percent, two-sided test g. 5-percent, one-sided positive test 5. Carefully distinguish between the following concepts: a. positive and negative serial correlation b. pure and impure serial correlation d 5 1.03, K 5 6, N 5 26, d 5 0.91, K 5 2, N 5 28, d 5 1.75, K 5 1, N 5 45, d 5 2.84, K 5 4, N 5 35, d 5 1.56, K 5 5, N 5 30, d 5 3.48, K 5 2, N 5 15, d 5 0.81, K 5 3, N 5 21, (HS 5 �0 1 �1P 1 �) R2 5 .857 N 5 28 �̂ 5 0.82 t 5 2.5 2 3.7 1.1 2 1.5 (14.1) (0.10) (0.09) (3.9) B̂t 5 2 193.3 1 35.2ln Yt 2 0.38PBt 1 0.10PPt 2 5.7Dt 345 SERIAL CORRELATION c. serially correlated observations of the error term and serially corre- lated residuals d. the Cochrane–Orcutt method and the AR(1) method e. GLS and Newey–West standard errors 6. In Statistical Table B-4, column is greater than 2 for the five smallest sample sizes in the table. What does it mean if ? 7. A study by M. Hutchinson and D. Pyle12 found some evidence of a link between short-term interest rates and the budget deficit in a sam- ple that pools annual time-series and cross-sectional data from six countries. a. Suppose you were told that the Durbin–Watson d from their best regression was 0.81. Test this DW for indications of serial correla- tion 5-percent one-sided test for positive serial correlation). b. Based on this result, would you conclude that serial correlation ex- isted in their study? Why or why not? (Hint: The six countries were the United Kingdom, France, Japan, Canada, Italy, and the United States; assume that the order of the data was United Kingdom, fol- lowed by France, etc.) c. How would you use GLS to correct for serial correlation in this case? 8. Suppose that the data in a time-series study were entered in reverse chronological order. Would this change in any way the testing or ad- justing for serial correlation? How? In particular: a. What happens to the Durbin–Watson statistic’s ability to detect serial correlation if the order is reversed? b. What happens to the GLS method’s ability to adjust for serial corre- lation if the order is reversed? c. What is the intuitive economic explanation of reverse serial correlation? 9. Suppose that a plotting of the residuals of a regression with respect to time indicates a significant outlier in the residuals. (Be careful here: this is not an outlier in the original data but is an outlier in the residuals of a regression.) a. How could such an outlier occur? What does it mean? b. Is the Durbin–Watson d statistic applicable in the presence of such an outlier? Why or why not? (N 5 60, K 5 4, dU . 2 K 5 5, dU 12. M. M. Hutchinson and D. H. Pyle, “The Real Interest Rate/Budget Deficit Link: International Evidence, 1973–82,” Federal Reserve Bank of San Francisco Economic Review, Vol. 4, pp. 26–35. 346 SERIAL CORRELATION 10. After GLS has been run on an equation, the are still good estimates of the original (nontransformed) equation except for the constant term: a. What must be done to the estimate of the constant term generated by GLS to compare it with the one estimated by OLS? b. Why is such an adjustment necessary? c. Return to Equation 22 and calculate the that would be comparable to the one in Equation 6.8. (Hint: Take a look at Equation 20.) d. The two estimates are different. Why? Does such a difference con- cern you? 11. Your friend is just finishing a study of attendance at Los Angeles Laker regular-season home basketball games when she hears that you’ve read a chapter on serial correlation and asks your advice. Before run- ning the equation on last season’s data, she “reviewed the literature” by interviewing a number of basketball fans. She found out that fans like to watch winning teams. In addition, she learned that while some fans like to watch games throughout the season, others are most inter- ested in games played late in the season. Her estimated equation (standard errors in parentheses) was: where: A t � the attendance at game t L t � the winning percentage (games won divided by games played) of the Lakers before game t Pt � the winning percentage before game t of the Lakers’ op- ponent in that game Wt � a dummy variable equal to one if game t was on Friday, Saturday, or Sunday, 0 otherwise a. Test for serial correlation using the Durbin–Watson d test at the 5-percent level. b. Make and test appropriate hypotheses about the slope coefficients at the 1-percent level. c. Compare the size and significance of the estimated coefficient of L with that for P. Is this difference surprising? Is L an irrelevant vari- able? Explain your answer. d. If serial correlation exists, would you expect it to be pure or impure serial correlation? Why? DW 5 0.85 N 5 40 R2 5 .46 (500) (1000) (300) Â t 5 14123 1 20L t 1 2600Pt 1 900Wt �̂0 �̂s 347 SERIAL CORRELATION e. Your friend omitted the first game of the year from the sample be- cause the first game is always a sellout and because neither team had a winning percentage yet. Was this a good decision? 12. About two thirds of the way through the 2008 season, the Los Angeles Dodgers baseball team traded for superstar Manny Ramirez, and the result was a divisional pennant and dramatically increased atten- dance. Suppose that you’ve been hired by Manny’s agent to help pre- pare for his upcoming contract negotiations by determining how much money Manny generated for the Dodgers. You decide to build a model of the Dodgers’ attendance, and, after learning as much as you can about such modeling, you collect data for 2008 (Table 1) and estimate the following equation: ATTi � 34857 � 4104MANNYi � 2282PMi � 5632WKNDi � 4029PROMi � 8081TEAMi (1021) (1121) (1096) (1068) (5819) t � 4.02 2.04 5.14 3.77 1.39 N � 81 � .54 DW � 1.30 where: ATTi � the number of tickets sold for the ith Dodger home game MANNYi � 1 after the trade for Manny Ramirez, 0 other- wise PMi � 1 if the ith game was a night game, 0 otherwise WKNDi � 1 if the ith game was on the weekend, 0 other- wise PROMi � 1 if the ith game included a major promotion (for example, fireworks or a free bobble-head), 0 otherwise TEAMi � the winning percentage of the Dodgers’ oppo- nent before the ith game (set equal to the 2007 percentage for the first three games of 2008) a. You expect each coefficient to be positive. Test these expectations at the 5-percent level. b. Test for serial correlation in this equation by running a Durbin– Watson test. c. What potential econometric problems (out of omitted variables, ir- relevant variables, incorrect functional form, multicollinearity, and serial correlation) do you see in this equation? Explain. d. Assume that your answer to part c is that you’re concerned with se- rial correlation. Use the data in Table 1 to estimate the equation with generalized least squares. R2 348 SERIAL CORRELATION Table 1 Data for the Dodger Attendance Exercise OBS VS ATT PM WKND PROM TEAM MANNY RIVAL 1 SF 56000 0 0 1 0.438 0 1 2 SF 44054 1 0 0 0.438 0 1 3 SF 43217 1 0 0 0.438 0 1 4 SD 54052 1 1 1 0.546 0 1 5 SD 54955 1 1 1 0.546 0 1 6 SD 47357 0 1 0 0.546 0 1 7 PIT 37334 1 0 0 0.420 0 0 8 PIT 37896 1 0 1 0.420 0 0 9 PIT 53629 1 0 1 0.420 0 0 10 ARI 42590 1 0 0 0.750 0 0 11 ARI 38350 1 0 0 0.714 0 0 12 COL 53205 1 1 1 0.455 0 0 13 COL 50469 1 1 0 0.435 0 0 14 COL 50670 0 1 1 0.417 0 0 15 NYM 44181 1 0 0 0.552 0 0 16 NYM 43927 1 0 0 0.533 0 0 17 NYM 40696 0 0 0 0.516 0 0 18 HOU 52658 1 1 1 0.514 0 0 19 HOU 45212 1 1 0 0.528 0 0 20 HOU 40217 0 1 1 0.541 0 0 21 CIN 34669 1 0 0 0.477 0 0 22 CIN 34306 1 0 0 0.467 0 0 23 CIN 33224 1 0 0 0.547 0 0 24 STL 52281 1 1 1 0.571 0 0 25 STL 44785 1 1 0 0.580 0 0 26 STL 46566 0 1 0 0.588 0 0 27 COL 39098 1 0 0 0.351 0 0 28 COL 38548 1 0 0 0.345 0 0 29 COL 36393 0 0 0 0.356 0 0 30 CHC 44998 1 0 1 0.633 0 0 31 CHC 52484 1 1 1 0.639 0 0 32 CHC 50020 0 1 0 0.629 0 0 33 CHC 49994 1 1 0 0.619 0 0 34 CLE 50667 1 1 1 0.452 0 0 35 CLE 45036 0 1 1 0.495 0 0 36 CLE 39993 0 1 0 0.467 0 0 37 CWS 43900 1 0 0 0.547 0 0 38 CWS 40162 1 0 0 0.553 0 0 39 CWS 37956 0 0 0 0.545 0 0 40 LAA 50419 1 1 0 0.608 0 1 41 LAA 55784 1 1 0 0.600 0 1 42 LAA 48155 0 1 0 0.593 0 1 (continued ) 349 43 ATL 39896 1 0 0 0.472 0 0 44 ATL 39702 1 0 0 0.467 0 0 45 ATL 39815 1 0 0 0.473 0 0 46 FLA 40417 1 0 0 0.516 0 0 47 FLA 49545 1 1 0 0.522 0 0 48 FLA 55220 1 1 1 0.527 0 0 49 FLA 42213 0 1 1 0.532 0 0 50 WSH 47313 1 1 1 0.373 0 0 51 WSH 42122 1 1 0 0.369 0 0 52 WSH 38660 0 1 0 0.365 0 0 53 SF 37483 1 0 0 0.413 0 1 54 SF 40110 1 0 0 0.419 0 1 55 SF 41282 1 0 0 0.415 0 1 56 ARIZ 42440 1 0 0 0.514 0 0 57 ARIZ 55239 1 1 1 0.519 1 0 58 ARIZ 54544 1 1 0 0.523 1 0 59 ARIZ 52972 0 1 1 0.518 1 0 60 PHI 45547 1 0 0 0.547 1 0 61 PHI 47587 1 0 1 0.542 1 0 62 PHI 45786 1 0 0 0.538 1 0 63 PHI 51064 1 0 0 0.533 1 0 64 MIL 44546 1 1 1 0.574 1 0 65 MIL 52889 1 1 1 0.569 1 0 66 MIL 45267 0 1 0 0.573 1 0 67 COL 46687 1 0 0 0.452 1 0 68 COL 48183 1 0 0 0.457 1 0 69 COL 44885 0 0 0 0.461 1 0 70 SD 44085 1 0 1 0.390 1 1 71 SD 39330 1 0 0 0.387 1 1 72 SD 48822 1 0 1 0.384 1 1 73 ARIZ 52270 1 1 1 0.511 1 0 74 ARIZ 47543 0 1 0 0.507 1 0 75 ARIZ 54137 0 1 1 0.504 1 0 76 SF 55135 1 1 1 0.444 1 1 77 SF 55452 1 1 1 0.448 1 1 78 SF 54841 0 1 1 0.445 1 1 79 SD 48907 1 0 0 0.391 1 1 80 SD 46741 1 0 0 0.389 1 1 81 SD 51783 1 0 1 0.386 1 1 Datafile = DODGERS9 Source: www.dodgers.com SERIAL CORRELATION Table 1 (continued) OBS VS ATT PM WKND PROM TEAM MANNY RIVAL 350 SERIAL CORRELATION e. Assume that your answer to part c is that you are more concerned with an omitted variable than with serial correlation, especially because an omitted variable can cause impure serial correlation. Add RIVALi (a dummy variable equal to 1 if the opponent in the ith game is an in-state rival of the Dodgers, 0 otherwise) to the equation and estimate your new specification using the data in Table 1. f. Which do you prefer, using GLS or adding RIVAL? Explain. g. Given your answer to part f, what’s your conclusion? How many fans per game did Manny Ramirez attract to Dodger Stadium? Was this result fairly robust (stable as the specification was changed)? 13. You’re hired by Farmer Vin, a famous producer of bacon and ham, to test the possibility that feeding pigs at night allows them to grow faster than feeding them during the day. You take 200 pigs (from new- born piglets to extremely old porkers) and randomly assign them to feeding only during the day or feeding only at night and, after six months, end up with the following (admittedly very hypothetical) equation: where: Wi � the percentage weight gain of the ith pig Gi � a dummy variable equal to 1 if the ith pig is a male, 0 otherwise Di � a dummy variable equal to 1 if the ith pig was fed only at night, 0 if only during the day Fi � the amount of food (pounds) eaten per day by the ith pig a. Test for serial correlation at the 5-percent level in this equation. b. What econometric problems appear to exist in this equation? (Hint: Be sure to make and test appropriate hypotheses about the slope coefficients.) c. The goal of your experiment is to determine whether feeding at night represents a significant improvement over feeding during the day. What can you conclude? R2 5 .70 N 5 200 DW 5 0.50 t 5 3.5 7.0 2 2.5 (1.0) (1.0) (0.10) Ŵi 5 12 1 3.5Gi 1 7.0Di 2 0.25Fi 351 SERIAL CORRELATION 13. Josef C. Brada and Ronald L. Graves, “The Slowdown in Soviet Defense Expenditures,” Southern Economic Journal, Vol. 54, No. 4, pp. 969–984. In addition to the variables used in this exercise, Brada and Graves also provide data for SFPt, the rate of Soviet factor productivity in year t, which we include in Table 2 because we suggest exercises using SFP in the instructor’s manual. d. The observations are ordered from the youngest pig to the oldest pig. Does this information change any of your answers to the previ- ous parts of this question? Is this ordering a mistake? Explain your answer. 14. In a 1988 article, Josef Brada and Ronald Graves built an interesting model of defense spending in the Soviet Union just before the breakup of that nation.13 The authors felt sure that Soviet defense spending was a function of U.S. defense spending and Soviet GNP but were less sure about whether defense spending also was a function of the ratio of Soviet nuclear warheads to U.S. nuclear warheads. Using a double-log functional form, the authors estimated a number of alter- native specifications, including (standard errors in parentheses): (26) (27) where: SDHt � the CIA’s “high” estimate of Soviet defense expendi- tures in year t (billions of 1970 rubles) USDt � U.S. defense expenditures in year t (billions of 1980 dollars) SYt � Soviet GNP in year t (billions of 1970 rubles) SPt � the ratio of the number of USSR nuclear warheads (NRt) to the number of U.S. nuclear warheads (NUt) in year t N 5 25 (annual 1960–1984) R2 5 .977 DW 5 0.43 t 5 1.44 28.09 (0.073) (0.038) ln SDHt 5 2 2.88 1 0.105lnUSDt 1 1.066lnSYt N 5 25 (annual 1960–1984) R2 5 .979 DW 5 0.49 t 5 0.76 14.98 1.80 (0.074) (0.065) (0.032) ln SDHt 5 2 1.99 1 0.056lnUSDt 1 0.969lnSYt 1 0.057lnSPt 352 SERIAL CORRELATION a. The authors expected positive signs for all the slope coefficients of both equations. Test these hypotheses at the 5-percent level. b. Use our four specification criteria to determine whether SP is an irrel- evant variable. Explain your reasoning. c. Test both equations for positive first-order serial correlation. Does the high probability of serial correlation cause you to reconsider your answer to part b? Explain. d. Someone might argue that because the DW statistic improved when lnSP was added, that the serial correlation was impure and Table 2 Data on Soviet Defense Spending Year SDH SDL USD SY SFP NR NU 1960 31 23 200.54 232.3 7.03 415 1734 1961 34 26 204.12 245.3 6.07 445 1846 1962 38 29 207.72 254.5 3.90 485 1942 1963 39 31 206.98 251.7 2.97 531 2070 1964 42 34 207.41 279.4 1.40 580 2910 1965 43 35 185.42 296.8 1.87 598 4110 1966 44 36 203.19 311.9 4.10 674 4198 1967 47 39 241.27 326.3 4.90 1058 4338 1968 50 42 260.91 346.0 4.07 1270 4134 1969 52 43 254.62 355.9 2.87 1662 4026 1970 53 44 228.19 383.3 4.43 2047 5074 1971 54 45 203.80 398.2 3.77 3199 6282 1972 56 46 189.41 405.7 2.87 2298 7100 1973 58 48 169.27 435.2 3.87 2430 8164 1974 62 51 156.81 452.2 4.30 2534 8522 1975 65 53 155.59 459.8 6.33 2614 9170 1976 69 56 169.91 481.8 0.63 3219 9518 1977 70 56 170.94 497.4 2.23 4345 9806 1978 72 57 154.12 514.2 1.03 5097 9950 1979 75 59 156.80 516.1 0.17 6336 9945 1980 79 62 160.67 524.7 0.27 7451 9668 1981 83 63 169.55 536.1 0.47 7793 9628 1982 84 64 185.31 547.0 0.07 8031 10124 1983 88 66 201.83 567.5 1.50 8730 10201 1984 90 67 211.35 578.9 1.63 9146 10630 Source: Josef C. Brada and Ronald L. Graves, “The Slowdown in Soviet Defense Expenditures,” Southern Economic Journal, Vol. 54, No. 4, p. 974. Datafile = DEFEND9 353 SERIAL CORRELATION that GLS was not called for. Do you agree with this conclusion? Why or why not? e. If we run a GLS version of Equation 26, we get Equation 28. Does this result cause you to reconsider your answer to part b? Explain: (28) 15. As an example of impure serial correlation caused by an incorrect functional form, let’s return to the equation for the percentage of putts made (Pi) as a function of the length of the putt in feet (Li) that we discussed originally in Exercise 6 in Chapter 1. The complete doc- umentation of that equation is (29) a. Test Equation 29 for serial correlation using the Durbin–Watson d test at the 1-percent level. b. Why might the linear functional form be inappropriate for this study? Explain your answer. c. If we now reestimate Equation 29 using a double-log functional form, we obtain: (30) Test Equation 30 for serial correlation using the Durbin–Watson d test at the 1-percent level. d. Compare Equations 29 and 30. Which equation do you prefer? Why? N 5 19 R2 5 .903 DW 5 1.22 t 5 2 13.0 (0.07) lnPi 5 5.50 2 0.92 lnLi N 5 19 R2 5 .861 DW 5 0.48 t 5 2 10.6 (0.4) P̂i 5 83.6 2 4.1Li N 5 24 (annual 1960–1984) R2 5 .994 �̂ 5 0.96 t 5 1.61 0.64 2 0.03 (0.067) (0.214) (0.027) ln SDHt 5 3.55 1 0.108lnUSDt 1 0.137 lnSYt 2 0.0008 lnSPt 354 Answers Exercise 2 a. Yt PBt PRPt Dt H0 �1 � 0 �2 � 0 �3 � 0 �4 � 0 HA �1 0 �2 0 �3 0 �4 0 tY � 6.6 tPB � �2.6 tPRP � 2.7 tD � �3.17 tc � 1.714 tc � 1.714 tc � 1.714 tc � 1.714 We can reject the null hypothesis for all four coefficients because the t-scores all are in the expected direction with absolute values greater than 1.714 (the 5-percent one-sided critical t-value for 23 degrees of freedom). b. With a 5-percent, one-sided test and N � 28, K � 4, the critical values are dL � 1.10 and du � 1.75. Since d � 0.94 1.10, we can reject the null hypothesis of no positive serial correlation. c. The probable positive serial correlation suggests GLS. d. We prefer the GLS equation, because we’ve rid the equation of much of the serial correlation while retaining estimated coeffi- cients that make economic sense. Note that the dependent vari- ables in the two equations are different, so an improved fit is not evidence of a better equation. SERIAL CORRELATION 355 356 We believe that econometrics is best learned by doing, not by reading books, listening to lectures, or taking tests. To us, learning the art of econometrics has more in common with learning to fly a plane or learning to play golf than it does with learning about history or literature. In fact, we developed the interactive exercises of this chapter precisely because of our confidence in learning by doing. Although interactive exercises are a good bridge between textbook exam- ples and running your own regressions, they don’t go far enough. You still need to “get your hands dirty.” We think that you should run your own regression project before you finish reading this text even if you’re not required to do so. We’re not alone. Some professors substitute a research proj- ect for the final exam as their class’s comprehensive learning experience. Running your own regression project has three major components: 1. Choosing a topic 2. Applying the six steps in regression analysis to that topic 3. Writing your research report The first and third of these components are the topics of Sections 1 and 5, re- spectively. The rest of the chapter focuses on helping you carry out the six steps in regression analysis. 1 Choosing Your Topic 2 Collecting Your Data 3 Advanced Data Sources 4 Practical Advice for Your Project 5 Writing Your Research Report 6 A Regression User’s Checklist and Guide 7 Summary 8 Appendix: The Housing Price Interactive Exercise Running Your Own Regression Project From Chapter 11 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011 by Pearson Education. Published by Addison-Wesley. All rights reserved. 357 RUNNING YOUR OWN REGRESSION PROJECT Choosing Your Topic The purpose of an econometric research project is to use regression analysis to build the best explanatory equation for a particular dependent variable for a particular sample. Often, though, the hardest part is getting started. How can you choose a good topic? There are at least three keys to choosing a topic. First, try to pick a field that you find interesting and/or that you know something about. If you enjoy working on your project, the hours involved will seem to fly by. In addition, if you know something about your subject, you’ll be more likely to make cor- rect specification choices and/or to notice subtle indications of data errors or theoretical problems. A second key is to make sure that data are readily avail- able with a reasonable sample (we suggest at least 25 observations). Nothing is more frustrating than searching through data source after data source in search of numbers for your dependent variable or one of your independent variables, so before you lock yourself into a topic, see if the data are there. The final key is to make sure that there is some substance to your topic. Try to avoid topics that are purely descriptive or virtually tautological in nature. In- stead, look for topics that address an inherently interesting economic or be- havioral question or choice. Perhaps the best place to look for ideas for topics is to review your text- books and notes from previous economics classes or to look over the exam- ples and exercises. Often, you can take an idea from a previous study and up- date the data to see if the idea can be applied in a different context. Other times, reading an example will spark an idea about a similar or related study that you’d be interested in doing. Don’t feel that your topic has to contain an original hypothesis or equation. On your first or second project, it’s more im- portant to get used to the econometrics than it is to create a publishable mas- terpiece. Another way to find a topic is to read through issues of economics jour- nals, looking for article topics that you find interesting and that might be possible to model. For example, Table 1 contains a list of the journals cited so far in this text (in order of the frequency of citation). These journals would be a great place to start if you want to try to replicate or update a previous re- search study. Although this is an excellent way to get ideas, it’s also frustrat- ing, because most current articles use econometric techniques that go beyond those that we’ve covered so far in this text. As a result, it’s often difficult to compare your results to those in the article. If you get stuck for a topic, go directly to the data sources themselves. That is, instead of thinking of a topic and then seeing if the data are available, look over what data are available and see if they help generate ideas for topics. Quite often, a reference will have data not only for a dependent variable but 1 358 RUNNING YOUR OWN REGRESSION PROJECT also for most of the relevant independent variables all in one place, minimiz- ing time spent collecting data. Once you pick a topic, don’t rush out and run your first regression. Re- member, the more time you spend reviewing the literature and analyzing your expectations on a topic, the better the econometric analysis and, ulti- mately, your research report will be. Collecting Your Data Before any quantitative analysis can be done, the data must be collected, orga- nized, and entered into a computer. Usually, this is a time-consuming and frus- trating task because of the difficulty of finding data, the existence of definitional differences between theoretical variables and their empirical counterparts, and 2 Table 1 Sources of Potential Topic Ideas American Economic Review Econometrica Journal of Applied Econometrics Journal of Urban Economics Southern Economic Journal Economica Economic Inquiry Journal of the American Statistical Association Journal of Econometrics Journal of Economic Education Journal of Money, Credit and Banking Review of Economics and Statistics World Development Biometrica The Annals of Statistics American Psychologist Annals of Mathematical Statistics Applied Economics Assessment and Evaluation of Higher Education Journal of Business and Economic Statistics Journal of Economic Literature Journal of Economic Perspectives Journal of Economic Surveys Journal of Financial and Quantitative Studies Journal of the Royal Statistical Society National Tax Review NBER (Working Papers) Scandinavian Journal of Economics 359 the high probability of data entry errors or data transmission errors. In general, though, time spent thinking about and collecting the data is well spent, since a researcher who knows the data sources and definitions is much less likely to make mistakes using or interpreting regressions run on that data. What Data to Look For Before you settle on a research topic, it’s good advice to make sure that data for your dependent variable and all relevant independent variables are avail- able. However, checking for data availability means deciding what specific variables you want to study. Half of the time that beginning researchers spend collecting data is wasted by looking for the wrong variables in the wrong places. A few minutes thinking about what data to look for will save hours of frustration later. For example, if the dependent variable is the quantity of television sets de- manded per year, then most independent variables should be measured an- nually as well. It would be inappropriate and possibly misleading to define the price of TVs as the price from a particular month. An average of prices over the year (usually weighted by the number of TVs sold per month) would be more meaningful. If the dependent variable includes all TV sets sold re- gardless of brand, then the price would appropriately be an aggregate based on prices of all brands. Calculating such aggregate variables, however, is not straightforward. Researchers typically make their best efforts to compute the respective aggregate variables and then acknowledge that problems still re- main. For example, if the price data for all the various brands are not avail- able, a researcher may be forced to compromise and use the price of one or a few of the major brands as a substitute for the proper aggregate price. Another issue is suggested by the TV example. Over the years of the sample, it’s likely that the market shares of particular kinds of TV sets have changed. For example, flat-screen HD TV sets might have made up a majority of the market in one decade, but black-and-white sets might have been the favorite 40 years be- fore. In cases where the composition of the market share, the size, or the quality of the various brands have changed over time, it would make little sense to mea- sure the dependent variable as the number of TV sets because a “TV set” from one year has little in common with a “TV set” from another. The approach usu- ally taken to deal with this problem is to measure the variable in dollar terms, under the assumption that value encompasses size and quality. Thus, we would work with the dollar sales of TVs rather than the number of sets sold. A third issue, whether to use nominal or real variables, usually depends on the underlying theory of the research topic. Nominal (or money) variables are measured in current dollars and thus include increases caused by inflation. RUNNING YOUR OWN REGRESSION PROJECT 360 If theory implies that inflation should be filtered out, then it’s best to state the variables in real (constant-dollar) terms by selecting an appropriate price deflator, such as the Consumer Price Index, and adjusting the money (or nominal) value by it. As an example, the appropriate price index for Gross Domestic Product is called the GDP deflator. Real GDP is calculated by multiplying nominal GDP by the ratio of the GDP deflator from the base year to the GDP deflator from the current year: In 2007, U.S. nominal GDP was $13,807.5 billion and the GDP deflator was 119.82 (for a base year of 2000 � 100), so real GDP was:1 That is, the goods and services produced in 2007 were worth $13,807.5 bil- lion if 2007 dollars were used but were worth only $11,523.9 billion if 2000 prices were used. Fourth, recall that all economic data are either time-series or cross-sectional in nature. Since time-series data are for the same economic entity from differ- ent time periods, whereas cross-sectional data are from the same time period but for different economic entities, the appropriate definitions of the variables depend on whether the sample is a time series or a cross-section. To understand this, consider the TV set example once again. A time-series model might study the sales of TV sets in the United States from 1967 to 2005, and a cross-sectional model might study the sales of TV sets by state for 2005. The time-series data set would have 39 observations, each of which would refer to a particular year. In contrast, the cross-sectional model data set would have 50 observations, each of which would refer to a particular state. A variable that might be appropriate for the time-series model might be com- pletely inappropriate for the cross-sectional model, and vice versa; at the very least, it would have to be measured differently. National advertising in a par- ticular year would be appropriate for the time-series model, for example, while advertising in or near each particular state would make more sense for the cross-sectional one. Finally, learn to be a critical reader of the descriptions of variables in economet- ric research. For instance, most readers breezed right through the equation on the demand for beef without asking some vital questions. Are prices and Real GDP 5 $13,807.5 (100>119.82) 5 $11,523.9 billion
Real GDP 5 nominal GDP 3 (base GDP deflator>current GDP deflator)
RUNNING YOUR OWN REGRESSION PROJECT
1. 2009 Economic Report of the President, pp. 282–285.
361

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income measured in nominal or real terms? Is the price of beef wholesale or
retail? Where did the data originate? A careful reader would want to know the
answers to these questions before analyzing the results of Equation 7 from
chapter 2. (For the record, Yd measures real income, P measures real whole-
sale prices, and the data come from various issues of Agricultural Statistics,
published in Washington, D.C., by the U.S. Department of Agriculture.)
Where to Look for Economic Data
Although some researchers generate their own data through surveys or other
techniques (and we’ll address this possibility in Section 3), the vast majority
of regressions are run on publicly available data. The best sources for such
data are government publications and machine-readable data files. In fact,
the U.S. government has been called the most thorough statistics-collecting
agency in history.
Excellent government publications include the annual Statistical Abstract of
the U.S., the annual Economic Report of the President, the Handbook of Labor
Statistics, and Historical Statistics of the U.S. (published in 1975). One of the
best places to start with U.S. data is the annual Census Catalog and Guide,
which provides overviews and abstracts of data sources and various statistical
products as well as details on how to obtain each item.2 Consistent interna-
tional data are harder to come by, but the United Nations publishes a number
of compilations of figures. The best of these are the U.N. Statistical Yearbook
and the U.N. Yearbook of National Account Statistics.
Most researchers use on-line computer databases to find data instead of
plowing through stacks of printed volumes. These on-line databases, avail-
able through most college and university libraries, contain complete series
on literally thousands of possible variables. A huge variety of data is available
directly on the Internet. The best guides to the data available in this rapidly
changing world are “Resources for Economists on the Internet,” Economagic,
and WebEC.3 Links to these sites and other good sources of data are on the
text’s Web site www.pearsonhighered.com/studenmund. Other good Internet
resources are EconLit (www.econlit.org), which is an on-line summary of the
Journal of Economic Literature, and “Dialog,” which provides on-line access to
a large number of data sets at a lower cost than many alternatives.
2. To obtain this guide, write the Superintendent of Documents, Government Printing Office,
Washington, D.C.
3. On the Web, the Resources for Economists location is http://www.rfe.org. The Economagic
location is www.economagic.com. The WebEC location is www.helsinki.fi/WebEc.
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Missing Data
Suppose the data aren’t there? What happens if you choose the perfect vari-
able and look in all the right sources and can’t find the data?
The answer to this question depends on how much data is missing. If a
few observations have incomplete data in a cross-sectional study, you usually
can afford to drop these observations from the sample. If the incomplete data
are from a time series, you can sometimes estimate the missing value by in-
terpolating (taking the mean of adjacent values). Similarly, if one variable is
available only annually in an otherwise quarterly model, you may want to
consider quarterly interpolations of that variable. In either case, interpolation
can be justified only if the variable moves in a slow and smooth manner. Ex-
treme caution should always be exercised when “creating” data in such a way
(and full documentation is required).
If no data at all exist for a theoretically relevant variable, then the problem
worsens significantly. Omitting a relevant variable runs the risk of biased co-
efficient estimates. After all, how can you hold a variable constant if it’s not
included in the equation? In such cases, most researchers resort to the use of
proxy variables.
Proxy variables can sometimes substitute for theoretically desired variables
for which data are missing. For example, the value of net investment is a vari-
able that is not measured directly in a number of countries. As a result, a re-
searcher might use the value of gross investment as a proxy, the assumption
being that the value of gross investment is directly proportional to the value of
net investment. This proportionality (which is similar to a change in units) is
required because the regression analyzes the relationship between changes
among variables, rather than the absolute levels of the variables.
In general, a proxy variable is a “good” proxy when its movements corre-
spond relatively well to movements in the theoretically correct variable. Since
the latter is unobservable whenever a proxy must be used, there is usually no
easy way to examine a proxy’s “goodness” directly. Instead, the researcher
must document as well as possible why the proxy is likely to be a good or
bad one. Poor proxies and variables with large measurement errors constitute
“bad” data, but the degree to which the data are bad is a matter of judgment
by the individual researcher.
Advanced Data Sources
So far, all the data sets in this text have been cross-sectional or time-series in
nature, and we have collected our data by observing the world around us, in-
stead of by creating the data ourselves. It turns out, however, that time-series
3
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and cross-sectional data can be pooled to form panel data, and that data can
be generated through surveys. The purpose of this short section is to introduce
you to these more advanced data sources and to explain why it probably
doesn’t make sense to use these data sources on your first regression project.
Surveys
Surveys are everywhere in our society. Marketing firms use surveys to learn
more about products and competition, political candidates use surveys to fine-
tune their campaign advertising or strategies, and governments use surveys for
all sorts of purposes, including keeping track of their citizens with instruments
like the U.S. Census. As a result, many beginning researchers (particularly those
who are having trouble obtaining data for their project) are tempted to run their
own surveys in the hope that it’ll be an easy way to generate the data they need.
However, running a survey is not as easy as it might seem. For example, the
topics to be covered in the survey need to be thought through carefully, be-
cause once a survey has been run, it’s virtually impossible to go back to the re-
spondents and add another question. In addition, the questions themselves
need to be worded precisely (and pretested) to avoid confusing the respondent
or “leading” the respondent to a particular answer. Perhaps most importantly,
it’s crucial for the sample to be random and to avoid the selection, survivor,
and nonresponse biases. In fact, running a survey properly is so difficult that
entire books and courses are devoted to the topic.
As a result, we don’t encourage beginning researchers to run their own sur-
veys, and we’re cautious when we analyze the results of surveys run by others.
As put by the American Statistical Association, “The quality of a survey is best
judged not by its size, scope, or prominence, but by how much attention is
given to preventing, measuring, and dealing with the many important prob-
lems that can arise.”4
Panel Data
As mentioned previously, panel data are formed when cross-sectional and
time-series data sets are pooled to create a single data set. Why would you
want to use panel data? In some cases, researchers use panel data to increase
4. As quoted in “Best Practices for Survey and Public Opinion Research,” on the web site of the
American Association for Public Opinion Research: www.aapor.org/bestpractices. The best prac-
tices outlined on this web site are a good place to start if you decide to create your own survey.
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their sample size, but the main reason for using panel data is to provide an
insight into an analytical question that can’t be obtained by using time-series
or cross-sectional data alone.
What’s an example of panel data? Suppose that we’re interested in the rela-
tionship between budget deficits and interest rates but that we have only
10 years’ worth of comparable annual data to study. Ten observations is too
small a sample for a reasonable regression, so it might seem as if we’re out
of luck. However, if we can find time-series data on the same economic
variables—interest rates and budget deficits—for the same ten years for six
different countries, we’ll end up with a sample of 10∗6 � 60 observations,
which is more than enough to use. The result is a pooled cross-section time-
series data set—a panel data set!
Unfortunately, panel data can’t be analyzed fully with the econometric
techniques you’ve learned to date in this text, so we don’t encourage be-
ginning researchers to attempt to run regressions on panel data. Instead,
we’ve devoted the majority of a chapter to panel data, and we urge you to
read that chapter if you’re interested.
Practical Advice for Your Project
“Econometrics is much easier without data.”5
The purpose of this section6 is to give the reader some practical advice
about actually doing applied econometric work. Such advice often is miss-
ing from econometrics textbooks and courses, but the advice is crucial be-
cause many of the skills of an applied econometrician are judgmental and
subjective in nature. No single text or course can teach these skills, and that’s
not our goal. Instead, we want to alert you to some technical suggestions
that a majority of experienced applied econometricians would be likely to
support.
4
5. M. Verbeek, A Guide to Modern Econometrics (New York: Wiley, 2000), p. 1.
6. This section was inspired by and heavily draws upon Chapter 22, “Applied Econometrics,” in
Peter Kennedy’s A Guide to Econometrics (Malden, MA: Blackwell, 2008), pp. 361–384. We are
extremely grateful to Prof. Kennedy, the MIT Press, and Blackwell Publishing for their kind per-
mission to reprint major portions of that chapter here.
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We start off with Peter Kennedy’s “10 commandments of applied econo-
metrics,” move on to discuss what to check if you get an unexpected sign, and
finish up by bringing together a dozen practical tips from other sections of
this text that are worth reiterating.
The 10 Commandments of Applied Econometrics
Rule 1: Use common sense and economic theory.
“Time and again I was thanked (and paid) for asking questions and
suggesting perspectives that seemed to me to be little more than common
sense. This common sense is an easily overlooked but extraordinarily
valuable commodity.”7
Common sense is not all that common. In fact, it sometimes seems as if
not much thought (let alone good thought) has gone into empirical work.
There are thousands of examples of common sense. For example, common
sense should cause researchers to match per capita variables with per capita
variables, to use real exchange rates to explain real imports or exports, to em-
ploy nominal interest rates to explain real money demand, and to never,
never infer causation from correlation.
Rule 2: Ask the right questions.
“Far better an approximate answer to the right question, which is often
vague, than an exact answer to the wrong question, which can always
be made precise.”8
Be sure that the question being asked is the relevant one. When a re-
searcher encounters a regression problem, the solution to that problem often
is quite simple. Asking simple questions about the context of the problem
can bring to light serious misunderstandings. For example, it may be that it is
the cumulative change in a variable that is relevant, not the most recent
change, or it may be that the null hypothesis should be that a coefficient is
equal to another coefficient, rather than equal to zero.
The main lesson here is a blunt one: Ask questions, especially seemingly
foolish questions, to ensure that you have a full understanding of the goal of
the research; it often turns out that the research question has not been for-
mulated appropriately.
7. M. W. Trosset, Comment, Statistical Science, 1998, p. 23.
8. J. W. Tukey, “The Future of Data Analysis,” Annals of Mathematical Statistics, Vol. 33, No. 1,
pp. 13–14.
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Rule 3: Know the context.
“Don’t try to model without understanding the non-statistical aspects
of the real-life system you are trying to subject to statistical analysis.
Statistical analysis done in ignorance of the subject matter is just that
—ignorant statistical analysis.”9
It’s crucial to become intimately familiar with the subject being investigated—
its history, institutions, operating constraints, measurement peculiarities,
cultural customs, and so on, going beyond a thorough literature review.
Questions must be asked: Exactly how were the data gathered? Did govern-
ment agencies impute the data using unknown formulas? What were the
rules governing the auction? How were the interviewees selected? What in-
structions were given to the participants? What accounting conventions
were followed? How were the variables defined? What is the precise word-
ing of the questionnaire? How closely do measured variables match their
theoretical counterparts? Another way of viewing this rule is to recognize
that you, the researcher, know more than the computer—you know, for ex-
ample, that water freezes at 0 degrees Celsius, that people tend to round
their incomes to the nearest five thousand, and that some weekends are
three-day weekends.
Rule 4: Inspect the data.
“Every number is guilty unless proved innocent.”10
Even if a researcher knows the subject, he or she needs to become inti-
mately familiar with the data. Economists are particularly prone to the com-
plaint that researchers do not know their data very well, a phenomenon
made worse by the computer revolution, which has allowed researchers to
obtain and work with data electronically by pushing buttons.
Inspecting the data involves summary statistics, graphs, and data cleaning,
to both check and “get a feel for” the data. Summary statistics tend to be very
simple, such as means, standard errors, maximums, minimums, and correla-
tion matrices, but they can help a researcher find data errors that otherwise
would have gone undetected. If in doubt, graph your data. The advantage of
graphing is that a picture can force us to notice what we never expected to
9. D. A. Belsley and R. E. Welch, “Modelling Energy Consumption—Using and Abusing Regres-
sion Diagnostics,” Journal of Business and Economic Statistics, Vol. 6, p. 47.
10. C. R. Rao, Statistics and Truth: Putting Chance to Work (Singapore: World Scientific, 1997),
p. 152.
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RUNNING YOUR OWN REGRESSION PROJECT
see. Researchers should supplement their summary statistics with simple
graphs: histograms, residual plots, scatterplots of residualized data, and
graphs against time. Data cleaning looks for inconsistencies in the data—are
any observations impossible, unrealistic, or suspicious? Do you know how
missing data were coded? Are dummies all coded 0 or 1? Are all observations
consistent with applicable minimum or maximum values? Do all observa-
tions obey the logical constraints that they must satisfy?
Rule 5: Keep it sensibly simple.
“Do not choose an analytic method to impress your readers or to deflect
criticism. If the assumptions and strength of a simpler method are rea-
sonable for your data and research problem, use it.”11
Progress in economics results from beginning with simple models, seeing
how they work in applications, and then modifying them if necessary. Begin-
ning with a simple model is referred to as a bottom-up (or specific-to-general)
approach to developing an econometric specification. Its main drawback is
that testing is biased if the simple model omits one or more relevant variables.
The competing top-down (or general-to-specific) approach is unrealistic in
that it requires the researcher to be able to think of the “right” general model
from the start.
Over time, a compromise methodology has evolved. Practitioners begin with
simple models which are expanded whenever they fail. When they fail, the gen-
eral-to-specific approach is used to create a new simple model that is subjected
to misspecification tests, and this process of discovery is repeated. In this way
simplicity is combined with the general-to-specific methodology, producing a
compromise process which, judging by its wide application, is viewed as an ac-
ceptable rule of behavior. Examples are the functional form specifications of
some Nobel Laureates—Tinbergen’s social welfare functions; Arrow’s and
Solow’s work on the CES production function; Friedman’s, Becker’s, Tobin’s,
and Modigliani’s consumer models; and Lucas’s rational expectations model.
Rule 6: Look long and hard at your results.
“Apply the ‘laugh’ test—if the findings were explained to a layperson,
could that person avoid laughing?”12
11. Leland Wilkinson and the Task Force on Statistical Inference, “Statistical Methods in Psy-
chology Journals,” American Psychologist, Vol. 54, No. 8, p. 598.
12. Peter Kennedy, A Guide to Econometrics (Malden, MA: Blackwell, 2008), p. 393.
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Part of this rule is to check whether the results make sense. Are the signs
of coefficients as expected? Are important variables statistically significant?
Are coefficient magnitudes reasonable? Are the implications of the results
consistent with theory? Are there any anomalies? Are any obvious restric-
tions evident?
But another part of this rule is more subtle and subjective. By looking long
and hard at reams of computer output, researchers should eventually recog-
nize the message they are conveying and become comfortable with it. This
subjective procedure should be viewed as separate from and complementary
to formal statistical testing procedures.
Rule 7: Understand the costs and benefits of data mining.
“Any attempt to allow data to play a role in model specification . . .
amounted to data mining, which was the greatest sin any researcher
could commit.”13
“Data mining is misunderstood, and once it is properly understood, it is
seen to be no sin at all.”14
There are two variants of “data mining”: one classified as the greatest of
the basement sins, but the other viewed as an important ingredient in data
analysis. The undesirable version of data mining occurs when one tailors
one’s specification to the data, resulting in a specification that is misleading
because it embodies the peculiarities of the particular data at hand. Further-
more, traditional testing procedures used to “sanctify” the specification are
no longer legitimate, because these data, since they have been used to gener-
ate the specification, cannot be judged impartial if used to test that specifica-
tion. The desirable version of “data mining” refers to experimenting with the
data to discover empirical regularities that can inform economic theory and
be tested on a second data set.
Data mining is inevitable; the art of the applied econometrician is to allow
for data-driven theories while avoiding the considerable danger inherent in
testing those data-driven theories on the same datasets that were used to cre-
ate them.
13. C. Mukherjee, H. White, and M. Wuyts, Econometrics and Data Analysis for Developing Coun-
tries (London: Routledge, 1998), p. 30.
14. K. D. Hoover, “In Defense of Data Mining: Some Preliminary Thoughts,” in K. D. Hoover
and S. M. Sheffrin (eds.), Monetarism and the Methodology of Economics: Essays in Honor of Thomas
Mayer (Aldershot: Edward Elgar, 1995), p. 243.
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Rule 8: Be prepared to compromise.
“The three most important aspects of real data analysis are to compro-
mise, compromise, compromise.”15
In virtually every econometric analysis there is a gap—usually a vast gulf—
between the problem at hand and the closest scenario to which standard
econometric theory is applicable. Very seldom does one’s problem even come
close to satisfying the Classical Assumptions under which econometric the-
ory delivers an optimal solution. A consequence of this is that practitioners
are always forced to compromise and adopt suboptimal solutions, the char-
acteristics of which are unknown.
The issue here is that in their econometric theory courses students are
taught standard solutions to standard problems, but in practice there are no
standard problems. Applied econometricians are continually faced with awk-
ward compromises and must be willing to make ad hoc modifications to
standard solutions.
Rule 9: Do not confuse statistical significance with meaningful
magnitude.
“Few would deny that in the hands of the masters the methodologies
perform impressively, but in the hands of their disciples it is all much
less convincing.”16
Very large sample sizes, such as those that have become common in cross-
sectional data, can give rise to estimated coefficients with very small standard
errors. A consequence of this is that coefficients of trivial magnitude may test
significantly different from zero, creating a misleading impression of what is
important. Because of this, researchers must always look at the magnitude of
coefficient estimates as well as their significance.
An even more serious problem associated with significance testing is that
there is a tendency to conclude that finding significant coefficients “sancti-
fies” a theory, with a resulting tendency for researchers to stop looking for
further insights. Sanctification via significance testing should be replaced by
continual searches for additional evidence, both corroborating evidence and,
especially, disconfirming evidence. If your theory is correct, are there testable
15. Ed Leamer, “Revisiting Tobin’s 1950 Study of Food Expenditure,” Journal of Applied Econo-
metrics, Vol. 12, No. 5, p. 552.
16. A. R. Pagan, “Three Econometric Methodologies: A Critical Appraisal,” Journal of Economic
Surveys, Vol. 1, p. 20.
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implications? Can you explain a range of interconnected findings? Can you
find a bundle of evidence consistent with your hypothesis but inconsistent
with alternative hypotheses? Can your theory “encompass” its rivals in the
sense that it can explain other models’ results?
Rule 10: Report a sensitivity analysis.
“Sinners are not expected to avoid sins; they need only confess their
errors openly.”17
It’s important to check whether regression results are sensitive to the assump-
tions upon which the estimation has been based. This is the purpose of a sensi-
tivity analysis, indicating to what extent the substantive results of the research
are affected by adopting different specifications about which reasonable people
might disagree. For example, are the results sensitive to the sample period, the
functional form, the set of explanatory variables, or the choice of proxies? If
they are, then this sensitivity casts doubt on the conclusions of the research.
There’s a second dimension to sensitivity analyses. Published research pa-
pers are typically notoriously misleading accounts of how the research actu-
ally was conducted. Because of this, it’s very difficult for readers of research
papers to judge the extent to which data mining may have unduly influenced
the results. Indeed, results tainted by subjective specification decisions un-
dertaken during the heat of econometric battle should be considered the rule,
rather than the exception. When reporting a sensitivity analysis, researchers
should explain fully their specification search so that readers can judge for
themselves how the results may have been affected.
What to Check If You Get an Unexpected Sign
An all-too-familiar problem for a beginning econometrician is to run a re-
gression and find that the sign of one or more of the estimated coefficients is
the opposite of what was expected. While an unexpected sign certainly is
frustrating, it’s not entirely bad news. Rather than considering this a disaster,
a researcher should consider it a blessing—this result is a friendly message
that some detective work needs to be done—there is undoubtedly some
shortcoming in one’s theory, data, specification, or estimation procedure. If
the “correct” signs had been obtained, odds are that the analysis would not
be double-checked. What should be checked?
17. Ed Leamer, Specification Searches: Ad Hoc Inference with Nonexperimental Data (New York:
John Wiley, 1978), p. vi.
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1. Recheck the expected sign. Every once in a while, a variable that is defined
“upside down” will cause a researcher to expect the wrong sign. For ex-
ample, in an equation for student SATs, the variable “high school rank
in class” (where a rank of 1 means that the student was first in his or
her class) can sometimes lure a beginning researcher into expecting a
positive coefficient for rank.
2. Check your data for input errors and/or outliers. If you have data errors or
oddball observations, the chances of getting an unexpected sign—even
a significant unexpected sign—increase dramatically.
3. Check for an omitted variable. The most frequent source of a significant
unexpected sign for the coefficient of a relevant independent variable is
an omitted variable. Think hard about what might have been omitted,
and, in particular, remember to use our equation for expected bias.
4. Check for an irrelevant variable. A frequent source of insignificant unex-
pected signs is that the variable doesn’t actually belong in the equation
in the first place. If the true coefficient for an irrelevant variable is zero,
then you’re likely to get an unexpected sign half the time.
5. Check for multicollinearity. Multicollinearity increases the variances and
standard errors of the estimated coefficients, increasing the chance that
a coefficient could have an unexpected sign. The sampling distribu-
tions will be widely spread and may straddle zero, implying that it is
quite possible that a draw from this distribution will produce an unex-
pected sign. Indeed, one of the casual indicators of multicollinearity is
the presence of unexpected signs.
6. Check for sample selection bias. An unexpected sign sometimes can be due
to the fact that the observations included in the data were not obtained
randomly.
7. Check your sample size. Multicollinearity isn’t the only source of high
variances; they could result from a small sample size or minimal varia-
tion in the explanatory variables. In some cases, all it takes to fix an un-
expected sign is to increase the sample.
8. Check your theory. If you’ve exhausted every logical econometric expla-
nation for your unexpected sign, there are only two likely remaining
explanations. Either your theory is wrong, or you’ve got a bad data set.
If your theory is wrong, then you of course have to change your ex-
pected sign, but remember to test this new expectation on a different
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data set. However, be careful! It’s amazing how economists can conjure
up rationales for unexpected signs after the regression has been run!
One theoretical source of bias, and therefore unexpected signs, is if the
underlying model is simultaneous in nature.
A Dozen Practical Tips Worth Reiterating
Here are a number of practical tips for applied econometrics that are worth
emphasizing. They work!
1. Don’t attempt to maximize .
2. Always review the literature and hypothesize the signs of your coeffi-
cients before estimating a model.
3. Remember to inspect and clean your data before estimating a model.
Know that outliers should not be automatically omitted; instead,
they should be investigated to make sure that they belong in the
sample.
4. Know the Classical Assumptions cold!
5. In general, use a one-sided t-test unless the expected sign of the coef-
ficient actually is in doubt.
6. Don’t automatically discard a variable with an insignificant t-score. In
general, be willing to live with a variable with a t-score lower than the
critical value in order to decrease the chance of omitting a relevant
variable.
7. Know how to analyze the size and direction of the bias caused by an
omitted variable.
8. Understand all the different functional form options and their com-
mon uses, and remember to choose your functional form primarily
on the basis of theory, not fit.
9. Remember that multicollinearity doesn’t create bias; the estimated
variances are large, but the estimated coefficients themselves are un-
biased. As a result, the most-used remedy for multicollinearity is to
do nothing.
10. If you get a significant Durbin–Watson, Park, or White test, remember
to consider the possibility that a specification error might be causing
impure serial correlation or heteroskedasticity. Don’t change your esti-
mation technique from OLS to GLS or use adjusted standard errors
until you have the best possible specification.
R2
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11. Remember that adjusted standard errors like Newey–West standard
errors or HC standard errors use the OLS coefficient estimates. It’s the
standard errors of the estimated coefficients that change, not the esti-
mated coefficients themselves.
12. Finally, and perhaps most importantly, if in doubt, rely on common
sense and economic theory, not on statistical tests.
The Ethical Econometrician
One conclusion that a casual reader of this text might draw from the large
number of specifications we include is that we encourage the estimation of
numerous regression results as a way of ensuring the discovery of these best
possible estimates.
Nothing could be further from the truth!
As every reader of this text should know by now, our opinion is that the
best models are those on which much care has been spent to develop the the-
oretical underpinnings and only a short time is spent pursuing alternative
estimations of that equation. Many econometricians, ourselves included,
would hope to be able to estimate only one specification of an equation for
each data set. Econometricians are fallible and our data are sometimes im-
perfect, however, so it is unusual for a first attempt at estimation to be totally
problem free. As a result, two or even more regressions are often necessary to
rid an estimation of fairly simple difficulties that perhaps could have been
avoided in a world of perfect foresight.
Unfortunately, a beginning researcher usually has little motivation to stop
running regressions until he or she likes the way the result looks. If running
another regression provides a result with a better fit, why shouldn’t one more
specification be tested?
The reason is a compelling one. Every time an extra regression is run and a
specification choice is made on the basis of fit or statistical significance, the
chances of making a mistake of inference increase dramatically. This can hap-
pen in at least two ways:
1. If you consistently drop a variable when its coefficient is insignificant
but keep it when it is significant, it can be shown that you bias your
estimates of the coefficients of the equation and of the t-scores.
2. If you choose to use a lag structure, or a functional form or an estima-
tion procedure other than OLS, on the basis of fit rather than on the
basis of previously theorized hypotheses, you run the risk that your
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equation will work poorly when it’s applied to data outside your
sample. If you restructure your equation to work well on one data
set, you might decrease the chance of it working well on another.
What might be thought of as ethical econometrics is also in reality good
econometrics. That is, the real reason to avoid running too many different
specifications is that the fewer regressions you run, the more reliable and
more consistently trustworthy are your results. The instance in which profes-
sional ethics come into play is when a number of changes are made (differ-
ent variables, lag structures, functional forms, estimation procedures, data
sets, dropped outliers, and so on), but the regression results are presented to
colleagues, clients, editors, or journals as if the final and best equation had
been the first and only one estimated. Our recommendation is that all esti-
mated equations be reported even if footnotes or an appendix have to be
added to the documentation.
We think that there are two reasonable goals for econometricians when es-
timating models:
1. Run as few different specifications as possible while still attempting
to avoid the major econometric problems. The only exception to our
recommendation to run as few specifications as possible is sensitivity
analysis.
2. Report honestly the number and type of different specifications esti-
mated so that readers of the research can evaluate how much weight
to give to your results.
Therefore, the art of econometrics boils down to attempting to find the best
possible equation in the fewest possible number of regression runs. Only care-
ful thinking and reading before estimating first regression can bring this about.
An ethical econometrician is honest and complete in reporting the different
specifications and/or data sets used.
Writing Your Research Report
Once you’ve finished your research, it’s important to write a report on your re-
sults so that others can benefit from what you found out (or didn’t find out)
or so that you can get feedback on your econometric techniques from some-
one else. Most good research reports have a number of elements in common:
● A brief introduction that defines the dependent variable and states the
goals of the research.
● A short review of relevant previous literature and research.
5
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● An explanation of the specification of the equation (model). This
should include explaining why particular independent variables and
functional forms were chosen as well as stating the expected signs of
(or other hypotheses about) the slope coefficients.
● A description of the data (including generated variables), data sources,
and any irregularities with the data.
● A presentation of each estimated specification, using our standard
documentation format. If you estimate more than one specification,
be sure to explain which one is best (and why).
● A careful analysis of the regression results that includes a discussion
of any econometric problems encountered and complete documen-
tation of all equations estimated and all tests run. (Beginning re-
searchers are well advised to test for every possible econometric
problem; with experience, you’ll learn to focus on the most likely
difficulties.)
● A short summary/conclusion that includes any policy recommendations
or suggestions for further research.
● A bibliography.
● An appendix that includes all data, all regression runs, and all relevant
computer output. Do this carefully; readers appreciate a well-organized
and labeled appendix.
We think that the easiest way to write such a research report is to keep a re-
search journal as you go along. In this journal, you can keep track of a priori
hypotheses, regression results, statistical tests, different specifications you
considered, and theoretical analyses of what you thought was going on in
your equation. You’ll find that when it comes time to write your research re-
port, this journal will almost write your paper for you! The alternative to
keeping a journal is to wait until you’ve finished all your econometric work
before starting to write your research report, but by doing this, you run the risk
of forgetting the thought process that led you to make a particular decision
(or some other important item).
A Regression User’s Checklist and Guide
Table 2 contains a list of the items that a researcher checks when reviewing
the output from a computer regression package. Not every item in the
checklist will be produced by your computer package, and not every item in
your computer output will be in the checklist, but the checklist can be a
very useful reference. In most cases, a quick glance at the checklist will
6
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remind you of the text sections that deal with the item, but if this is not the
case, the fairly minimal explanation in the checklist should not be relied on
to cover everything needed for complete analysis and judgment. Instead,
you should look up the item in the index. In addition, note that the actions
in the right-hand column are merely suggestions. The circumstances of each
individual research project are much more reliable guides than any dog-
matic list of actions.
There are two ways to use the checklist. First, you can refer to it as a “glos-
sary of packaged computer output terms” when you encounter something in
your regression result that you don’t understand. Second, you can work your
way through the checklist in order, finding the items in your computer out-
put and marking them. As with the Regression User’s Guide (Table 3), the use
of the Regression User’s Checklist will be most helpful for beginning re-
searchers, but we also find ourselves referring back to it once in a while even
after years of experience.
Be careful. All simplified tables, like the two in this chapter, must trade
completeness for ease of use. As a result, strict adherence to a set of rules is
not recommended even if the rules come from one of our tables. Someone
who understands the purpose of the research, the exact definitions of the
variables, and the problems in the data is much more likely to make a correct
judgment than is someone equipped with a set of rules created to apply to a
wide variety of possible applications.
Table 3, the Regression User’s Guide, contains a brief summary of the
major econometric maladies discussed so far in this text. For each economet-
ric problem, we list:
1. Its nature.
2. Its consequences for OLS estimation.
3. How to detect it.
4. How to attempt to get rid of it.
How might you use the guide? If an estimated equation has a particular
problem, such as an insignificant coefficient estimate, a quick glance at the
guide can give some idea of the econometric problems that might be causing
the symptom. Both multicollinearity and irrelevant variables can cause re-
gression coefficients to have insignificant t-scores, for example, and someone
who remembered only one of these potential causes might take the wrong
correction action. After some practice, the use of this guide will decrease until
it eventually will seem fairly limiting and simplistic. Until then, however, our
experience is that those about to undertake their first econometric research
can benefit by referring to this guide.
377

RUNNING YOUR OWN REGRESSION PROJECT
Table 2 Regression User’s Checklist
Symbol Checkpoint Reference Decision
X, Y Data observations Check for errors, espe-
cially outliers, in the
data. Spot-check
transformations of
variables. Check
means, maximums,
and minimums.
Correct any errors. If
the quality of the data
is poor, may want to
avoid regression
analysis or use just
OLS.
number of
observations
number of explana-
tory variables
If
equation cannot be
estimated, and if the
degrees of freedom
are low, precision is
low. In such a case,
try to include more
observations.
N 2 K 2 1 # 0,
K 5
N 5
N 2 K 2 1 . 0df Degrees of freedom
Compare signs and
magnitudes to ex-
pected values.
If they are unexpected,
respecify model if ap-
propriate or assess
other statistics for
possible correct
procedures.
Estimated coefficient�̂
Two-sided test:
One-sided test:
, the hypothesized
is supplied by the re-
searcher, and is often
zero.
�,�H0
HA: �k . �H0
H0: �k # �H0
HA: �k 2 �H0
H0: �k 5 �H0
t t-statistic
or
for computer-
supplied t-scores
or whenever
�H0
5 0
tk 5
�̂k
SE(�̂k)
tk 5
�̂k 2 �H0
SE(�̂k)
Measures the degree of
overall fit of the
model to the data.
A guide to the overall
fit.
R2 Coefficient of determi-
nation
Same as R2. Also at-
tempts to show the
contribution of an ad-
ditional explanatory
variable.
R2 adjusted for degrees
of freedom
R 2
Reject
and if the estimate is
of the expected sign.
tc is the critical value of
level of significance
and de-
grees of freedom.
N 2 K 2 1
�
H0 if u tk u . tc
One indication that an
explanatory variable
is irrelevant is if the
falls when it is
included.
R 2
378

RUNNING YOUR OWN REGRESSION PROJECT
TSS Total sum of squares
An estimate of Com-
pare with for a
measure of overall fit.
A guide to the overall
fit.Y
�.SEE Standard error of the
equation
Check for transcription
errors.
Check for heteroskedas-
ticity by examining
the pattern of the
residuals.
Correct the data.
May take appropriate
corrective action, but
test first.
ei Residual
Tests:
For positive serial
correlation.
Reject H0 if
Inconclusive if
(dL
and dU are critical
DW values.)
dL # DW # dU.
DW , dL.
HA: p . 0
H0: p # 0DW Durbin–Watson d
statistic
To test
HA: H0 not true
Calculate special
F-statistic to test joint
hypotheses.
Reject the
critical value for
level of significance
and K numerator and
denomi-
nator d.f.
N 2 K 2 1
�
H0 if F $ Fc,
5 �k 5 0
H0: �1 5 �2 5 . . .
F F-statistic
Table 2 (continued)
Symbol Checkpoint Reference Decision
RSS Residual sum of
squares
Used in t-statistic. A guide to statistical
significance.
Standard error of �̂kSE(�̂k)
Usually provided by
an autoregressive
routine.
If negative, implies a
specification error.
Estimated first-order
autocorrelation
coefficient
�̂
Used to detect multi-
collinearity.
Suspect severe multi-
collinearity if
r12 . .8.
r12 Simple correlation
coefficient between
X1 and X2
Used to detect multi-
collinearity.
Suspect severe multi-
collinearity if
VIF . 5.
VIF Variance inflation
factor
Used to compute F, R2,
and .R 2
Same as above. Also
used in hypothesis
testing.
TSS 5 g
i
(Yi 2 Y)
2
RSS 5 g
i
(Yi 2 Ŷi)
2
379

Table 3 Regression User’s Guide
What Can Go What Are the How Can It Be How Can It Be
Wrong? Consequences? Detected? Corrected?
Omitted Variable
The omission of a
relevant indepen-
dent variable
Bias in the coeffi-
cient estimates
(the s) of the
included Xs.
Theory, significant
unexpected signs,
or surprisingly
poor fits.
Include the omitted
variable or a
proxy.�̂
The inclusion of a
variable that
does not belong
in the equation
Decreased preci-
sion in the form
of higher stan-
dard errors and
lower t-scores.
1. Theory
2. t-test on
3.
4. Impact on other
coefficients if X is
dropped.
Delete the variable
if its inclusion is
not required by
the underlying
theory.
R 2
�̂
Irrelevant Variable
Some of the inde-
pendent variables
are (imperfectly)
correlated
No universally ac-
cepted rule or
test is available.
Use high r12s or
the VIF test.
Drop redundant
variables, but to
drop others
might introduce
bias. Often doing
nothing is best.
Multicollinearity
The variance of the
error term is not
constant for all
observations,
as in:
Same as for serial
correlation.
Use the Park or
White tests.
If impure, add the
omitted variable.
Otherwise, use
HC standard
errors or reformu-
late the variables.VAR(�i) 5 �
2Z2i
Heteroskedasticity
The functional form
is inappropriate
Biased estimates,
poor fit, and diffi-
cult interpretation.
Examine the theory
carefully; think
about the rela-
tionship between
X and Y.
Transform the vari-
able or the equa-
tion to a different
functional form.
Observations of the
error term are
correlated, as in:
Use Durbin–Watson
d test; if signifi-
cantly less than
2, positive serial
correlation
exists.
If impure, add the
omitted variable
or change the
functional form.
Otherwise, con-
sider Generalized
Least Squares or
Newey– West
standard errors.
�t5 ��t21 1 ut
Serial Correlation
Incorrect Functional Form
No biased but
OLS no longer is
minimum vari-
ance, and hy-
pothesis testing
is unreliable.
�̂s,
No biased but
estimates of the
separate effects
of the Xs are not
reliable, i.e., high
SEs (and low
t-scores).
�̂s,
380

RUNNING YOUR OWN REGRESSION PROJECT
Summary
1. Running your own regression project involves choosing your de-
pendent variable, applying the six steps in applied regression to
that dependent variable, and then writing a research report that sum-
marizes your work.
2. A great research topic is one that you know something about, one
that addresses an inherently interesting economic or behavioral
question or choice, and one for which data are available not only
for the dependent variable but also for the obvious independent
variables.
3. Don’t underestimate the difficulty and importance of collecting a
complete and accurate data set. It’s a lot of work, but it’s worth it!
4. The art of econometrics boils down to finding the best possible
equation in the fewest possible number of regression runs. The only
way to do this is to spend quite a bit of time thinking through the
underlying principles of your research project before you run your
first regression.
5. Before you complete your research project, be sure to review the prac-
tical hints and regression user’s guide and checklist in Sections 5
and 6.
Appendix: The Housing Price
Interactive Exercise
Our goal here is to bridge the gap between textbook and computer. As a re-
sult, this interactive exercise will provide you with a short literature review
and the data, but you’ll be asked to calculate your own estimates. Feedback
on your specification choices will once again be found in the hints in at the
end of the chapter.
8
7
381

RUNNING YOUR OWN REGRESSION PROJECT
18. G. M. Grether and Peter Mieszkowski, “Determinants of Real Estate Values,” Journal of Urban
Economics, Vol. 1, pp. 127–146. Another classic article of the same era is J. Kain and J. Quigley,
“Measuring the Value of Housing Quality,” Journal of American Statistical Association, Vol. 45,
pp. 532–548.
Since the only difference between this interactive exercise and the first one
is that this one requires you to estimate your chosen specification(s) with the
computer, our guidelines for interactive exercises still apply:
1. Take the time to look over a portion of the reading list before choosing
a specification.
2. Try to estimate as few regression runs as possible.
3. Avoid looking at the hints until after you’ve reached what you think is
your best specification.
We believe that the benefits you get from an interactive exercise are di-
rectly proportional to the effort you put into it. If you have to delay this ex-
ercise until you have the time and energy to do your best, that’s probably a
good idea.
Building a Hedonic Model of Housing Prices
We’re going to ask you to specify the independent variables and functional
form for an equation whose dependent variable is the price of a house in
Southern California. Before making these choices, it’s vital to review the
housing price literature and to think through the theory behind such models.
Such a review is especially important in this case because the model we’ll be
building will be hedonic in nature.
What is a hedonic model? We estimated an equation for the price of a
house as a function of the size of that house. Such a model is called hedonic
because it uses measures of the quality of a product as independent variables
instead of measures of the market for that product (like quantity demanded,
income, etc.). Hedonic models are most useful when the product being ana-
lyzed is heterogeneous in nature because we need to analyze what causes
products to be different and therefore to have different prices. With a homo-
geneous product, hedonic models are virtually useless.
Perhaps the most-cited early hedonic housing price study is that of
G. Grether and P. Mieszkowski.18 Grether and Mieszkowski collected a seven-
year data set and built a number of linear models of housing price using
382

RUNNING YOUR OWN REGRESSION PROJECT
19. Peter Linneman, “Some Empirical Results on the Nature of the Hedonic Price Functions for
the Urban Housing Market,” Journal of Urban Economics, Vol. 8, No. 1, pp. 47–68.
20. Keith Ihlanfeldt and Jorge Martinez-Vasquez, “Alternate Value Estimates of Owner-Occu-
pied Housing: Evidence on Sample Selection Bias and Systematic Errors,” Journal of Urban Eco-
nomics, Vol. 20, No. 3, pp. 356–369. Also see Eric Cassel and Robert Mendelsohn, “The Choice
of Functional Forms for Hedonic Price Equations: Comment,” Journal of Urban Economics, Vol. 18,
No. 2, pp. 135–142.
21. Allen C. Goodman, “An Econometric Model of Housing Price, Permanent Income, Tenure
Choice, and Housing Demand,” Journal of Urban Economics, Vol. 23, pp. 327–353.
different combinations of variables. They included square feet of space, the
number of bathrooms, and the number of rooms, although the number of
rooms turned out to be insignificant. They also included lot size and the age
of the house as variables, specifying a quadratic function for the age variable.
Most innovatively, they used several slope dummies in order to capture the
interaction effects of various combinations of variables (like a hardwood-
floors dummy times the size of the house).
Peter Linneman19 estimated a housing price model on data from Los Angeles,
Chicago, and the entire United States. His goal was to create a model that
worked for the two individual cities and then to apply it to the nation to
test the hypothesis of a national housing market. Linneman did not in-
clude any lot characteristics, nor did he use any interaction variables. His
only measures of the size of the living space were the number of bath-
rooms and the number of nonbathrooms. Except for an age variable, the
rest of the independent variables were dummies describing quality charac-
teristics of the house and neighborhood. Although many of the dummy
variables were quite fickle, the coefficients of age, number of bathrooms,
and the number of nonbathrooms were relatively stable and significant.
Central air conditioning had a negative, insignificant coefficient for the
Los Angeles regression.
K. Ihlanfeldt and J. Martinez-Vasquez20 investigated sample bias in various
methods of obtaining house price data and concluded that the house’s sales
price is the least biased of all measures. Unfortunately, they went on to esti-
mate an equation by starting with a large number of variables and then drop-
ping all those that had t-scores below 1, almost surely introducing bias into
their equation.
Finally, Allen Goodman21 added some innovative variables to an estimate
on a national data set. He included measures of specific problems like rats,
cracks in the plaster, holes in the floors, plumbing breakdowns, and the level of
property taxes. Although the property tax variable showed the capitalization of
383

RUNNING YOUR OWN REGRESSION PROJECT
low property taxes, as would be expected, the rats coefficient was insignificant,
and the cracks variable’s coefficient asserted that cracks significantly increase
the value of a house.
The Housing Price Interactive Exercise
Now that we’ve reviewed at least a portion of the literature, it’s time to build
your own model. Recall that in Chapter 1. We built a simple model of the
price of a house as a function of the size of that house, Equation of Chapter 1
where: Pi � the price (in thousands of dollars) of the ith house
Si � the size (in square feet) of the ith house
Equation of Chapter 1 was estimated on a sample of 43 houses that were
purchased in the same Southern California town (Monrovia) within a few
weeks of each other. It turns out that we have a number of additional inde-
pendent variables for the data set we used to estimate Equation of Chapter 1.
Also available are:
Ni � the quality of the neighborhood of the ith house (1 � best,
4 � worst) as rated by two local real estate agents
Ai � the age of the ith house in years
BEi � the number of bedrooms in the ith house
BAi � the number of bathrooms in the ith house
CAi � a dummy variable equal to 1 if the ith house has central air
conditioning, 0 otherwise
SPi � a dummy variable equal to 1 if the ith house has a pool, 0
otherwise
Yi � the size of the yard around the ith house (in square feet)
Read through the list of variables again, developing your own analyses of the
theory behind each variable. What are the expected signs of the coefficients?
Which variables seem potentially redundant? Which variables must you
include?
In addition, there are a number of functional form modifications that can
be made. For example, you might consider a quadratic polynomial for age, as
Grether and Mieszkowski did, or you might consider creating slope dummies
such as SP S or CA S. Finally, you might consider interactive variables that
involve the neighborhood proxy variable such as N S or N BA. What hy-
potheses would each of these imply?
??
??
P̂i 5 40.0 1 0.138Si
384

RUNNING YOUR OWN REGRESSION PROJECT
Table 4 Data for the Housing Price Interactive Exercise
P S N A BE BA CA SP Y
107 736 4 39 2 1 0 0 3364
133 720 3 63 2 1 0 0 1780
141 768 2 66 2 1 0 0 6532
165 929 3 41 3 1 0 0 2747
170 1080 2 44 3 1 0 0 5520
173 942 2 65 2 1 0 0 6808
182 1000 2 40 3 1 0 0 6100
200 1472 1 66 3 2 0 0 5328
220 1200 1.5 69 3 1 0 0 5850
226 1302 2 49 3 2 0 0 5298
260 2109 2 37 3 2 1 0 3691
275 1528 1 41 2 2 0 0 5860
280 1421 1 41 3 2 0 1 6679
289 1753 1 1 3 2 1 0 2304
295 1528 1 32 3 2 0 0 6292
300 1643 1 29 3 2 0 1 7127
310 1675 1 63 3 2 0 0 9025
315 1714 1 38 3 2 1 0 6466
350 2150 2 75 4 2 0 0 14825
365 2206 1 28 4 2.5 1 0 8147
503 3269 1 5 4 2.5 1 0 10045
135 936 4 75 2 1 0 0 5054
147 728 3 40 2 1 0 0 1922
165 1014 3 26 2 1 0 0 6416
175 1661 3 27 3 2 1 0 4939
190 1248 2 42 3 1 0 0 7952
191 1834 3.5 40 3 2 0 1 6710
195 989 2 41 3 1 0 0 5911
205 1232 1 43 2 2 0 0 4618
210 1017 1 38 2 1 0 0 5083
215 1216 2 77 2 1 0 0 6834
Develop your specification carefully. Think through each variable and/or
functional form decision, and take the time to write out your expectations for
the sign and size of each coefficient. Don’t take the attitude that you should
include every possible variable and functional form modification and then
drop the insignificant ones. Instead, try to design the best possible hedonic
model of housing prices you can the first time around.
Once you’ve chosen a specification, estimate your equation, using the data
in Table 4 and analyze the result.
(continued)
385

1. Test your hypotheses for each coefficient with the t-test. Pay special atten-
tion to any functional form modifications.
2. Decide what econometric problems exist in the equation, testing, if ap-
propriate, for multicollinearity, serial correlation, or heteroskedasticity.
3. Decide whether to accept your first specification as the best one or to
make a modification in your equation and estimate again. Make sure
you avoid the temptation to estimate an additional specification “just
to see what it looks like.”
Once you’ve decided to make no further changes, you’re finished—
congratulations! Now turn to the hints at the end of the chapter for feedback
on your choices.
RUNNING YOUR OWN REGRESSION PROJECT
Table 4 (continued)
P S N A BE BA CA SP Y
228 1447 2 44 2 2 0 0 4143
242 1974 1.5 65 4 2 0 1 5499
250 1600 1.5 63 3 2 1 0 4050
250 1168 1.5 63 3 1 0 1 5182
255 1478 1 50 3 2 0 0 4122
255 1756 2 36 3 2 0 1 6420
265 1542 2 38 3 2 0 0 6833
265 1633 1 32 4 2 0 1 7117
275 1500 1 42 2 2 1 0 7406
285 1734 1 62 3 2 0 1 8583
365 1900 1 42 3 2 1 0 19580
397 2468 1 10 4 2.5 1 0 6086
Datafile � HOUSE11
386

Answers
Exercise 2
Hints for the Housing Price Interactive Exercise
The biggest problem most students have with this interactive exercise
is that they run far too many different specifications “just to see”
what the results look like. In our opinion, all but one or two of the
specification decisions involved in this exercise should be made be-
fore the first regression is estimated, so one measure of the quality
of your work is the number of different equations you estimated.
Typically, the fewer the better.
As to which specification to run, most of the decisions involved
are matters of personal choice and experience. Our favorite model
on theoretical grounds is:
We think that BE and BA are redundant with S. In addition, we can
justify both positive and negative coefficients for SP, giving it an
ambiguous expected sign, so we’d avoid including it. We would not
quibble with someone who preferred a linear functional form for A
to our quadratic. In addition, we recognize that CA is quite insignif-
icant for this sample, but we’d retain it, at least in part because it
gets quite hot in Monrovia in the summer.
As to interactive variables, the only one we can justify is between
S and N. Note, however, that the proper variable is not but
instead is or something similar, to account for the
different expected signs. This variable turns out to improve the fit
while being quite collinear (redundant) with N and S.
In none of our specifications did we find evidence of serial
correlation or heteroskedasticity, although the latter is certainly a
possibility in such cross-sectional data.
S ? (5 2 N),
S ? N
P 5 f( S
1
, N
2
, A
2
, A
1
2, Y
1
, CA
1
)
RUNNING YOUR OWN REGRESSION PROJECT
387

388

1 Dynamic Models
2 Serial Correlation and Dynamic Models
3 Granger Causality
4 Spurious Correlation and Nonstationarity
5 Summary and Exercises
Time-Series Models
The purpose of this chapter is to provide an introduction to a number of in-
teresting models that have been designed to cope with and take advantage of
the special properties of time-series data. Working with time-series data often
causes complications that simply can’t happen with cross-sectional data.
Most of these complications involve the order of the observations because
order matters quite a bit in time-series data but doesn’t matter much (if at all)
in cross-sectional data.
The most important of the topics concerns a class of dynamic models in
which a lagged value of the dependent variable appears on the right-hand
side of the equation. As you will see, the presence of a lagged dependent vari-
able on the right-hand side of the equation implies that the impact of the in-
dependent variables can be spread out over a number of time periods.
Why would you want to distribute the impact of an independent variable
over a number of time periods? To see why, consider the impact of advertis-
ing on sales. Most analysts believe that people remember advertising for
more than one time period, so advertising affects sales in the future as well as
in the current time period. As a result, models of sales should include current
and lagged values of advertising, thus distributing the impact of advertising
over a number of different lags.
While this chapter focuses on such dynamic models, you’ll also learn about
models in which different numbers of lags appear and we’ll investigate how
the presence of these lags affects our estimators. The chapter concludes with a
brief introduction to a topic called nonstationarity. If variables have signifi-
cant changes in basic properties (like their mean or variance) over time, they
From Chapter 12 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
389

TIME-SERIES MODELS
are said to be nonstationary, and it turns out that nonstationary variables have
the potential to inflate t-scores and measures of overall fit in an equation.
Dynamic Models
Distributed Lag Models
Lagged independent variables can be used whenever you expect X to affect Y
after a period of time. For example, if the underlying theory suggests that X1
affects Y with a one-time-period lag (but X2 has an instantaneous impact on
Y), we use equations like:
(1)
Such lags are called simple lags, and the estimation of �1 with OLS is no
more difficult than the estimation of the coefficients of nonlagged equations,
except for possible impure serial correlation if the lag is misspecified. Remem-
ber, however, that the coefficients of such equations should be interpreted
carefully. For example, �2 in Equation 1 measures the effect of a one-unit in-
crease in this time’s X2 on this time’s Y holding last time’s X1 constant.
A case that’s more complicated than this simple lag model occurs when
the impact of an independent variable is expected to be spread out over a
number of time periods. For example, suppose we’re interested in studying
the impact of a change in the money supply on GDP. Theoretical and empir-
ical studies have provided evidence that because of rigidities in the market-
place, it takes time for the economy to react completely to a change in the
money supply. Some of the effect on GDP will take place in the first quarter,
some more in the second quarter, and so on. In such a case, the appropriate
econometric model would be a distributed lag model:
(2)
A distributed lag model explains the current value of Y as a function of cur-
rent and past values of X, thus “distributing” the impact of X over a number
of time periods. Take a careful look at Equation 2. The coefficients �0, �1, and
�2 through �p measure the effects of the various lagged values of X on the
current value of Y. In most economic applications, including our money sup-
ply example, we’d expect the impact of X on Y to decrease as the length of the
lag (indicated by the subscript of the �) increases. That is, although �0 might
be larger or smaller than �1, we certainly would expect either �0 or �1 to be
larger in absolute value than �6 or �7.
Yt 5 �0 1 �0Xt 1 �1Xt 2 1 1 �2Xt 2 2 1
c 1 �pXt 2 p 1 �t
Yt 5 �0 1 �1X1t 2 1 1 �2X2t 1 �t
1
390

TIME-SERIES MODELS
Unfortunately, the estimation of Equation 2 with OLS causes a number of
problems:
1. The various lagged values of X are likely to be severely multicollinear,
making coefficient estimates imprecise.
2. In large part because of this multicollinearity, there is no guarantee
that the estimated �s will follow the smoothly declining pattern that
economic theory would suggest. Instead, it’s quite typical for the esti-
mated coefficients of Equation 2 to follow a fairly irregular pattern, for
example:
3. The degrees of freedom tend to decrease, sometimes substantially, for
two reasons. First, we have to estimate a coefficient for each lagged X,
thus increasing K and lowering the degrees of freedom (N � K � 1).
Second, unless data for lagged Xs outside the sample are available, we
have to decrease the sample size by one for each lagged X we calculate,
thus lowering the number of observations, N, and therefore the degrees
of freedom.
As a result of these problems with OLS estimation of functions like Equa-
tion 2, called ad hoc distributed lag equations, it’s standard practice to use a
simplifying assumption in such situations. The most commonly used sim-
plification is to replace all the lagged independent variables with a lagged
value of the dependent variable, and we’ll call that kind of equation a
dynamic model.
What Is a Dynamic Model?
The simplest dynamic model is:
�̂0 5 0.26 �̂1 5 0.07 �̂2 5 0.17 �̂3 5 2 0.03 �̂4 5 0.08
Note that Y is on both sides of the equation! Luckily, the subscripts are differ-
ent in that the Y on the left-hand side is Yt, and the Y on the right-hand side
is Yt�1. It’s this difference in time period that makes the equation dynamic.
Thus, the simplest dynamic model is an equation in which the current value
of the dependent variable Y is a function of the current value of X and a
Yt � �0 � �0Xt � �Yt�1 � ut (3)
391

TIME-SERIES MODELS
lagged value of Y itself. Such a model with a lagged dependent variable is
often called an autoregressive equation.
Let’s take a look at Equation 3 to try to see why it can be used to represent
a distributed lag model or any model in which the impact of X on Y is
distributed over a number of lags. Suppose that we lag Equation 3 one time
period:
(4)
If we now substitute Equation 4 into Equation 3, we get:
(5)
or
(6)
If we do this one more time (that is, if we lag Equation 3 two time periods,
substitute it into Equation 5 and rearrange), we get:
(7)
where is the new (combined) intercept and is the new (combined)
error term. In other words, Yt � f(Xt, Xt�1, Xt�2). We’ve shown that a dy-
namic model can indeed be used to represent a distributed lag model!
In addition, note that the coefficients of the lagged Xs follow a clear pat-
tern. To see this, let’s go back to Equation 2:
(2)
and compare the coefficients in Equation 2 to those in Equation 7,
we get:
�1 � ��0 (8)
�2 � �
2�0
�3 � �
3�0
�p � �
p�0
?
?
?
Yt 5 �0 1 �0Xt 1 �1Xt 2 1 1 �2Xt 2 2 1
c 1 �pXt 2 p 1 �t
ut*�0*
Yt 5 �0* 1 �0Xt 1 ��0Xt 2 1 1 �
2�0Xt 2 2 1 �
3Yt 2 3 1 ut*
Yt 5 (�0 1 ��0) 1 �0Xt 1 ��0Xt 2 1 1 �
2Yt 2 2 1 (�ut 2 1 1 ut)
Yt 5 �0 1 �0Xt 1 �(�0 1 �0Xt 2 1 1 �Yt 2 2 1 ut 2 1) 1 ut
Yt 2 1 5 �0 1 �0Xt 2 1 1 �Yt 2 2 1 ut 2 1
392

TIME-SERIES MODELS
0 1 2 3 4
= 0.75
= 0.50
= 0.02
λ
λ
λ
5 6 7
1.0
0.5
R
e
la
ti
v
e
W
e
ig
h
t
o
f
L
a
g
g
e
d
V
a
ri
a
b
le
Time Period of Lag
Figure 1 Geometric Weighting Schemes for Various Dynamic Models
As long as � is between 0 and 1, a dynamic model has the impact of the independent
variable declining as the length of the lag increases.
1. This model sometimes is referred to as a Koyck distributed lag model because it was origi-
nally developed by L. M. Koyck in Distributed Lags and Investment Analysis (Amsterdam: North-
Holland Publishing, 1954).
As long as � is between 0 and 1, these coefficients will indeed smoothly de-
cline,1 as shown in Figure 1.
Dynamic models like Equation 3 avoid the three major problems with ad
hoc distributed lag equations that we outlined. The degrees of freedom have
increased dramatically, and the multicollinearity problem has disappeared. If
ut is well behaved, OLS estimation of Equation 3 can be shown to have desir-
able properties for large samples. How large is “large enough”? Our recom-
mendation, based more on experience than proof, is to aim for a sample of at
least 50 observations. The smaller the sample, the more likely you are to
encounter bias. Samples below 25 in size should be avoided entirely, in part
because of the bias and in part because hypothesis testing becomes
untrustworthy.
393

TIME-SERIES MODELS
In addition to this sample size issue, dynamic models face another serious
problem. They are much more likely to encounter serial correlation than are
equations without a lagged dependent variable as an independent variable.
To make things worse, serial correlation almost surely will cause bias in the
OLS estimates of dynamic models no matter how large the sample size is.
This problem will be discussed in Section 2.
An Example of a Dynamic Model
As an example of a dynamic model, let’s look at an aggregate consumption
function from a macroeconomic equilibrium GDP model. Many economists
argue that in such a model, consumption (COt) is not just an instantaneous
function of disposable income (YDt). Instead, they believe that current con-
sumption is also influenced by past levels of disposable income (YDt�1,
YDt�2, etc.):
(9)
Such an equation fits well with simple models of consumption, but it makes
sense only if the weights given past levels of income decrease as the length of
the lag increases. That is, the impact of lagged income on current consump-
tion should decrease as the lag gets bigger. Thus we’d expect the coefficient of
YDt�2 to be less than the coefficient of YDt�1, and so on.
As a result, most econometricians would model Equation 9 with a dy-
namic model:
(10)
To estimate Equation 10, where we will build a small macromodel of the
U.S. economy from 1976 through 2007. The OLS estimates of Equation 10
for this data set are (standard errors in parentheses):
(11)
(0.10) (0.10)
4.70 5.66
R2 5 .999 N 5 32 (annual 1976–2007)
COt 5 2 266.6 1 0.46YDt 1 0.56COt 2 1
COt 5 �0 1 �0YDt 1 �COt 2 1 1 ut
COt 5 f(YD
1
t, YD
1
t 2 1, YD
1
t 2 2, etc.) 1 �t
394

TIME-SERIES MODELS
2. Note that the constant term equals �0/(1 � �).
If we substitute into Equation 3 for i � 1, we obtain
If we continue this process, it turns out
that Equation 11 is equivalent to:2
(12)
As can be seen, the coefficients of YD in Equation 12 do indeed decline as
we’d expect in a dynamic model.
To compare this estimate with an OLS estimate of the same equation with-
out the dynamic model format, we’d need to estimate an ad hoc distributed
lag equation with the same number of lagged variables.
(13)
If we estimate Equation 13 using the same data set, we get:
(14)
How do the coefficients of Equation 14 look? As the lag increases, the coeffi-
cients of YD decrease sharply, actually going negative for t�3. Neither eco-
nomic theory nor common sense leads us to expect this pattern. Such a poor
result is due to the severe multicollinearity between the lagged Xs. Most
econometricians therefore estimate consumption functions with a lagged
dependent variable simplification scheme like the dynamic model in Equa-
tion 10.
An interesting interpretation of the results in Equation 11 concerns the
long-run multiplier implied by the model. The long-run multiplier measures
the total impact of a change in income on consumption after all the lagged
effects have been felt. One way to get this estimate would be to add up all the
s, but an easier alternative is to calculate 0[1/(1� )], which in this case �̂�̂�̂
COt 5 2 695.89 1 0.73YDt 1 0.38YDt 2 1 1 0.006YDt 2 2 2 0.08YDt 2 3
COt 5 �0 1 �0YDt 1 �1YDt 2 1 1 �2YDt 2 2 1 �3YDt 2 3 1 �t
1 0.08YDt 2 3 1
c
COt 5 2 605.91 1 0.46YDt 1 0.26YDt 2 1 1 0.14YDt 2 2
�̂1 5 �̂0�̂
1 5 (0.46) (0.56)1 5 0.26.
�̂0 5 0.46 and �̂ 5 0.56
equals 0.46[1/(1�0.56)] or 1.05. A sample of this size is likely to encounter
small sample bias, however, so we shouldn’t overanalyze the results. For
more on testing and adjusting dynamic equations like Equation 11 for serial
correlation, let’s move on to the next section.
395

TIME-SERIES MODELS
Serial Correlation and Dynamic Models
The consequences of serial correlation depend crucially on the type of model
we’re talking about. For an ad hoc distributed lag model such as Equation 2, se-
rial correlation has the effects outlined: Serial correlation causes OLS to no
longer be the minimum variance unbiased estimator, serial correlation
causes the SE( )s to be biased, and serial correlation causes no bias in the
OLS s themselves.
For dynamic models such as Equation 3, however, all this changes, and se-
rial correlation does indeed cause bias in the s produced by OLS. Com-
pounding this is the fact that the consequences, detection, and remedies for
serial correlation are all either incorrect or need to be modified in the pres-
ence of a lagged dependent variable.
Serial Correlation Causes Bias in Dynamic Models
If an equation with a lagged dependent variable as an independent variable
has a serially correlated error term, then OLS estimates of the coefficients of
that equation will be biased, even in large samples. To see where this bias
comes from, let’s look at a dynamic model like Equation 3 (ignore the ar-
rows for a bit):
(3)
and assume that the error term ut is serially correlated: ut � �ut�1 � �t where
�t is a classical error term. If we substitute this serially correlated error term
into Equation 3, we get:
(15)
Let’s also look at Equation 3 lagged one time period:
(16)
What happens when the previous time period’s error term (ut�1) is posi-
tive? In Equation 16, the positive ut�1 causes Yt�1 to be larger than it
would have been otherwise (these changes are marked by upward-pointing
arrows for ut�1 in Equation 16 and for Yt�1 in Equations 3, 15, and 16).
In addition, the positive ut�1 is quite likely to cause ut to be positive
Yt 2 1 5 �0 1 �0Xt 2 1 1 �Yt 2 2 1 ut 2 1
cc
Yt 5 �0 1 �0Xt 1 �Yt 2 1 1 �ut 2 1 1 �t
cc
Yt 5 �0 1 �0Xt 1 �Yt 2 1 1 ut
cc
�̂
�̂
�̂
2
396

TIME-SERIES MODELS
3. The reason that pure serial correlation doesn’t cause bias in the coefficient estimates of equa-
tions that don’t include a lagged dependent variable is that the “omitted variable” ut�1 isn’t
correlated with any of the included independent variables.
4. The opposite is not a problem. A Durbin�Watson d test that indicates serial correlation in
the presence of a lagged dependent variable, despite the bias toward 2, is an even stronger affir-
mation of serial correlation.
in Equation 3 because ut � �ut�1 � �t and � usually is positive (these
changes are marked by upward-pointing arrows in Equation 15 and
Equation 3).
Take a look at the arrows in Equation 3. Yt�1 and ut are correlated! Such a
correlation violates Classical Assumption III, which assumes that the error
term is not correlated with any of the explanatory variables.
The consequences of this correlation include biased estimates, in particu-
lar of the coefficient �, because OLS attributes to Yt�1 some of the change in
Yt actually caused by ut. In essence, the uncorrected serial correlation acts like
an omitted variable (ut�1). Since an omitted variable causes bias whenever it
is correlated with one of the included independent variables, and since ut�1
is correlated with Yt�1, the combination of a lagged dependent variable and
serial correlation causes bias in the coefficient estimates.3
Serial correlation in a dynamic model also causes estimates of the stan-
dard errors of the estimated coefficients and the residuals to be biased. The
former bias means that hypothesis testing is invalid, even for large samples.
The latter bias means that tests based on the residuals, like the Durbin�
Watson d test, are potentially invalid.
Testing for Serial Correlation in Dynamic Models
Until now, we’ve relied on the Durbin�Watson d test to test for serial correla-
tion, but, as mentioned above, the Durbin�Watson d statistic is potentially
invalid for an equation that contains a lagged dependent variable as an inde-
pendent variable. This is because the biased residuals described in the previ-
ous paragraph cause the DW d statistic to be biased toward 2. This bias to-
ward 2 means that the Durbin�Watson test sometimes fails to detect the
presence of serial correlation in a dynamic model.4
The widely used alternative is to use a special case of a general testing pro-
cedure called the Lagrange Multiplier Serial Correlation (LMSC) Test, which
is a method that can be used to test for serial correlation by analyzing how
well the lagged residuals explain the residuals of the original equation (in an
equation that includes all the explanatory variables of the original model).
397

TIME-SERIES MODELS
5. For example, some readers may remember that the White test of Section 10.3 is a Lagrange
Multiplier test. For a survey of the various uses to which Lagrange Multiplier tests can be put
and a discussion of the LM test’s relationship to the Wald and Likelihood Ratio tests, see Rob
Engle, “Wald, Likelihood Ratio, and Lagrange Multiplier Tests in Econometrics,” in Z. Griliches
and M. D. Intriligator (eds.), Handbook of Econometrics, Volume II (Amsterdam: Elsevier Science
Publishers, 1984).
If the lagged residuals are significant in explaining this time’s residuals (as
shown by the chi-square test), then we can reject the null hypothesis of no se-
rial correlation. Interestingly, although we suggest using the LMSC test for dy-
namic models, it also could have been used instead of the Durbin�Watson
test to test for serial correlation in equations without a lagged dependent
variable. Other applications of the general Lagrange Multiplier test approach
are as a specification test and as a test for heteroskedasticity and other econo-
metric problems.5
Using the Lagrange Multiplier to test for serial correlation for a typical dy-
namic model involves three steps:
1. Obtain the residuals from the estimated equation:
(17)
2. Use these residuals as the dependent variable in an auxiliary equation
that includes as independent variables all those on the right-hand side
of the original equation as well as the lagged residuals:
(18)
3. Estimate Equation 18 using OLS and then test the null hypothesis that
a3 � 0 with the following test statistic:
(19)
where N is the sample size and R2 is the unadjusted coefficient of de-
termination, both of the auxiliary equation, Equation 18. For large
samples, LM has a chi-square distribution with degrees of freedom
equal to the number of restrictions in the null hypothesis (in this case,
one). If LM is greater than the critical chi-square value from Statistical
Table B-8, then we reject the null hypothesis that a3 � 0 and conclude
that there is indeed serial correlation in the original equation.
To run an LMSC test for second-order or higher-order serial correlation,
add lagged residuals (et�2 for second order, et�2 and et�3 for third order) to
the auxiliary equation, Equation 18. This latter change makes the null
hypothesis a3 � a4 � a5 � 0. Such a null hypothesis raises the degrees of
LM 5 N* R2
et 5 a0 1 a1Xt 1 a2Yt 2 1 1 a3et 2 1 1 ut
et 5 Yt 2 Ŷt 5 Yt 2 �̂0 2 �̂0X1t 2 �̂Yt 2 1
398

TIME-SERIES MODELS
6. For more on these complications, see R. Betancourt and H. Kelejian, “Lagged Endogenous
Variables and Cochrane-Orcutt Procedure,” Econometrica, Vol. 49, No. 4, pp. 1073�1078.
freedom in the chi-square test to three because we have imposed three restric-
tions on the equation (three coefficients are jointly set equal to zero). To run
an LMSC test with more than one lagged dependent variable, add the lagged
variables (Yt�2, Yt�3, etc.) to the original equation. For practice with the LM
test, see Exercise 6; for practice with testing for higher-order serial correlation,
see Exercise 7.
Correcting for Serial Correlation in Dynamic Models
There are three strategies for attempting to rid a dynamic model of serial
correlation: improving the specification, instrumental variables, and modi-
fied GLS.
The first strategy is to consider the possibility that the serial correlation
could be impure, caused by either omitting a relevant variable or by failing to
capture the actual distributed lag pattern accurately. Unfortunately, finding
an omitted variable or an improved lag structure is easier said than done. Be-
cause of the dangers of sequential specification searches, this option should
be considered only if an alternative specification exists that has a theoreti-
cally sound justification.
The second strategy, called instrumental variables, consists of substituting
an “instrument” (a variable that is highly correlated with Yt�1 but is uncorre-
lated with ut) for Yt�1 in the original equation, thus eliminating the correla-
tion between Yt�1 and ut. Although using an instrument is a reasonable
option that is straightforward in principle, it’s not always easy to find a proxy
that retains the distributed lag nature of the original equation.
The final solution to serial correlation in dynamic models (or in models
with lagged dependent variables and similar error term structures) is to use
an iterative maximum likelihood technique to estimate the components of
the serial correlation and then to transform the original equation so that the
serial correlation has been eliminated. This technique is not without its com-
plications. In particular, the sample needs to be large, the standard errors of
the estimated coefficients potentially need to be adjusted, and the estimation
techniques are flawed under some circumstances.6
399

TIME-SERIES MODELS
7. See C. W. J. Granger, “Investigating Causal Relations by Econometric Models and Cross-Spectral
Methods,” Econometrica, Vol. 37, No. 3, pp. 424�438.
8. In the fifth edition, we ended this paragraph by saying, “For example, Christmas cards typi-
cally arrive before Christmas, but it’s clear that Christmas wasn’t caused by the arrival of the
cards.” However, this isn’t a true example of Granger causality, because the date of Christmas is
fixed and therefore isn’t a “time-series variable.” See Erdal Atukeren, “Christmas cards, Easter
bunnies, and Granger-causality,” Quality & Quantity, Vol. 42, No. 6, Dec. 2008, pp. 835–844.
For an in-depth discussion of causality, see Kevin Hoover, Causality in Macroeconomics
(Cambridge: Cambridge University Press, 2001).
In essence, serial correlation causes bias in dynamic models, but ridding
the equation of that serial correlation is not an easy task.
Granger Causality
One application of ad hoc distributed lag models is to provide evidence
about the direction of causality in economic relationships. Such a test is use-
ful when we know that two variables are related but we don’t know which
variable causes the other to move. For example, most economists believe that
increases in the money supply stimulate GDP, but others feel that increases
in GDP eventually lead the monetary authorities to increase the money sup-
ply. Who’s right?
One approach to such a question of indeterminate causality is to theorize
that the two variables are determined simultaneously. A second approach to
the problem is to test for what is called “Granger causality.”
How can we claim to be able to test for causality? After all, didn’t we say
in Chapter 1 that even though most economic relationships are causal in
nature, regression analysis can’t prove such causality? The answer is that
we don’t actually test for theoretical causality; instead, we test for Granger
causality.
Granger causality, or precedence, is a circumstance in which one time-
series variable consistently and predictably changes before another variable.7
Granger causality is important because it allows us to analyze which variable
precedes or “leads” the other, and, as we shall see, such leading variables are
extremely useful for forecasting purposes.
Despite the value of Granger causality, however, we shouldn’t let ourselves
be lured into thinking that it allows us to prove economic causality in any
rigorous way. If one variable precedes (“Granger causes”) another, we can’t be
sure that the first variable “causes” the other to change.8
3
400

TIME-SERIES MODELS
9. See John Geweke, R. Meese, and W. Dent, “Comparing Alternative Tests of Causality in Tem-
poral Systems,” Journal of Econometrics, Vol. 21, pp. 161�194, and Rodney Jacobs, Edward
Leamer, and Michael Ward, “Difficulties with Testing for Causation,” Economic Inquiry, Vol. 17,
No. 3, pp. 401�413.
As a result, even if we’re able to show that event A always happens before
event B, we have not shown that event A “causes” event B.
There are a number of different tests for Granger causality, and all the vari-
ous methods involve distributed lag models in one way or another.9 Our
preference is to use an expanded version of a test originally developed by
Granger. Granger suggested that to see if A Granger-caused Y, we should run:
(20)
and test the null hypothesis that the coefficients of the lagged As (the �s)
jointly equal zero. If we can reject this null hypothesis using the F-test, then
we have evidence that A Granger-causes Y. Note that if p � 1, Equation 20 is
similar to the dynamic model, Equation 3.
Applications of this test involve running two Granger tests, one in each di-
rection. That is, run Equation 20 and also run:
(21)
testing for Granger causality in both directions by testing the null hypothesis
that the coefficients of the lagged Ys (again, the �s) jointly equal zero. If the
F-test is significant for Equation 20 but not for Equation 21, then we can con-
clude that A Granger-causes Y. For practice with this dual version of the
Granger test, see Exercise 8.
Spurious Correlation and Nonstationarity
One problem with time-series data is that independent variables can appear
to be more significant than they actually are if they have the same underlying
trend as the dependent variable. In a country with rampant inflation, for ex-
ample, almost any nominal variable will appear to be highly correlated with
4
At 5 �0 1 �1At 2 1 1
c 1 �pAt 2 p 1 �1Yt 2 1 1
c 1 �pYt 2 p 1 �t
Yt 5 �0 1 �1Yt 2 1 1
c 1 �pYt 2 p 1 �1At 2 1 1
c 1 �pAt 2 p 1 �t
401

TIME-SERIES MODELS
10. There are two different definitions of stationarity. The particular definition we use here is a
simplification of the most frequently cited definition, referred to by various authors as weak,
wide-sense, or covariance stationarity.
all other nominal variables. Why? Nominal variables are unadjusted for infla-
tion, so every nominal variable will have a powerful inflationary component.
This inflationary component will usually outweigh any real causal relation-
ship, making nominal variables appear to be correlated even if they aren’t.
Such a problem is an example of spurious correlation, a strong relation-
ship between two or more variables that is not caused by a real underlying
causal relationship. If you run a regression in which the dependent variable
and one or more independent variables are spuriously correlated, the result is
a spurious regression, and the t-scores and overall fit of such spurious regres-
sions are likely to be overstated and untrustworthy.
There are many causes of spurious correlation. In a cross-sectional data set,
for example, spurious correlation can be caused by dividing both the depen-
dent variable and one independent variable by a third variable that varies
considerably more than do the first two. The focus of this section, however,
will be on time-series data and in particular on spurious correlation caused
by nonstationary time series.
Stationary and Nonstationary Time Series
A stationary series is one whose basic properties, for example its mean and its
variance, do not change over time. In contrast, a nonstationary series has one
or more basic properties that do change over time. For instance, the real per
capita output of an economy typically increases over time, so it’s nonstation-
ary. By contrast, the growth rate of real per capita output often does not in-
crease over time, so this variable is stationary even though the variable it’s
based on, real per capita output, is nonstationary. A time series can be non-
stationary even with a constant mean if another property, such as the vari-
ance, changes over time.
More formally, a time-series variable, Xt, is stationary if:
1. the mean of Xt is constant over time,
2. the variance of Xt is constant over time, and
3. the simple correlation coefficient between Xt and Xt�k depends on the
length of the lag (k) but on no other variable (for all k).10
If one or more of these properties is not met, then Xt is nonstationary. If a
series is nonstationary, that problem is often referred to as nonstationarity.
402

TIME-SERIES MODELS
11. See, for example, C. R. Nelson and C. I. Plosser, “Trends and Random Walks in Macroeco-
nomics Time Series: Some Evidence and Implication,” Journal of Monetary Economics, Vol. 10,
pp. 169�182, and J. Campbell and N. G. Mankiw, “Permanent and Transitory Components in
Macroeconomic Fluctuations,” American Economic Review, Vol. 77, No. 2, pp. 111�117.
Although our definition of a stationary series focuses on stationary and
nonstationary variables, it’s important to note that error terms (and, therefore,
residuals) also can be nonstationary. In fact, we’ve already had experience
with a nonstationary error term. Many cases of heteroskedasticity in time-
series data involve an error term with a variance that tends to increase with
time. That kind of heteroskedastic error term is also nonstationary!
The major consequence of nonstationarity for regression analysis is spuri-
ous correlation that inflates R2 and the t-scores of the nonstationary inde-
pendent variables, which in turn leads to incorrect model specification. This
occurs because the regression estimation procedure attributes to the nonsta-
tionary Xt changes in Yt that were actually caused by some factor (trend, for
example) that also affects Xt. Thus, the variables move together because of
the nonstationarity, increasing R2 and the relevant t-scores. This is especially
important in macroeconometrics, and the macroeconomic literature is dom-
inated by articles that examine various series for signs of nonstationarity.11
Some variables are nonstationary mainly because they increase rapidly
over time. Spurious regression results involving these kinds of variables often
can be avoided by the addition of a simple time trend to
the equation as an independent variable.
Unfortunately, many economic time-series variables are nonstationary
even after the removal of a time trend. This nonstationarity typically takes the
form of the variable behaving as though it were a “random walk.” A random
walk is a time-series variable where next period’s value equals this period’s
value plus a stochastic error term. A random-walk variable is nonstationary
because it can wander up and down without an inherent equilibrium and
without approaching a long-term mean of any sort.
To get a better understanding of the relationship between nonstationarity
and a random walk, let’s suppose that Yt is generated by an equation that in-
cludes only past values of itself (an autoregressive equation):
Yt � �Yt�1 � vt (22)
where vt is a classical error term.
Take a look at Equation 22. Can you see that if u�u � 1, then the expected
value of Yt will eventually approach 0 (and therefore be stationary) as the sam-
ple size gets bigger and bigger? (Remember, since vt is a classical error term, its
(t 5 1, 2, 3, c , T)
403

TIME-SERIES MODELS
12. See C. W. J. Granger and P. Newbold, “Spurious Regression in Econometrics,” Journal of
Econometrics, Volume 2, pp. 111–120.
expected value � 0.) Similarly, can you see that if u�u � 1, then the expected
value of Yt will continuously increase, making Yt nonstationary? This is nonsta-
tionarity due to a trend, but it still can cause spurious regression results.
Most importantly, what about if u�u � 1? In this case,
Yt � Yt�1 � vt (23)
It’s a random walk! The expected value of Yt does not converge on any value,
meaning that it is nonstationary. This circumstance, where � � 1 in Equa-
tion 23 (or similar equations), is called a unit root. If a variable has a unit
root, then Equation 23 holds, and the variable follows a random walk and is
nonstationary. The relationship between unit roots and nonstationarity is so
strong that most econometricians use the words interchangeably, even though
they recognize that both trends and unit roots can cause nonstationarity.
Spurious Regression
As noted at the beginning of Section 4, if the dependent variable and at least
one independent variable in an equation are nonstationary, it’s possible for
the results of an OLS regression to be spurious.12
Consider the linear regression model
(24)
If both X and Y are nonstationary, then they can be highly correlated for non-
causal reasons, and our standard regression inference measures will almost
surely be very misleading in that they’ll overstate and the t-score for 0.
For example, take a look at the following estimated equation:
PRICEt � �27.8 � 0.070TUITIONt (25)
(0.006)
t � 11.4
� .94 T � 10 (annual)
The R2 of this equation and the t-score for the coefficient of TUITION are
clearly significant, but what are the definitions of the variables? Well, PRICE is
the price of a gallon of gasoline in Portland, Oregon, and TUITION is the tu-
ition for a semester of study at Occidental College (Oxy) in Los Angeles (both
measured in nominal dollars). Is it possible that an increase in the tuition at
R 2
�̂R 2
Yt 5 �0 1 �0Xt 1 ut
404

TIME-SERIES MODELS
13. D. A. Dickey and W. A. Fuller, “Distribution of the Estimators for Autoregressive Time-Series
with a Unit Root,” Journal of the American Statistical Association, Vol. 74, pp. 427–431. The
Dickey–Fuller test comes in a variety of forms, including an augmented test to use in cases of a
serially correlated error term.
14. For more on unit roots, see John Y. Campbell and Pierre Peron, “Pitfalls and Opportunities:
What Macroeconomists Should Know About Unit Roots,” NBER Macroeconomics Annual
(Cambridge, MA: MIT Press, 1991), pp. 141–219.
Oxy caused gas prices in Portland to go up? Not unless every Oxy student was
the child of a Portland gas station owner! What’s going on? Well, the 1970s
were a decade of inflation, so any nominally measured variables are likely to
result in an equation that fits as well as Equation 25. Both variables are non-
stationary, and this particular regression result clearly is spurious.
To avoid spurious regression results, it’s crucial to be sure that time-series
variables are stationary before running regressions.
The Dickey–Fuller Test
To ensure that the equations we estimate are not spurious, it’s important to
test for nonstationarity. If we can be reasonably sure that all the variables are
stationary, then we need not worry about spurious regressions. How can you
tell if a time series is nonstationary? The first step is to visually examine the
data. For many time series, a quick glance at the data (or a diagram of the
data) will tell you that the mean of a variable is increasing dramatically over
time and that the series is nonstationary.
After this trend has been removed, the standard method of testing for non-
stationarity is the Dickey–Fuller test,13 which examines the hypothesis that
the variable in question has a unit root14 and, as a result, is likely to benefit
from being expressed in first-difference form.
To best understand how the Dickey–Fuller test works, let’s return to the
discussion of the role that unit roots play in the distinction between station-
arity and nonstationarity. Recall that we looked at the value of � in Equation
22 to help us determine if Y was stationary or nonstationary:
Yt � � Yt�1 � vt (22)
We decided that if u�u � 1 then Y is stationary, and that if u�u � 1, then Yt is
nonstationary. However, if u�u � 1, then Yt is nonstationary due to a unit
root. Thus we concluded that the autoregressive model is stationary if u�u � 1
and nonstationary otherwise.
405

TIME-SERIES MODELS
From this discussion of stationarity and unit roots, it makes sense to esti-
mate Equation 22 and determine if u�u � 1 to see if Y is stationary, and that’s
almost exactly how the Dickey–Fuller test works. First, we subtract Yt�1 from
both sides of Equation 22, yielding:
(Yt � Yt�1) � (� � 1) Yt�1 � vt (26)
If we define �Yt � Yt � Yt�1 then we have the simplest form of the
Dickey–Fuller test:
where �1 � � � 1. The null hypothesis is that Yt contains a unit root and the
alternative hypothesis is that Yt is stationary. If Yt contains a unit root, � � 1
and �1 � 0. If Yt is stationary, u�u � 1 and �1 � 0. Hence we construct a one-
sided t-test on the hypothesis that �1 � 0:
H0: �1 � 0
HA: �1 � 0
Interestingly, the Dickey–Fuller test actually comes in three versions:
1. Equation 27,
2. Equation 27 with a constant term added (Equation 28), and
3. Equation 27 with a constant term and a trend term added
(Equation 29).
The form of the Dickey–Fuller test in Equation 27 is correct if Yt follows
Equation 22, but the test must be changed if Yt doesn’t follow Equation
22. For example, if we believe that Equation 22 includes a constant, then the
appropriate Dickey–Fuller test equation is:
�Yt � �0 � �1Yt�1 � vt (28)
In a similar fashion, if we believe Yt contains a trend “t”
then we’d add “t” to the equation as a variable with a coefficient, and the ap-
propriate Dickey–Fuller test equation is:
�Yt � �0 � �1Yt�1 � �2t � vt (29)
No matter what form of the Dickey–Fuller test we use, the decision rule is
based on the estimate of �1. If 1 is significantly less than 0, then we can�̂
(t 5 1, 2, 3, c , T)
�Yt � �1Yt�1 � vt (27)
406

TIME-SERIES MODELS
Table 1 Large-Sample Critical Values for the Dickey–Fuller Test
One-Sided Significance Level: .01 .025 .05 .10
tc 3.43 3.12 2.86 2.57
15. Most sources list negative critical values for the Dickey–Fuller test, because the unit root test
is one sided with a negative expected value. However, the t-test decision rule of this text is based
on the absolute value of the t-score, so negative critical values would cause every null hypothe-
sis to be rejected. As a result, the critical values in Table 1 are positive. For adjusted critical
t-values for the Dickey–Fuller test, including those in Table 1, see J. G. MacKinnon, “Critical
Values of Cointegration Tests,” in Rob Engle and C. W. J. Granger, eds., Long-Run Economic
Relationships: Readings in Cointegration (New York: Oxford University Press, 1991). Most soft-
ware packages provide these critical values with the output from a Dickey–Fuller test.
reject the null hypothesis of nonstationarity. If 1 is not significantly less
than 0, then we cannot reject the null hypothesis of nonstationarity.
Be careful, however. The standard t-table does not apply to Dickey–Fuller
tests. The critical values depend on the version of the Dickey–Fuller test that
is applicable. For the case of no constant and no trend (Equation 27) the
large-sample values for tc are listed in Table 1.
15 Although not displayed in
Table 1, the critical t-values for smaller samples are about 60 percent larger in
magnitude than those in Statistical Table B-1. For example, a 2.5 percent one-
sided t-test of �1 from Equation 27 with 50 degrees of freedom has a critical
t-value of 3.22, compared to 2.01 for a standard t-test. For practice in running
Dickey–Fuller tests, see Exercises 10 and 11.
Note that the equation for the Dickey–Fuller test and the critical values
for each of the specifications are derived under the assumption that the error
term is serially uncorrelated. If the error term is serially correlated, then the
regression specification must be modified to take this serial correlation into
account. This adjustment takes the form of adding in several lagged first dif-
ferences as independent variables in the equation for the Dickey–Fuller test.
There are several good methods for choosing the number of lags to add, but
there currently is no universal agreement as to which of these methods is
optimal.
Cointegration
If the Dickey–Fuller test reveals nonstationarity, what should we do?
�̂
407

TIME-SERIES MODELS
The traditional approach has been to take the first differences (�Y � Yt �
Yt�1 and �X � Xt � Xt�1) and use them in place of Yt and Xt in the equa-
tion. With economic data, taking a first difference usually is enough to
convert a nonstationary series into a stationary one. Unfortunately, using
first differences to correct for nonstationarity throws away information
that economic theory can provide in the form of equilibrium relationships
between the variables when they are expressed in their original units (Xt and
Yt). As a result, first differences should not be used without carefully weigh-
ing the costs and benefits of that shift, and in particular first differences
should not be used until the residuals have been tested for cointegration.
Cointegration consists of matching the degree of nonstationarity of the
variables in an equation in a way that makes the error term (and residuals) of
the equation stationary and rids the equation of any spurious regression re-
sults. Even though individual variables might be nonstationary, it’s possible
for linear combinations of nonstationary variables to be stationary, or
cointegrated. If a long-run equilbrium relationship exists between a set of vari-
ables, those variables are said to be cointegrated. If the variables are cointe-
grated, then you can avoid spurious regressions even though the dependent
variable and at least one independent variable are nonstationary.
To see how this works, let’s return to Equation 24:
Yt � �0 � �0Xt � ut (24)
As we saw in the previous section, if Xt and Yt are nonstationary, it’s likely
that we’ll get spurious regression results. To understand how it’s possible to
get sensible results from Equation 24 if the nonstationary variables are coin-
tegrated, let’s focus on the case in which both Xt and Yt contain one unit root.
The key to cointegration is the behavior of ut.
If we solve Equation 24 for ut, we get:
ut � Yt � �0 � �0Xt (30)
In Equation 30, ut is a function of two nonstationary variables, so you’d cer-
tainly expect ut also to be nonstationary, but that’s not necessarily the case. In
particular, suppose that Xt and Yt are related? More specifically, if economic
theory supports Equation 24 as an equilibrium, then departures from that
equilibrium should not be arbitrarily large.
Hence, if Yt and Xt are related, then the error term ut may well be sta-
tionary even though Xt and Yt are nonstationary. If ut is stationary, then
408

TIME-SERIES MODELS
16. For more on cointegration, see Peter Kennedy, A Guide to Econometrics (Malden, MA: Black-
well, 2008), pp. 309–313 and 327–330, and B. Bhaskara Rau, ed., Cointegration for the Applied
Economist (New York: St. Martin’s Press, 1994).
17. See J. G. MacKinnon, “Critical Values of Cointegration Tests,” in Rob Engle and C. W. J.
Granger, eds., Long-Run Economic Relationships: Readings in Cointegration (New York: Oxford Uni-
versity Press, 1991) and Rob Engle and C. W. J. Granger, “Co-integration and Error Correction:
Representation, Estimation and Testing,” Econometrica, Vol. 55, No. 2.
18. In this case, it’s common practice to use a version of the original equation called the Error
Correction Model (ECM). While the equation for the ECM is fairly complex, the model itself is
a logical extension of the cointegration concept. If two variables are cointegrated, then there is
an equilibrium relationship connecting them. A regression on these variables therefore is an es-
timate of this equilibrium relationship along with a residual, which is a measure of the extent
to which these variables are out of equilibrium. When formulating a dynamic relationship be-
tween the variables, economic theory suggests that the current change in the dependent vari-
able should be affected not only by the current change in the independent variable but also by
the extent to which these variables were out of equilibrium in the preceding period (the resid-
ual from the cointegrating process). The resulting equation is the ECM. For more on the ECM,
see Peter Kennedy, A Guide to Econometrics (Malden, MA: Blackwell, 2008), pp. 299–301 and
322–323.
the unit roots in Yt and Xt have “cancelled out” and Yt and Xt are said to be
cointegrated.16
We thus see that if Xt and Yt are cointegrated then OLS estimation of the
coefficients in Equation 24 can avoid spurious results. To determine if Xt and
Yt are cointegrated, we begin with OLS estimation of Equation 24 and calcu-
late the OLS residuals:
(31)
We then perform a Dickey–Fuller test on the residuals. Once again, the
standard t-values do not apply to this application, so adjusted critical
t-values should be used.17 However, these adjusted critical values are only
slightly higher than standard critical t-values, so the numbers in Statistical
Table B-1 can be used as rough estimates of the more accurate figures. If we
are able to reject the null hypothesis of a unit root in the residuals, we can
conclude that Yt and Xt are cointegrated and our OLS estimates are not
spurious.
To sum, if the Dickey–Fuller test reveals that our variables have unit roots,
the first step is to test for cointegration in the residuals. If the nonstationary
variables are not cointegrated, then the equation should be estimated using
first differences (�Y and �X). However, if the nonstationary variables are
cointegrated, then the equation can be estimated in its original units.18
et 5 Yt 2 �̂0 2 �̂0Xt
409

TIME-SERIES MODELS
1. Specify the model. This model might be a time-series equation
with no lagged variables, it might be a dynamic model in its
simplest form (Equation 3), or it might be a dynamic model that
includes lags in both the dependent and the independent
variables.
2. Test all variables for nonstationarity (technically unit roots) using
the appropriate version of the Dickey–Fuller test.
3. If the variables don’t have unit roots, estimate the equation in its
original units (Y and X).
4. If the variables have unit roots, test the residuals of the equation
for cointegration using the Dickey–Fuller test.
5. If the variables have unit roots but are not cointegrated, then
change the functional form of the model to first differences (�Y
and �X) and estimate the equation.
6. If the variables have unit roots and also are cointegrated, then
estimate the equation in its original units
A Standard Sequence of Steps for Dealing
with Nonstationary Time Series
This material is fairly complex, so let’s pause for a moment to summarize the
various steps suggested in Section 4. To deal with the possibility that nonsta-
tionary time series might be causing regression results to be spurious, most
empirical work in time series follows a standard sequence of steps:
Summary
1. A distributed lag explains the current value of Y as a function of cur-
rent and past values of X, thus “distributing” the impact of X over a
number of lagged time periods. OLS estimation of distributed lag
equations without any constraints (ad hoc distributed lags) encoun-
ters problems with multicollinearity, degrees of freedom, and a non-
continuous pattern of coefficients over time.
2. A dynamic model avoids these problems by assuming that the coeffi-
cients of the lagged independent variables decrease in a geometric
5
410

TIME-SERIES MODELS
fashion the longer the lag. Given this, the dynamic model is:
Yt � �0 � �0Xt � �Yt�1 � ut
where Yt�1 is a lagged dependent variable and 0 � � � 1.
3. In small samples, OLS estimates of a dynamic model are biased and
have unreliable hypothesis testing properties. Even in large samples,
OLS will produce biased estimates of the coefficients of a dynamic
model if the error term is serially correlated.
4. In a dynamic model, the Durbin–Watson d test sometimes can fail to
detect the presence of serial correlation because d is biased toward 2.
The most-used alternative is the Lagrange Multiplier test.
5. Granger causality, or precedence, is a circumstance in which one time-
series variable consistently and predictably changes before another
variable does. If one variable precedes (Granger-causes) another, we
still can’t be sure that the first variable “causes” the other to change.
6. A nonstationary series is one that exhibits significant changes (for ex-
ample, in its mean and variance) over time. If the dependent variable
and at least one independent variable are nonstationary, a regression
may encounter spurious correlation that inflates and the t-scores of
the nonstationary independent variable(s).
7. Nonstationarity can be detected using the Dickey–Fuller test. If the
variables are nonstationary (have unit roots) then the residuals of the
equation should be tested for cointegration using the Dickey–Fuller
test. If the variables are nonstationary but are not cointegrated, then
the equation should be estimated with first differences. If the vari-
ables are nonstationary and also are cointegrated, then the equation
can be estimated in its original units.
EXERCISES
(The answer to Exercise 2 is at the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or your notes), and then compare your definition with the
version in the text for each:
a. dynamic model
b. ad hoc distributed lag model
R2
411

TIME-SERIES MODELS
c. Lagrange Multiplier Serial Correlation test
d. Granger causality
e. nonstationary series
f. Dickey–Fuller test
g. unit root
h. random walk
i. cointegration
2. Consider the following equation aimed at estimating the demand for
real cash balances in Mexico (standard errors in parentheses):
where: Mt � the money stock in year t (millions of pesos)
Rt � the long-term interest rate in year t (percent)
Yt � the real GNP in year t (millions of pesos)
a. What economic relationship between Y and M is implied by the
equation?
b. How are Y and R similar in terms of their relationship to M?
c. Does this equation seem likely to have serial correlation? Explain.
3. Calculate and graph the pattern of the impact of a lagged X on Y as
the lag increases for each of the following estimated dynamic models:
a.
b.
c.
d.
e. Look over your graphs for parts c and d. What restriction do they
combine to show the wisdom of?
4. Consider the following equation for the determination of wages in
the United Kingdom (standard error in parentheses):
where: Wt � wages and salaries per employee in year t
Pt � the price level in year t
Ut � the percent unemployment in year t
R2 5 .87 N 5 19
(0.080) (0.072) (0.658)
Wt 5 8.562 1 0.364Pt 1 0.004Pt 2 1 2 2.56Ut
�
Yt 5 13.0 1 12.0Xt 2 0.4Yt 2 1
Yt 5 13.0 1 12.0Xt 1 2.0Yt 2 1
Yt 5 13.0 1 12.0Xt 1 0.08Yt 2 1
Yt 5 13.0 1 12.0Xt 1 0.04Yt 2 1
R2 5 .90 DW 5 1.80 N 5 26
(0.10) (0.35) (0.10)
lnMt 5 2.00 2 0.10lnRt 1 0.70lnYt 1 0.60lnMt 2 1
412

TIME-SERIES MODELS
a. Develop and test your own hypotheses with respect to the individ-
ual slope coefficients at the 10-percent level.
b. Discuss the theoretical validity of and how your opinion
of that validity has been changed by its statistical significance.
Should be dropped from the equation? Why or why not?
c. If is dropped from the equation, the general functional form
of the equation changes radically. Why?
5. You’ve been hired to determine the impact of advertising on gross
sales revenue for “Four Musketeers” candy bars. Four Musketeers has
the same price and more or less the same ingredients as competing
candy bars, so it seems likely that only advertising affects sales. You
decide to build a distributed lag model of sales as a function of adver-
tising, but you’re not sure whether an ad hoc or a dynamic model is
more appropriate.
Using data on Four Musketeers candy bars from Table 2, estimate
both of the following distributed lag equations from 1985–2009 and
compare the lag structures implied by the estimated coefficients. (Hint:
Be careful to use the correct sample.)
a. an ad hoc distributed lag model (4 lags)
b. a dynamic model
6. Test for serial correlation in the estimated dynamic model you got as
your answer to Exercise 5b.
7. Suppose you’re building a dynamic model and are concerned with
the possibility that serial correlation, instead of being first order, is
second order:
a. What is the theoretical meaning of such second-order serial
correlation?
b. Carefully write out the formula for the Lagrange Multiplier Serial
Correlation (LMSC) test auxiliary equation (similar to Equa-
tion 18) that you would have to estimate to test such a possibil-
ity. How many degrees of freedom would there be in such an
LMSC test?
c. Test for second-order serial correlation in the estimated dynamic
model you got as your answer to Exercise 5b.
8. Most economists consider investment and output to be jointly (simulta-
neously) determined. One test of this simultaneity would be to see
whether one of the variables could be shown to Granger-cause the other.
Take the data set from the small macroeconomic model in Table 1 from
Chapter 14 and test the possibility that investment (I) Granger-causes
ut 5 f(ut 2 1, ut 2 2).
Pt 2 1
Pt 2 1
Pt 2 1
413

TIME-SERIES MODELS
GDP (Y) (or vice versa) with a two-sided Granger test with four
lagged Xs.
9. Some farmers were interested in predicting inches of growth of corn
as a function of rainfall on a monthly basis, so they collected data
from the growing season and estimated an equation of the following
form:
Gt 5 �0 1 �1Rt 1 �2Gt 2 1 1 �t
Table 2 Data for the Four Musketeers Exercise
Year Sales Advertising
1981 * 30
1982 * 35
1983 * 36
1984 320 39
1985 360 40
1986 390 45
1987 400 50
1988 410 50
1989 400 50
1990 450 53
1991 470 55
1992 500 60
1993 500 60
1994 490 60
1995 580 65
1996 600 70
1997 700 70
1998 790 60
1999 730 60
2000 720 60
2001 800 70
2002 820 80
2003 830 80
2004 890 80
2005 900 80
2006 850 75
2007 840 75
2008 850 75
2009 850 75
Datafile � MOUSE12
414

TIME-SERIES MODELS
where: Gt � inches of growth of corn in month t
Rt � inches of rain in month t
� a normally distributed classical error term
The farmers expected a negative sign for (they felt that since corn
can only grow so much, if it grows a lot in one month, it won’t grow
much in the next month), but they got a positive estimate instead.
What suggestions would you have for this problem?
10. Run 2.5 percent Dickey–Fuller tests (of the form in Equation 27) for
the following variables using the data in Table 2 from Chapter 6
from the chicken demand equation and determine which variables,
if any, you think are nonstationary. (Hint: Use 3.12 as your critical
t-value.)
a. Yt
b. PCt
c. PBt
d. YDt
11. Run 2.5 percent Dickey–Fuller tests (of the form in Equation 27) for
the following variables using the data from the small macroeco-
nomic model in Table 1 from Chapter 4 and determine which vari-
ables, if any, you think are nonstationary. (Hint: Use 3.12 as your
critical t-value.)
a. Y (GDP)
b. r (the interest rate)
c. CO (consumption)
d. I (investment)
12. In 2001, Heo and Tan published an article19 in which they used the
Granger causality model to test the relationship between economic
growth and democracy. For years, political scientists have noted a strong
positive relationship between economic growth and democracy, but the
authors of previous studies (which included Granger causality studies)
disagreed about the causality involved. Heo and Tan studied 32 devel-
oping countries and found that economic growth “Granger-caused”
democracy in 11 countries, while democracy “Granger-caused” economic
growth in 10 others.
�2
�t
19. Uk Heo and Alexander Tan, “Democracy and Economic Growth: a Causal Analysis,”
Comparative Politics, Vol. 33, No. 4 (July 2001), pp. 463–473.
415

a. How is it possible to get significant Granger causality results in two
different directions in the same study? Is this evidence that the
study was done incorrectly? Is this evidence that Granger causality
tests cannot be applied to this topic?
b. Based on the evidence presented, what’s your conclusion about the
relationship between economic growth and democracy? Explain.
c. If this were your research project, what would your next step be?
(Hint: In particular, is there anything to be gained by learning more
about the countries in the two different Granger causality groups?20)
TIME-SERIES MODELS
20. For the record, the 11 countries in which growth Granger caused democracy were Costa Rica,
Egypt, Guatemala, India, Israel, South Korea, Mexico, Nicaragua, Thailand, Uruguay, and
Venezuela, and the 10 countries in which democracy Granger caused growth were Bolivia, Burma,
Colombia, Ecuador, El Salvador, Indonesia, Iran, Paraguay, the Philippines, and South Africa.
Answers
Exercise 2
a. The double-log functional form doesn’t change the fact that this
is a dynamic model. As a result, Y and M almost surely are related
by a distributed lag.
b. In their relationship to M, both Y and R have the same distrib-
uted lag pattern over time, since the lambda of 0.60 applies to
both. (The equation is in double-log form, so technically the re-
lationships are between the logs of those variables.)
c. Serial correlation is always a concern in a dynamic model. Many
students will look at the Durbin–Watson statistic of 1.80 and
conclude that there is no evidence of positive serial correlation in
this equation, but the d-statistic is biased toward 2 in the pres-
ence of a lagged dependent variable. Ideally, we would use the
Lagrange Multiplier Serial Correlation Test, but we don’t have the
data to do so. Durbin’s h test, which is beyond the scope of this
text, provides evidence that there is indeed serial correlation in
the equation. For more, see Robert Raynor, “Testing for Serial
Correlation in the Presence of Lagged Dependent Variables,” The
Review of Economics and Statistics, Vol. 75, No. 4, pp. 716–721.
416

Until now, our discussion of dummy variables has been restricted to dummy
independent variables. However, there are many important research topics
for which the dependent variable is appropriately treated as a dummy, equal
only to 0 or 1.
In particular, researchers analyzing consumer choice often must cope
with dummy dependent variables (also called qualitative dependent vari-
ables). For example, how do high school students decide whether to go to
college? What distinguishes Pepsi drinkers from Coke drinkers? How can we
convince people to use public transportation instead of driving? For an
econometric study of these topics, or of any topic that involves a discrete
choice of some sort, the dependent variable is typically a dummy variable.
In the first two sections of this chapter, we’ll present two frequently used
ways to estimate equations that have dummy dependent variables: the linear
probability model and the binomial logit model. In the last section, we’ll
briefly discuss two other useful dummy dependent variable techniques: the
binomial probit model and the multinomial logit model.
The Linear Probability Model
What Is a Linear Probability Model?
The most obvious way to estimate a model with a dummy dependent variable
is to run OLS on a typical linear econometric equation. A linear probability
1
Dummy Dependent
Variable Techniques
1 The Linear Probability Model
2 The Binomial Logit Model
3 Other Dummy Dependent Variable Techniques
4 Summary and Exercises
From Chapter 13 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011
by Pearson Education. Published by Addison-Wesley. All rights reserved.
417

DUMMY DEPENDENT VARIABLE TECHNIQUES
model is just that, a linear-in-the-coefficients equation used to explain a
dummy dependent variable:
(1)
where Di is a dummy variable and the Xs, are typical independent
variables, regression coefficients, and an error term, respectively.
For example, suppose you’re interested in understanding why some states
have female governors and others don’t. In such a model, the appropriate de-
pendent variable would be a dummy, for example Di equal to one if the ith
state has a female governor and equal to zero otherwise. If we hypothesize
that states with a high percentage of females and a low percentage of social
conservatives would be likely to have a female governor, then a linear proba-
bility model would be:
(2)
where: Di 5 1 if the ith state has a female governor, 0 otherwise
Fi 5 females as a percent of the ith state’s population
Ri 5 conservatives as a percent of the ith state’s registered voters
The term linear probability model comes from the fact that the right side
of the equation is linear while the expected value of the left side measures
the probability that To understand this second statement, let’s as-
sume that we estimate Equation 2 and get a of 0.10 for a particular state.
What does that mean? Well, since if the governor is female and
if the governor is male, a state with a of 0.10 can perhaps best be
thought of as a state in which there is a 10-percent chance that the gover-
nor will be female, based on the state’s values for the independent vari-
ables. Thus measures the probability that for the ith observa-
tion, and:
(3)
where indicates the probability that for the ith observation.
How should we interpret the coefficients of Equation 3? Since meas-
ures the probability that then a coefficient in a linear probability
model tells us the percentage point change in the probability that Di 5 1
Di 5 1,
D̂i
Di 5 1Pr(Di 5 1)
D̂i 5 Pr(Di 5 1) 5 �̂0 1 �̂1Fi 1 �̂2Ri
Di 5 1D̂i
D̂iD 5 0
D 5 1
D̂i
Di 5 1.
Di 5 �0 1 �1Fi 1 �2Ri 1 �i
�s, and �
Di 5 �0 1 �1X1i 1 �2X2i 1 �i
418

DUMMY DEPENDENT VARIABLE TECHNIQUES
1. In addition, the error term of a linear probability model is neither homoskedastic nor nor-
mally distributed, mainly because D takes on just two values (0 and 1). In practice, however,
the impact of these problems on OLS estimation is minor, and many researchers ignore poten-
tial heteroskedasticity and nonnormality and apply OLS directly to the linear probability
model. See R. G. McGillvray, “Estimating the Linear Probability Function,” Econometrica, Vol. 38,
pp. 775–776.
caused by a one-unit increase in the independent variable in question, hold-
ing constant the other independent variables in the equation.
We can never observe the actual probability, however, because it reflects
the situation before a discrete decision is made. After the choice is made, we
can observe only the outcome of that choice, and so the dependent variable
Di can take on the values of only 0 or 1. Thus, even though the expected
value can be anywhere from 0 to 1, we can only observe the two extremes
(0 and 1) in our dependent variable (Di).
Problems with the Linear Probability Model
Unfortunately, the use of OLS to estimate the coefficients of an equation with
a dummy dependent variable encounters two major problems:1
1. is not an accurate measure of overall fit. For models with a dummy de-
pendent variable, tells us very little about how well the model ex-
plains the choices of the decision makers. To see why, take a look at
Figure 1. Di can equal only 1 or 0, but must move in a continuous
fashion from one extreme to the other. This means that is likely to
be quite different from Di for some range of Xi. Thus, is likely to be
much lower than 1 even if the model actually does an exceptional job
of explaining the choices involved. As a result, (or R2) should not be
relied on as a measure of the overall fit of a model with a dummy de-
pendent variable.
2. is not bounded by 0 and 1. Since Di is a dummy variable, we’d expect
to be limited to a range of 0 to 1. After all, the prediction that a
probability equals 2.6 (or for that matter) is almost meaningless.
However, take another look at Equation 3. Depending on the values of
the Xs and the the right-hand side might well be outside the mean-
ingful range. For instance, if all the Xs and s in Equation 3 equal 1.0,
then equals 3.0, substantially greater than 1.0.
The first of these two major problems isn’t impossible to deal with, because
there are a variety of alternatives to for equations with dummy-dependentR2
D̂i
�̂
�̂s,
22.6,
D̂i
D̂i
R2
R2
D̂i
D̂i
R2
R2
419

DUMMY DEPENDENT VARIABLE TECHNIQUES
Di
Di > 1
Di = 1
1 > Di > 0
Di = 0
Di = �0 + �1X1i + �2X2i
X1i
(Holding X2i Constant)
Di < 0 variables.2 Our preference is to create a measure based on the percentage of the observations in the sample that a particular estimated equation ex- plains correctly. To use this approach, consider a to predict that and a to predict that If we then compare these Di 5 0.D̂i , .5Di 5 1 D̂i . .5 Figure 1 A Linear Probability Model In a linear probability model, all the observed Dis equal either 0 or 1 but moves linearly from one extreme to the other. As a result, is often quite low even if the model does an excellent job of explaining the decision maker’s choice. In addition, exception- ally large or small values of X1i (holding X2i constant), can produce values of outside the meaningful range of 0 to 1. D̂i R2 D̂i 2. See M. R. Veal and K. F. Zimmerman, “Pseudo-R2 Measures for Some Common Limited Dependent Variables Models,” Journal of Economic Surveys, Vol. 10, No. 3, pp. 241–259 and C. S. McIntosh and J. J. Dorfman, “Qualitative Forecast Evaluation: A Comparison of Two Perfor- mance Measures,” American Journal of Agricultural Economics, Vol. 74, pp. 209–214. 420 DUMMY DEPENDENT VARIABLE TECHNIQUES 3. Although it’s standard to use as the value that distinguishes a prediction of from a prediction of , there’s no rule that requires that .5 be used. This is because it’s possible to imagine circumstances in which .5 is too high or too low. For example, if the payoff when you’re right if you classify is much lower than the payoff when you’re right if you classify , then a value lower than .5 might make sense. We’re grateful to Peter Kennedy for this observation. Di 5 0 Di 5 1 Di 5 0 Di 5 1D̂i 5 .5 predictions3 with the actual Di, we can calculate the percentage of observa- tions explained correctly. Unfortunately, using the percentage explained correctly as a substitute for for the entire sample has a flaw. Suppose that 85 percent of your ob- servations are ones and 15 percent are zeroes. Explaining 85 percent of the sample correctly sounds good, but your results are no better than naively guessing that every observation is a one! A better way might be to calculate the percentage of ones explained correctly, calculate the percentage of zeroes explained correctly, and then report the average of these two percentages. As a shorthand, we’ll call this average . That is, we’ll define to be the average of the percentage of ones explained correctly and the percentage of zeroes explained correctly. Since is a new statistic, we’ll calculate and discuss both and throughout this chapter. For most researchers, therefore, the major difficulty with the linear proba- bility model is the unboundedness of the predicted . Take another look at Figure 1 for a graphical interpretation of the situation. Because of the linear relationship between the can fall well outside the relevant range of 0 to 1. Using the linear probability model, despite this unbounded- ness problem, may not cause insurmountable difficulties. In particular, the signs and general significance levels of the estimated coefficients of the linear probability model are often similar to those of the alternatives we will dis- cuss later in this chapter. One simplistic way to get around the unboundedness problem is to assign to all values of above 1 and to all negative val- ues. This approach copes with the problem by ignoring it, since an obser- vation for which the linear probability model predicts a probability of 2.0 has been judged to be more likely to be equal to 1.0 than an observa- tion for which the model predicts a 1.0, and yet they are lumped together. Even isn’t very useful, because it implies that events will happen with certainty, surely a foolish prediction to make. What is needed is a systematic method of forcing the to range from 0 to 1 in a smooth and meaningful fashion. We’ll present such a method, the binomial logit, in Section 2. D̂is D̂i 5 1 D̂i 5 0.0D̂iD̂i 5 1.0 Xis and D̂i, D̂i Dis R2R2p R2p R2pR 2 p R2 421 DUMMY DEPENDENT VARIABLE TECHNIQUES An Example of a Linear Probability Model Before moving on to investigate the logit, however, let’s take a look at an exam- ple of a linear probability model: a disaggregate study of the labor force par- ticipation of women. A person is defined as being in the labor force if she either has a job or is actively looking for a job. Thus, a disaggregate (cross-sectional by person) study of women’s labor force participation is appropriately modeled with a dummy dependent variable: Di � 1 if the ith woman has or is looking for a job, 0 otherwise (not in the labor force) A review of the literature reveals that there are many potentially relevant independent variables. Two of the most important are the marital status and the number of years of schooling of the woman. The expected signs for the coefficients of these variables are fairly straightforward, since a woman who is unmarried and well educated is much more likely to be in the labor force than her opposite: (4) where: Mi 5 1 if the ith woman is married and 0 otherwise Si 5 the number of years of schooling of the ith woman The data are presented in Table 1. The sample size is limited to 30 in order to make it easier for readers to enter the dataset on their own. Unfortunately, such a small sample will make hypothesis testing fairly unreliable. Table 1 also includes the age of the ith woman for use in Exercises 8 and 9. Another typi- cally used variable, Oi 5 other income available to the ith woman, is not avail- able for this sample, introducing possible omitted variable bias. If we choose a linear functional form for both independent variables, we’ve got a linear probability model: (5) If we now estimate Equation 5 with the data on the labor force participation of women from Table 1, we obtain (standard errors in parentheses): (6) N 5 30 R2 5 .32 R2p 5 .81 (0.15) (0.03) D̂i 5 2 0.28 2 0.38Mi 1 0.09Si Di 5 �0 1 �1Mi 1 �2Si 1 �i Di 5 f(M 2 i, S 1 i) 1 �i 422 DUMMY DEPENDENT VARIABLE TECHNIQUES How do these results look? Despite the small sample and the possible bias due to omitting Oi, both independent variables have estimated coefficients that are significant in the expected direction. In addition, the 2 of .32 is fairly high for a linear probability model (since Di equals only 0 or 1, it’s almost im- possible to get an 2 much higher than .70). Further evidence of good fit is the fairly high 2p of .81, meaning that an average of 81 percent of the choices were explained “correctly” by Equation 6. R R R Table 1 Data on the Labor Force Participation of Women Observation # Di Mi Ai Si i 1 1.0 0.0 31.0 16.0 1.20 2 1.0 1.0 34.0 14.0 0.63 3 1.0 1.0 41.0 16.0 0.82 4 0.0 0.0 67.0 9.0 0.55 5 1.0 0.0 25.0 12.0 0.83 6 0.0 1.0 58.0 12.0 0.45 7 1.0 0.0 45.0 14.0 1.01 8 1.0 0.0 55.0 10.0 0.64 9 0.0 0.0 43.0 12.0 0.83 10 1.0 0.0 55.0 8.0 0.45 11 1.0 0.0 25.0 11.0 0.73 12 1.0 0.0 41.0 14.0 1.01 13 0.0 1.0 62.0 12.0 0.45 14 1.0 1.0 51.0 13.0 0.54 15 0.0 1.0 39.0 9.0 0.17 16 1.0 0.0 35.0 10.0 0.64 17 1.0 1.0 40.0 14.0 0.63 18 0.0 1.0 43.0 10.0 0.26 19 0.0 1.0 37.0 12.0 0.45 20 1.0 0.0 27.0 13.0 0.92 21 1.0 0.0 28.0 14.0 1.01 22 1.0 1.0 48.0 12.0 0.45 23 0.0 1.0 66.0 7.0 20.01 24 0.0 1.0 44.0 11.0 0.35 25 0.0 1.0 21.0 12.0 0.45 26 1.0 1.0 40.0 10.0 0.26 27 1.0 0.0 41.0 15.0 1.11 28 0.0 1.0 23.0 10.0 0.26 29 0.0 1.0 31.0 11.0 0.35 30 1.0 1.0 44.0 12.0 0.45 D̂ Datafile � WOMEN13 423 DUMMY DEPENDENT VARIABLE TECHNIQUES We need to be careful when we interpret the estimated coefficients in Equation 6, however. Remember that the slope coefficient in a linear proba- bility model represents the change in the probability that Di equals one caused by a one-unit increase in the independent variable (holding the other independent variables constant). Viewed in this context, do the estimated coefficients make economic sense? The answer is yes: the probability of a woman participating in the labor force falls by 38 percent if she is married (holding constant schooling). Each year of schooling increases the probability of labor force participation by 9 percent (holding constant marital status). The values for have been included in Table 1. Note that is indeed often outside the meaningful range of 0 and 1, causing most of the problems cited earlier. To attack this problem of the unboundedness of however, we need a new estimation technique, so let’s take a look at one. The Binomial Logit Model What Is the Binomial Logit? The binomial logit is an estimation technique for equations with dummy dependent variables that avoids the unboundedness problem of the linear probability model by using a variant of the cumulative logistic function: (7) Are the produced by a logit now limited by 0 and 1? The answer is yes, but to see why we need to take a close look at Equation 7. What is the largest that can be? Well, if equals infinity, then: (8) because e to the minus infinity equals zero. What’s the smallest that can be? If equals minus infinity, then: (9) Thus, is bounded by 1 and 0. As can be seen in Figure 2, approaches 1 and 0 very slowly (asymptotically). The binomial logit model therefore D̂iD̂i D̂i 5 1 1 1 e ` 5 1 ` 5 0 �̂0 1 �̂1X1i 1 �̂2X2i D̂i D̂i 5 1 1 1 e 2` 5 1 1 5 1 �̂0 1 �̂1X1i 1 �̂2X2iD̂i D̂is Di 5 1 1 1 e 2f�0 1 �1X1i 1 �2X2i 1 �ig 2 D̂i, D̂iD̂i 424 DUMMY DEPENDENT VARIABLE TECHNIQUES Di Di = 1 1 > Di > 0
Di = 0
X1
(Holding X2 Constant)
Logit
Linear Probability Model
(for comparison purposes)
Figure 2 i Is Bounded by 0 and 1 in a Binomial Logit Model
In a binomial logit model, is nonlinearly related to X1, so even exceptionally large
or small values of X1i, holding X2i constant, will not produce values of outside the
meaningful range of 0 to 1.
D̂i
D̂i
D̂
avoids the major problem that the linear probability model encounters in
dealing with dummy dependent variables. In addition, the logit is quite sat-
isfying to most researchers because it turns out that real-world data often
are described well by S-shape patterns like that in Figure 2.
Logits cannot be estimated using OLS. Instead, we use maximum likeli-
hood (ML), an iterative estimation technique that is especially useful for
equations that are nonlinear in the coefficients. ML estimation is inher-
ently different from least squares in that it chooses coefficient estimates that
425

DUMMY DEPENDENT VARIABLE TECHNIQUES
4. Actually, the ML program chooses coefficient estimates that maximize the log of the proba-
bility (or likelihood) of observing the particular set of values of the dependent variable in the
sample (Y1, Y2, . . . , YN) for a given set of Xs. For more on maximum likelihood, see Robert S.
Pindyck and Daniel L. Rubinfeld, Economic Models and Economic Forecasts (New York: McGraw-Hill,
1998), pp. 51–53 and 329–330.
5. The constant term, however, needs to be adjusted. Multiply where
p1 is the proportion of the observations chosen if Di � 1 and
p2 is the proportion of the observations chosen if Di � 0. See G. S. Maddala, Limited-Dependent
and Qualitative Variables in Econometrics (Cambridge: Cambridge University Press, 1983), pp.
90–91.
�̂0 by fln(p1) 2 ln(p2)g,
maximize the likelihood of the sample data set being observed.4 Interestingly,
OLS and ML estimates are not necessarily different; for a linear equation that
meets the Classical Assumptions (including the normality assumption), ML
estimates are identical to the OLS ones.
One of the reasons that maximum likelihood is used is that ML has a
number of desirable large sample properties; ML is consistent and asymptoti-
cally efficient (unbiased and minimum variance for large samples). With very
large samples, ML has the added advantage of producing normally distrib-
uted coefficient estimates, allowing the use of typical hypothesis testing tech-
niques. As a result, sample sizes for logits should be substantially larger than
for linear regressions. Some researchers aim for samples of 500 or more.
It’s also important to make sure that a logit sample contains a reasonable
representation of both alternative choices. For instance, if 98 percent of a
sample chooses alternative A and 2 percent chooses B, a random sample of
500 would have only 10 observations that choose B. In such a situation, our
estimated coefficients would be overly reliant on the characteristics of those
10 observations. A better technique would be to disproportionately sample
from those who choose B. It turns out that using different sampling rates for
subgroups within the sample does not cause bias in the slope coefficients of
a logit model,5 even though it might do so in a linear regression.
When we estimate a logit, we apply the ML technique to Equation 7, but
that equation’s functional form is complex, so let’s try to simplify it a bit.
First, a few mathematical steps can allow us to rewrite Equation 7 so that the
right side of the equation looks identical to the linear probability model:
(10)
where is the dummy variable. If you’re interested in the math behind this
transformation, see Exercise 4.
Di
lna
Di
f1 2 Dig
b 5 �0 1 �1X1i 1 �2X2i 1 �i
426

DUMMY DEPENDENT VARIABLE TECHNIQUES
Even Equation 10 is a bit cumbersome, however, since the left side of the
equation contains the log of the ratio of to , sometimes called the
“log of the odds.” To make things simpler still, let’s adopt a shorthand for
the logit functional form on the left side of Equation 10. Let’s define:
(11)
The L indicates that the equation is a logit of the functional form in
Equation 10 (derived from Equation 7), and the is a reminder
that the dependent variable is a dummy and that a produced by an esti-
mated logit equation is an estimate of the probability that . If we now
substitute Equation 11 into Equation 10, we get:
(12)
Equation 12 will be our standard documentation format for estimated logit
equations.
Interpreting Estimated Logit Coefficients
Once you’ve estimated a binomial logit, then hypothesis testing and the
analysis of potential econometric problems can be undertaken using the tech-
niques. The signs of the coefficients have the same meaning as they do in a
linear probability model, and the t-test can be used for tests of hypotheses
about logit coefficients.
When it comes to the economic interpretation of the estimated logit coef-
ficients, however, all this changes. In particular, the absolute sizes of esti-
mated logit coefficients tend to be quite different from the absolute sizes of
estimated linear probability model coefficients for the same specification
and the same data. What’s going on?
There are two powerful reasons for these differences. First, as you can see by
comparing Equations 1 and 10, the dependent variable in a logit equation isn’t
the same as the dependent variable in a linear probability model. Since the de-
pendent variable is different, it makes complete sense that the coefficients are
different. The second reason that logit coefficients are different is even more
dynamic. Take a look at Figure 2. The slope of the graph of the logit changes as
moves from 0 to 1! Thus the change in the probability that caused by
a one-unit increase in an independent variable (holding the other indepen-
dent variables constant) will vary as we move from to .D̂i 5 1D̂i 5 0
D̂i 5 1D̂i
L:Pr (Di 5 1) 5 �0 1 �1X1i 1 �2X2i 1 �i
Di 5 1
D̂i
“Pr(Di 5 1)”
L:Pr(Di 5 1) 5 lna
Di
f1 2 Dig
b
(1 2 Di)Di
427

DUMMY DEPENDENT VARIABLE TECHNIQUES
6. Ramu Ramanathan, Introductory Econometrics (Fort Worth: Harcourt Brace, 1998), p. 607.
7. See, for example, Jeff Wooldridge, Introductory Econometrics (Mason, OH: Southwestern,
2009), p. 584. Wooldridge also suggests a multiple of 0.40 for converting a probit coefficient
into a linear probability coefficient. We’ll briefly cover probits in Section 3.
Given all this, how can we interpret estimated logit coefficients? How can
we use them to measure the impact of an independent variable on the proba-
bility that ? It turns out that there are three reasonable ways of answer-
ing this question:
1. Change an average observation. Create an “average” observation by plug-
ging the means of all the independent variables into the estimated logit
equation and then calculating an “average” . Then increase the inde-
pendent variable of interest by one unit and recalculate the . The dif-
ference between the two tells you the impact of a one-unit increase
in that independent variable on the probability that (holding
constant the other independent variables) for an average observation.
This approach has the weakness of not being very meaningful when
one or more of the independent variables is a dummy variable (after
all, what is an average gender?), but it’s possible to work around this
weakness if you estimate the impact for an “average female” and an
“average male” by setting the dummy independent variable equal first
to zero and then to one.
2. Use a partial derivative. It turns out that if you take a derivative6 of the
logit, you’ll find that the change in the expected value of caused by a
one unit increase in holding constant the other independent vari-
ables in the equation equals . To use this formula, plug
in your estimates of and . As you can see, the marginal impact of
X does indeed depend on the value of .
3. Use a rough estimate of 0.25. The previous two methods are reasonably
accurate, but they’re hardly very handy. However, if you plug
into the previous equation, you get the much more useful result that if
you multiply a logit coefficient by 0.25, you’ll get an equivalent linear
probability model coefficient.7
On balance, which approach do we recommend? For all situations except
those requiring precise accuracy, we find ourselves gravitating toward the third
approach. To get a rough approximation of the economic meaning of a logit co-
efficient, multiply by 0.25 (or, equivalently, divide by 4). Remember, however,
that the dependent variable in question still is the probability that .D̂i 5 1
D̂i 5 0.5
D̂i
Di�1
�̂1D̂i(1 2 D̂i)
X1i
D̂i
D̂i 5 1
D̂is
D̂i
D̂i
D̂i 5 1
428

DUMMY DEPENDENT VARIABLE TECHNIQUES
Measuring the overall fit also is not straightforward. Recall that since the
functional form of the dependent variable has been changed, should not
be used to compare the fit of a logit with an otherwise comparable linear
probability model. In addition, don’t forget the general faults
inherent in using with equations with dummy dependent variables. Our
suggestion is to use the mean percentage of correct predictions, , from
Section 1.
To get some practice interpreting logit estimates, let’s estimate a logit on
the same women’s labor force participation data that we used in the pre-
vious section. The OLS linear probability model estimate of that model,
Equation 6, was:
(6)
where: Di 5 1 if the ith woman is in the labor force, 0 otherwise
Mi 5 1 if the ith woman is married, 0 otherwise
Si 5 the number of years of schooling of the ith woman
If we estimate a logit on the same data (from Table 1) and the same indepen-
dent variables, we obtain:
(13)
Let’s compare Equations 6 and 13. As expected, the signs and general signifi-
cance of the slope coefficients are the same. Even if we divide the logit coeffi-
cients by 4, as suggested earlier, they still are larger than the linear probability
model coefficients. Despite these differences, the overall fits are roughly com-
parable, especially after taking account of the different dependent variables
and estimation techniques. In this example, then, the two estimation proce-
dures differ mainly in that the logit does not produce outside the range of
0 and 1.
However, if the size of the sample in this example is too small for a linear
probability model, it certainly is too small for a logit, making any in-depth
analysis of Equation 13 problematic. Instead, we’re better off finding an ex-
ample with a much larger sample.
D̂is
N 5 30 R2p 5 .81 iterations 5 5
t 5 2 2.19 2.19
(1.18) (0.31)
L :Pr (Di 5 1) 5 2 5.89 2 2.59Mi 1 0.69Si
N 5 30 R2 5 .32 R2p 5 .81
(0.15) (0.03)
D̂i 5 2 0.28 2 0.38Mi 1 0.09Si
R2p
R2
R2
429

DUMMY DEPENDENT VARIABLE TECHNIQUES
A More Complete Example of the Use of the Binomial Logit
For a more complete example of the binomial logit, let’s look at a model of
the probability of passing the California State Department of Motor Vehicles
drivers’ license test. To obtain a license, each driver must pass a written and a
behind-the-wheel test. Even though the tests are scored from 0 to 100, all that
matters is that you pass and get your license.
Since the test requires some boning up on traffic and safety laws, driving
students have to decide how much time to spend studying. If they don’t study
enough, they waste time because they have to retake the test. If they study too
much, however, they also waste time, because there’s no bonus for scoring
above the minimum, especially since there is no evidence that doing well on
the test has much to do with driving well after the test (this, of course, might
be worth its own econometric study).
Recently, two students decided to collect data on test takers in order to
build an equation explaining whether someone passed the Department of
Motor Vehicles test. They hoped that the model, and in particular the esti-
mated coefficient of study time, would help them decide how much time to
spend studying for the test. (Of course, it took more time to collect the data
and run the model than it would have taken to memorize the entire traffic
code, but that’s another story.)
After reviewing the literature, choosing variables, and hypothesizing signs,
the students realized that the appropriate functional form was a binomial
logit because their dependent variable was a dummy variable:
Their hypothesized equation was:
(14)
where: Ai 5 the age of the ith test taker
Hi 5 the number of hours the ith test taker studied (usually less
than one hour!)
Ei 5 a dummy variable equal to 1 if the ith test taker’s primary
language was English, 0 otherwise
Ci 5 a dummy variable equal to 1 if the ith test taker had any
college experience, 0 otherwise
Di 5 f( A
1
i, H
1
i, E
1
i, C
1
i) 1 �i
Di 5 e
1 if the ith test taker passed the test on the first try
0 if the ith test taker failed the test on the first try
430

DUMMY DEPENDENT VARIABLE TECHNIQUES
8. For more, see G. S. Maddala, Limited Dependent Variables and Qualitative Variables in Econometrics
(Cambridge: Cambridge University Press, 1983) and T. Amemiya, “Qualitative Response Mod-
els: A Survey,” Journal of Economic Literature, Vol. 19, pp. 1483–1536. These surveys also cover
additional techniques, like the Tobit model, that are useful with bounded dependent variables
or other special situations.
After collecting data from 480 test takers, the students estimated the follow-
ing equation:
(15)
Note how similar these results look to a typical linear regression result. All the
estimated coefficients have the expected signs, and all but one are signifi-
cantly different from zero. Remember that the logit coefficients need to be
divided by 4 to get meaningful estimates of the impact of the independent
variables on the probability of passing the test. If we divide by 4, for exam-
ple, the impact of an hour’s studying turns out to be huge: according to our
estimates, the probability of passing the test would go up by 67.5 percent,
holding constant the other three independent variables. Note that
indicating that the equation correctly explained almost three quarters of the
sample based on nothing but the four variables in Equation 15.
And what about the two students? Did the equation help them? How much
did they end up deciding to study? They found that given their ages, their col-
lege experience, and their English-speaking backgrounds, the expected value
of for each of them was quite high, even if Hi was set equal to zero. So what
did they actually do? They studied for a half hour “just to be on the safe side”
and passed with flying colors, having devoted more time to passing the test
than anyone else in the history of the state.
Other Dummy Dependent Variable Techniques
Although the binomial logit is the most frequently used estimation technique
for equations with dummy dependent variables, it’s by no means the only one.
In this section, we’ll mention two alternatives, the binomial probit and the
multinomial logit, that are useful in particular circumstances. Our main goal is
to briefly describe these estimation techniques, not to cover them in any detail.8
3
D̂i
R2p is .74,
�̂H
N 5 480 R2p 5 .74 iterations 5 5
t 5 1.23 4.97 4.65 4.00
(0.009) (0.54) (0.34) (0.99)
L:Pr(Di 5 1) 5 2 1.18 1 0.011Ai 1 2.70Hi 1 1.62Ei 1 3.97Ci
431

DUMMY DEPENDENT VARIABLE TECHNIQUES
The Binomial Probit Model
The binomial probit model is an estimation technique for equations with
dummy dependent variables that avoids the unboundedness problem of the lin-
ear probability model by using a variant of the cumulative normal distribution.
(16)
where: Pi 5 the probability that the dummy variable Di 5 1
Zi 5
s 5 a standardized normal variable
As different as this probit looks from the logit that we examined in the pre-
vious section, it can be rewritten to look quite familiar:
(17)
where is the inverse of the normal cumulative distribution function.
Probit models typically are estimated by applying maximum likelihood tech-
niques to the model in the form of Equation 16, but the results often are pre-
sented in the format of Equation 17.
The fact that both the logit and the probit are cumulative distributive func-
tions means that the two have similar properties. For example, a graph of the
probit looks almost exactly like the logit in Figure 2. In addition, the probit
has the same requirement of a fairly large sample before hypothesis testing
becomes meaningful. Finally, continues to be of questionable value as a
measure of overall fit.
From a researcher’s point of view, the probit is theoretically appealing be-
cause many economic variables are normally distributed. With extremely large
samples, this advantage falls away, since maximum likelihood procedures can
be shown to be asymptotically normal under fairly general conditions.
For an example of a probit, let’s estimate one on the same women’s labor
force participation data we used in the previous logit and linear probability
examples (standard errors in parentheses):
(18)
Compare this result with Equation 13 from the previous section. Note that
except for a slight difference in the scale of the coefficients, the logit and pro-
bit models provide virtually identical results in this example.
N 5 30 R2p 5 .81 iterations 5 5
(0.62) (0.17)
Ẑi 5 �
21(Pi) 5 2 3.44 2 1.44Mi 1 0.40Si
R2
�21
Zi 5 �
21(Pi) 5 �0 1 �1X1i 1 �2X2i
�0 1 �1X1i 1 �2X2i
Pi 5
1
“2�
3
Zi
2`
e2s
2>2 ds
432

DUMMY DEPENDENT VARIABLE TECHNIQUES
The Multinomial Logit Model
In many cases, there are more than two qualitative choices available. In some
cities, for instance, a commuter has a choice of car, bus, or subway for the trip
to work. How could we build and estimate a model of choosing from more
than two different alternatives?
One answer is to hypothesize that choices are made sequentially and to
model a multichoice decision as a series of binary decisions. For example,
we might hypothesize that the commuter would first decide whether to
drive to work, and we could build a binary model of car versus public trans-
portation. For those commuters who choose public transportation, the next
step would be to choose whether to take the bus or the subway, and we
could build a second binary model of that choice. This method, called a
sequential binary logit, is cumbersome and at times unrealistic, but it does
allow a researcher to use a binary technique to model an inherently multi-
choice decision.
If a decision between multiple alternatives is truly made simultaneously, a
better approach is to build a multinomial logit model of the decision. A
multinomial logit model is an extension of the binomial logit technique
that allows several discrete alternatives to be considered at the same time.
If there are N different alternatives, we need dummy variables to
describe the choice, with each dummy equalling 1 only when that particular
alternative is chosen. For example, D1i would equal 1 if the ith person chose
alternative number 1 and would equal 0 otherwise. As before, the probability
that D1i is equal to 1, P1i, cannot be observed.
In a multinomial logit, one alternative is selected as the “base” alternative,
and then each other possible choice is compared to this base alternative with
a logit equation. A key distinction is that the dependent variable of these
equations is the log of the odds of the ith alternative being chosen compared
to the base alternative:
(19)
where: P1i 5 the probability of the ith person choosing the first alternative
Pbi 5 the probability of the ith person choosing the base alternative
If there are N alternatives, there should be different logit equations in
the multinomial logit model system, because the coefficients of the last equa-
tion can be calculated from the coefficients of the first equations. (If
you know that then you can calculate that A>B 5 3.)A>C 5 6 and B>C 5 2,
N 2 1
N 2 1
lna
P
1i
Pbi
b
N 2 1
433

DUMMY DEPENDENT VARIABLE TECHNIQUES
For example, if N � 3, as in the commuter-work-trip example cited previ-
ously, and the base alternative is taking the bus, then a multinomial logit
model would have a system of two equations:
(20)
(21)
where s � subway, c � car, and b � bus.
Summary
1. A linear probability model is a linear-in-the-coefficients equation
used to explain a dummy dependent variable (Di). The expected value
of Di is the probability that Di equals 1.
2. The estimation of a linear probability model with OLS encounters
two major problems:
a. is not an accurate measure of overall fit.
b. The expected value of Di is not limited by 0 and 1.
3. When measuring the overall fit of equations with dummy dependent
variables, an alternative to the average percentage of the
observations in the sample that a particular estimated equation
would have explained correctly.
4. The binomial logit is an estimation technique for equations with
dummy dependent variables that avoids the unboundedness problem
of the linear probability model by using a variant of the cumulative
logistic function:
5. The binomial logit is best estimated using the maximum likelihood
technique and a large sample. A slope coefficient from a logit mea-
sures the impact of a one-unit increase of the independent variable in
question (holding the other explanatory variables constant) on the
log of the odds of a given choice.
L:Pr (Di 5 1) 5 lna
D
i
f1 2 Dig
b 5 �0 1 �1X1i 1 �2X2i 1 �i
R2 is R2p ,
R2
4
lna
P
ci
Pbi
b 5 �0 1 �1X1i 1 �2X3i
lna
P
si
Pbi
b 5 �0 1 �1X1i 1 �2X2i
434

DUMMY DEPENDENT VARIABLE TECHNIQUES
6. The binomial probit model is an estimation technique for equations
with dummy dependent variables that uses the cumulative normal
distribution function. The binomial probit has properties quite simi-
lar to the binomial logit.
7. The multinomial logit model is an extension of the binomial logit that al-
lows more than two discrete alternatives to be considered simultaneously.
EXERCISES
(The answer to Exercise 2 is at the end of the chapter.)
1. Write the meaning of each of the following terms without referring to
the book (or your notes), and compare your definition with the ver-
sion in the text for each:
a. linear probability model
b.
c. binomial logit model
d. The interpretation of an estimated logit coefficient
e. binomial probit model
f. sequential binary model
g. multinomial logit model
2. R. Amatya9 estimated the following logit model of birth control for
1,145 continuously married women aged 35 to 44 in Nepal:
where: Di 5 1 if the ith woman has ever used a recognized form
of birth control, 0 otherwise
WNi 5 1 if the ith woman wants no more children,
0 otherwise
MEi 5 number of methods of birth control known to the
ith woman
t 5 5.64 10.36
(0.36) (0.14)
L:Pr(Di 5 1) 5 2 4.47 1 2.03WNi 1 1.45MEi
R2p
9. Ramesh Amatya, “Supply-Demand Analysis of Differences in Contraceptive Use in Seven
Asian Nations” (paper presented at the Annual Meetings of the Western Economic Association,
1988, Los Angeles).
435

DUMMY DEPENDENT VARIABLE TECHNIQUES
a. Explain the theoretical meaning of the coefficients for WN and ME.
How would your answer differ if this were a linear probability model?
b. Do the signs, sizes, and significance of the estimated slope coeffi-
cients meet your expectations? Why or why not?
c. What is the theoretical significance of the constant term in this
equation?
d. If you could make one change in the specification of this equation,
what would it be? Explain your reasoning.
3. Bond ratings are letter ratings (Aaa 5 best) assigned to firms that issue
debt. These ratings measure the quality of the firm from the point of
view of the likelihood of repayment of the bond. Suppose you’ve been
hired by an arbitrage house that wants to predict Moody’s Bond Ratings
before they’re published in order to buy bonds whose ratings are going
to improve. In particular, suppose your firm wants to distinguish be-
tween A-rated bonds (high quality) and B-rated bonds (medium qual-
ity) and has collected a data set of 200 bonds with which to estimate a
model. As you arrive on the job, your boss is about to buy bonds based
on the results of the following model (standard errors in parentheses):
where: Yi 5 1 if the rating of the ith bond 5 A, 0 otherwise
Pi 5 the profit rate of the firm that issued the ith bond
PVi 5 the standard deviation of Pi over the last five years
Di 5 the ratio of debt to total capitalization of the firm
that issued the ith bond
a. What econometric problems, if any, exist in this equation?
b. What suggestions would you have for a rerun of this equation with
a different specification?
c. Suppose that your boss rejects your suggestions, saying, “This is the
real world, and I’m sure that my model will forecast bond ratings
just as well as yours will.” How would you respond? (Hint: Saying
“Okay, boss, you win,” is sure to keep your job for you, but it won’t
get much credit on this question.)
4. Show that the logistic function, is indeed equiva-
lent to the binomial logit model, , where
Z 5 �0 1 �1X1 1 �2X2 1 �.
lnfD>(1 2 D)g 5 Z
D 5 1>(1 1 e2Z),
R2 5 .69 DW 5 0.50 N 5 200
(0.05) (0.02) (0.002)
Ŷi 5 0.70 1 0.05Pi 1 0.05PVi 2 0.020Di
436

DUMMY DEPENDENT VARIABLE TECHNIQUES
5. On graph paper, plot each of the following models. For what range of
Xi is
a.
b.
c.
d.
e.
f.
6. Because their college had just upgraded its residence halls, two seniors
decided to build a model of the decision to live on campus. They col-
lected data from 533 upper-class students (first-year students were
required to live on campus) and estimated the following equation:
L:Pr(Di � 1) � 3.26 � 0.03UNITi � 0.13ALCOi � 0.99YEARi � 0.39GREKi
(0.04) (0.08) (0.12) (0.21)
t � � 0.84 � 1.55 � 8.25 �1.38
N � 533 R2p � .668 iterations � 4
where: Di � 1 if the ith student lived on campus, 0 otherwise
UNITi � the number of academic units the ith student was
taking
ALCOi � the nights per week that the ith student consumed
alcohol
YEARi � 2 if the ith student was a sophomore, 3 if a junior,
and 4 if a senior
GREKi � 1 if the ith student was a member of a fraternity/
sorority, 0 otherwise
a. The two seniors expected UNIT to have a positive coefficient and
the other variables to have negative coefficients. Test these hypothe-
ses at the 10-percent level.
b. What problem do you see with the definition of the YEAR vari-
able? What constraint does this definition place on the estimated
coefficients?
c. Carefully state the meaning of the coefficient of ALCO and analyze
the size of the coefficient. (Hint: Be sure to discuss how the size of
the coefficient compares with your expectations.)
d. If you could add one variable to this equation, what would it be?
Explain.
7. What happens if we define a dummy dependent variable over a range
other than 0 to 1? For example, suppose that in the research cited in
lnfDi>(1 2 Di)g 5 2 1.0 1 0.3Xi
lnfDi>(1 2 Di)g 5 3.0 2 0.2Xi
lnfDi>(1 2 Di)g 5 0.3 1 0.1Xi
D̂i 5 21.0 1 0.3Xi
D̂i 5 3.0 2 0.2Xi
D̂i 5 0.3 1 0.1Xi
1 , D̂i? How about D̂i , 0?
437

DUMMY DEPENDENT VARIABLE TECHNIQUES
Exercise 2, Amatya had defined Di as being equal to 2 if the ith
woman had ever used birth control, 0 otherwise.
a. What would happen to the size and theoretical meaning of the esti-
mated logit coefficients? Would they stay the same? Would they
change? (If so, how?)
b. How would your answers to part a change if Amatya had estimated
a linear probability model instead of a binomial logit?
8. Return to our data on women’s labor force participation and consider
the possibility of adding Ai, the age of the ith woman, to the equation.
Be careful when you develop your expected sign and functional form
because the expected impact of age on labor force participation is diffi-
cult to pin down. For instance, some women drop out of the labor force
when they get married, but others continue working even while they’re
raising their children. Still others work until they get married, stay at
home to have children, and then return to the workforce once the chil-
dren reach school age. Malcolm Cohen et al., for example, found the
age of a woman to be relatively unimportant in determining labor force
participation, except for women who were 65 and older and were likely
to have retired.10 The net result for our model is that age appears to be a
theoretically irrelevant variable. A possible exception, however, is a
dummy variable equal to 1 if the ith woman is 65 or over, 0 otherwise.
a. Look over the data set in Table 1. What problems do you see with
adding an independent variable equal to 1 if the ith woman is 65
or older and 0 otherwise?
b. If you go ahead and add the dummy implied to Equation 13
and reestimate the model, you obtain Equation 22. Which
equation do you prefer, Equation 13 or Equation 22? Explain your
answer.
(22)
where: ADi 5 1 if the age of the ith woman is 0 otherwise
9. To get practice in actually estimating your own linear probability,
logit, and probit equations, test the possibility that age (Ai) is actually
.65,
N 5 30 R2p 5 .82 iterations 5 5
t 5 2 2.19 2.19 2 0.01
(1.18) (0.31) (0.30)
L:Pr(Di 5 1) 5 2 5.89 2 2.59Mi 1 0.69Si 2 0.03ADi
10. Malcolm Cohen, Samuel A. Rea, Jr., and Robert I. Lerman, A Micro Model of Labor Supply
(Washington, D.C.: U.S. Bureau of Labor Statistics, 1970), p. 212.
438

DUMMY DEPENDENT VARIABLE TECHNIQUES
a relevant variable in our women’s labor force participation model.
That is, take the data from Table 1 and estimate each of the following
equations. Then use our specification criteria to compare your equa-
tion with the parallel version in the text (without Ai). Explain why you
do or do not think that age is a relevant variable. (Hint: Be sure to cal-
culate .)
a. the linear probability model D 5 f(M,A,S)
b. the logit D 5 f(M,A,S)
c. the probit D 5 f(M,A,S)
10. An article published in a book edited by A. Kouskoulaf and B. Lytle11
presents coefficients from an estimated logit model of the choice be-
tween the car and public transportation for the trip to work in Boston.
All three public transportation modes in Boston (bus, subway, and
train, of which train is the most preferred) were lumped together as a
single alternative to the car in a binomial logit model. The dependent
variable was the log of the odds of taking public transportation for
the trip to work, so the first coefficient implies that as income rises,
the log of the odds of taking public transportation falls, and so on.
Independent Variable Coefficient
Family income (9 categories with 20.12
1 5 low and 9 5 high)
Number employed in the family 21.09
Out-of-pocket costs (cents) 23.16
Wait time (tenths of minutes) 0.18
Walk time (tenths of minutes) 20.03
In-vehicle travel time (tenths of minutes) 20.01
The last four variables are defined as the difference between the value
of the variable for taking public transportation and its value for taking
the car.
a. Do the signs of the estimated coefficients agree with your prior ex-
pectations? Which one(s) differ?
b. The transportation literature hypothesizes that people would rather
spend time traveling in a vehicle than waiting for or walking to that
vehicle. Do the sizes of the estimated coefficients of time support
this hypothesis?
R2p
11. “The Use of the Multinomial Logit in Transportation Analysis,” in A. Kouskoulaf and B. Lytle,
eds. Urban Housing and Transportation (Detroit: Wayne State University, 1975), pp. 87–90.
439

DUMMY DEPENDENT VARIABLE TECHNIQUES
c. Since trains run relatively infrequently, the researchers set wait time
for train riders fairly high. Most trains run on known schedules,
however, so the average commuter learns that schedule and attempts
to hold down wait time. Does this fact explain any of the unusual
results indicated in your answers to parts a and b?
11. Suppose that you want to build a multinomial logit model of how
students choose which college to attend. For the sake of simplicity,
let’s assume that there are only four colleges to choose from: your col-
lege (c), the state university (u), the local junior college (j), and the
nearby private liberal arts college (a). Further assume that everyone
agrees that the important variables in such a model are the family
income (Y) of each student, the average SAT scores of each college
(SAT), and the tuition (T) of each college.
a. How many equations should there be in such a multinomial logit
system?
b. If your college is the base, write out the definition of the dependent
variable for each equation.
12. In 2008, Goldman and Romley12 studied hospital demand by analyzing
how 8,721 Medicare-covered pneumonia patients chose from among
117 hospitals in the greater Los Angeles area. The authors concluded that
clinical quality (as measured by a low pneumonia mortality rate) played
a smaller role in hospital choice than did a variety of other factors.
Let’s focus on a subset of the Goldman–Romley sample: the 499
patients who chose either the UCLA Medical Center or the nearby
Cedars Sinai Medical Center. Typically, economists would expect
price to have a major influence on such a choice, but Medicare patients
pay roughly the same price no matter what hospital they choose.
Instead, factors like the distance the patient lives from the hospital
and the age and income of the patient become potentially important
factors:
L:Pr(Di � 1) � 4.41 � 0.38DISTANCEi � 0.072INCOMEi � 0.29OLDi (23)
(0.05) (0.036) (0.31)
t � � 8.12 � 2.00 � 0.94
N � 499 R2p � .66 iterations � 8
12. Dana Goldman and John Romley, “Hospitals as Hotels: The Role of Patient Amenities in
Hospital Demand,” NBER Working Paper 14619, December 2008. We appreciate the permission
of the authors to use a portion of their data set.
440

DUMMY DEPENDENT VARIABLE TECHNIQUES
where: Di � 1 if the ith patient chose Cedars Sinai, 0 if they
chose UCLA
DISTANCEi � the distance from the ith patient’s (according
to zip code) to Cedars Sinai minus the distance
from that point to the UCLA Medical Center
(in miles)
INCOMEi � the income of the ith patient (as measured
by the average income of their zip code in
thousands of dollars)
OLDi � 1 if the ith patient was older than 75, 0
otherwise
a. Create and test appropriate hypotheses about the coefficient of
DISTANCE at the 5-percent level.
b. Carefully state the meaning of the estimated coefficient of DISTANCE
in terms of the “per mile” impact on the probability of choosing
Cedars Sinai Medical Center.
c. Think about the definition of DISTANCE. Why do you think we de-
fined DISTANCE as the difference between the distances as op-
posed to entering the distance to Cedars and the distance to UCLA
as two different independent variables?
d. This data set is available on our Web site (www.pearsonhighered.
com/studenmund) and data disc as datafile � HOSPITAL13. Load
the data into your computer and use EViews, Stata, or your com-
puter’s regression program to estimate the linear probability model
and probit versions of this equation. What is the coefficient of DIS-
TANCE in your two estimates? Which model do you prefer? Ex-
plain. (Hint: It also makes sense to estimate a logit, just to make
sure that you’re using the same sample.)
e. (optional) Now create a slope dummy by adding OLD∗DISTANCE
to Equation 23 and estimating a new logit equation. Why do you
think we’re suggesting this particular slope dummy? Create and test
the appropriate hypotheses about the slope dummy at the
5-percent level. Which equation do you prefer, Equation 23 or the
new slope dummy logit? Explain.
441

Answers
Exercise 2
a. WN: The log of the odds that a woman has used a recognized
form of birth control is 2.03 higher if she doesn’t want any
more children than it is if she wants more children, holding ME
constant.
ME: A one-unit increase in the number of methods of birth con-
trol known to a woman increases the log of the odds that she has
used a form of birth control by 1.45, holding WN constant.
LPM: If the model were a linear probability model, then each in-
dividual slope coefficient would represent the impact of a one-
unit increase in the independent variable on the probability that
the ith woman had ever used a recognized form of birth control,
holding the other independent variable constant.
b. Yes, but we didn’t expect ME to be more significant than WN.
c. As we’ve said before, �0 has virtually no theoretical signifi-
cance. See Section 7.1.
d. We’d add one of a number of potentially relevant variables; for
instance, the educational level of the ith woman, whether the ith
woman lives in a rural area, and so on.
�̂�̂
DUMMY DEPENDENT VARIABLE TECHNIQUES
442

1 Structural and Reduced-Form Equations
2 The Bias of Ordinary Least Squares (OLS)
3 Two-Stage Least Squares (2SLS)
4 The Identification Problem
5 Summary and Exercises
6 Appendix: Errors in the Variables
Simultaneous Equations
The most important models in economics and business are simultaneous in
nature. Supply and demand, for example, is obviously simultaneous. To
study the demand for chicken without also looking at the supply of chicken
is to take a chance on missing important linkages and thus making signifi-
cant mistakes. Virtually all the major approaches to macroeconomics, from
Keynesian aggregate demand models to rational expectations schemes, are
inherently simultaneous. Even models that appear to be inherently single-
equation in nature often turn out to be much more simultaneous than you
might think. The price of housing, for instance, is dramatically affected by the
level of economic activity, the prevailing rate of interest in alternative assets,
and a number of other simultaneously determined variables.
All this wouldn’t mean much to econometricians if it weren’t for the fact
that the estimation of simultaneous equations systems with OLS causes a
number of difficulties that aren’t encountered with single equations. Most
important, Classical Assumption III, which states that all explanatory vari-
ables should be uncorrelated with the error term, is violated in simultaneous
models. Mainly because of this, OLS coefficient estimates are biased in si-
multaneous models. As a result, an alternative estimation procedure called
Two-Stage Least Squares usually is employed in such models instead of OLS.
You’re probably wondering why we’ve waited until now to discuss simulta-
neous equations if they’re so important in economics and if OLS encounters
bias when estimating them. The answer is that the simultaneous estimation
444

SIMULTANEOUS EQUATIONS
1. This also depends on how hungry you are, which is a function of how hard you’re working,
which depends on how many chickens you have to take care of. (Although this chicken/egg ex-
ample is simultaneous in an annual model, it would not be truly simultaneous in a quarterly or
monthly model because of the time lags involved.)
of an equation changes every time the specification of any equation in the
entire system is changed, so a researcher must be well equipped to deal with
specification problems. As a result, it does not make sense to learn how to es-
timate a simultaneous system until you are fairly adept at estimating a single
equation.
Structural and Reduced-Form Equations
Before we can study the problems encountered in the estimation of simulta-
neous equations, we need to introduce a few concepts.
The Nature of Simultaneous Equations Systems
Which came first, the chicken or the egg? This question is impossible to an-
swer satisfactorily because chickens and eggs are jointly determined; there is
a two-way causal relationship between the variables. The more eggs you have,
the more chickens you’ll get, but the more chickens you have, the more eggs
you’ll get.1 More realistically, the economic world is full of the kind of
feedback effects and dual causality that require the application of simultaneous
equations. Besides the supply and demand and simple macroeconomic
model examples mentioned previously, we could talk about the dual causal-
ity of population size and food supply, the joint determination of wages and
prices, or the interaction between foreign exchange rates and international
trade and capital flows. In a typical econometric equation:
(1)
a simultaneous system is one in which Y clearly has an effect on at least one
of the Xs in addition to the effect that the Xs have on Y.
Such topics are usually modeled by distinguishing between variables that
are simultaneously determined (the Ys, called endogenous variables) and
those that are not (the Xs, called exogenous variables):
(2)
(3)Y2t 5 �0 1 �1Y1t 1 �2X3t 1 �3X2t 1 �2t
Y1t 5 �0 1 �1Y2t 1 �2X1t 1 �3X2t 1 �1t
Yt 5 �0 1 �1X1t 1 �2X2t 1 �t
1
445

SIMULTANEOUS EQUATIONS
For example, Y1 and Y2 might be the quantity and price of chicken (respec-
tively), X1 the income of the consumers, X2 the price of beef (beef is a sub-
stitute for chicken in both consumption and production), and X3 the price
of chicken feed. With these definitions, Equation 2 would characterize the
behavior of consumers of chickens and Equation 3 the behavior of suppli-
ers of chickens. These behavioral equations are also called structural equa-
tions. Structural equations characterize the underlying economic theory
behind each endogenous variable by expressing it in terms of both endoge-
nous and exogenous variables. Researchers must view them as an entire sys-
tem in order to see all the feedback loops involved. For example, the Ys are
jointly determined, so a change in Y1 will cause a change in Y2, which will
in turn cause Y1 to change again. Contrast this feedback with a change in
X1, which will not eventually loop back and cause X1 to change again. The
and the in the equation are structural coefficients, and hypotheses
should be made about their signs just as we did with the regression coeffi-
cients of single equations.
Note that a variable is endogenous because it is jointly determined, not
just because it appears in both equations. That is, X2, which is the price of
beef but could be another factor beyond our control, is in both equations
but is still exogenous in nature because it is not simultaneously determined
within the chicken market. In a large general equilibrium model of the entire
economy, however, such a price variable would also likely be endogenous.
How do you decide whether a particular variable should be endogenous or
exogenous? Some variables are almost always exogenous (the weather, for
example), but most others can be considered either endogenous or exoge-
nous, depending on the number and characteristics of the other equations
in the system. Thus, the distinction between endogenous and exogenous
variables usually depends on how the researcher defines the scope of the
research project.
Sometimes, lagged endogenous variables appear in simultaneous sys-
tems, usually when the equations involved are distributed lag equations.
Be careful! Such lagged endogenous variables are not simultaneously de-
termined in the current time period. They thus have more in common
with exogenous variables than with nonlagged endogenous variables. To
avoid problems, we’ll define the term predetermined variable to include
all exogenous variables and lagged endogenous variables. “Predeter-
mined” implies that exogenous and lagged endogenous variables are de-
termined outside the system of specified equations or prior to the current
period. Endogenous variables that are not lagged are not predetermined,
because they are jointly determined by the system in the current time
period. Therefore, econometricians tend to speak in terms of endogenous
�s�s
446

SIMULTANEOUS EQUATIONS
and predetermined variables when discussing simultaneous equations
systems.
Let’s look at the specification of a simple supply and demand model, say
for the “cola” soft-drink industry:
(4)
(5)
where: QDt 5 the quantity of cola demanded in time period t
QSt 5 the quantity of cola supplied in time period t
Pt 5 the price of cola in time period t
X1t 5 dollars of advertising for cola in time period t
X2t 5 another “demand-side” exogenous variable (e.g., income
or the prices or advertising of other drinks)
X3t 5 a “supply-side” exogenous variable (e.g., the price of artifi-
cial flavors or other factors of production)
5 classical error terms (each equation has its own error term,
subscripted “D” and “S” for demand and supply)
In this case, price and quantity are simultaneously determined, but price,
one of the endogenous variables, is not on the left side of any of the equa-
tions. It’s incorrect to assume automatically that the endogenous variables
are those that appear on the left side of at least one equation; in this case, we
could have just as easily written Equation 5 with price on the left side and
quantity supplied on the right side, as we did in the chicken example in
Equations 2 and 3. Although the estimated coefficients would be different,
the underlying relations would not. Note also that there must be as many
equations as there are endogenous variables. In this case, the three endoge-
nous variables are QD, QS, and P.
What would be the expected signs for the coefficients of the price vari-
ables in Equations 4 and 5? We’d expect price to enter negatively in the
demand equation but to enter positively in the supply equation. The
higher the price, after all, the less quantity will be demanded, but the
more quantity will be supplied. These signs would result in the typical
supply and demand diagram (Figure 1) that we’re all used to. Look at Equa-
tions 4 and 5 again, however, and note that they would be identical but for
the different predetermined variables. What would happen if we accidentally
�t
QSt 5 QDt (equilibrium condition)
QSt 5 �0 1 �1Pt 1 �2X3t 1 �St
QDt 5 �0 1 �1Pt 1 �2X1t 1 �3X2t 1 �Dt
447

SIMULTANEOUS EQUATIONS
0 QQD = QS
S = Equation 14.5
�1 > 0
D = Equation 14.4
α1 < 0 P Pe put a supply-side predetermined variable in the demand equation or vice versa? We’d have a very difficult time identifying which equation was which, and the expected signs for the coefficients of the endogenous variable P would become ambiguous. As a result, we must take care when specifying the structural equations in a system. Simultaneous Systems Violate Classical Assumption III Recall that Classical Assumption III states that the error term and each ex- planatory variable must be uncorrelated with each other. If there is such a correlation, then the OLS regression estimation program is likely to attribute to the particular explanatory variable variations in the dependent variable that are actually being caused by variations in the error term. The result will be biased estimates. To see why simultaneous equations violate the assumption of indepen- dence between the error term and the explanatory variables, look again Figure 1 Supply and Demand Simultaneous Equations An example of simultaneous equations that jointly determine two endogenous variables is the supply and demand for a product. In this case, Equation 4, the downward-sloping de- mand function, and Equation 5, the upward-sloping supply function, intersect at the equi- librium price and quantity for this market. 448 SIMULTANEOUS EQUATIONS 2. This assumes that is negative, Y2 will decrease and there will be a negative correlation between and Y2, but this negative correlation will still violate Classical Assump- tion III. Also note that both Equations 2 and 3 could have Y1t on the left side; if two variables are jointly determined, it doesn’t matter which variable is considered dependent and which ex- planatory, because they are actually mutually dependent. We used this kind of simultaneous system in the cola model portrayed in Equations 4 and 5. �1 �1 is positive. If �1 at a simultaneous system, Equations 2 and 3 (repeated with directional errors): (2) (3) Let’s work through the system and see what happens when one of the error terms increases, holding everything else in the equations constant: 1. If increases in a particular time period, Y1 will also increase due to Equation 2. 2. If Y1 increases, Y2 will also rise 2 due to Equation 3. 3. But if Y2 increases in Equation 3, it also increases in Equation 2 where it is an explanatory variable. Thus, an increase in the error term of an equation causes an increase in an explanatory variable in the same equation: If increases, Y1 increases, and then Y2 increases, violating the assumption of independence between the error term and the explanatory variables. This is not an isolated result that depends on the particular equations in- volved. Indeed, as you’ll find in Exercise 3, this result works for other error terms, equations, and simultaneous systems. All that is required for the viola- tion of Classical Assumption III is that there be endogenous variables that are jointly determined in a system of simultaneous equations. Reduced-Form Equations An alternative way of expressing a simultaneous equations system is through the use of reduced-form equations, equations that express a par- ticular endogenous variable solely in terms of an error term and all the predetermined (exogenous plus lagged endogenous) variables in the si- multaneous system. �1 �1 Y2t 5 �0 1 �1Y1t 1 �2X3t 1 �3X2t 1 �2t c c Y1t 5 �0 1 �1Y2t 1 �2X1t 1 �3X2t 1 �1t c c c 449 SIMULTANEOUS EQUATIONS The reduced-form equations for the structural Equations 2 and 3 would thus be: (6) (7) where the vs are stochastic error terms and the are called reduced-form coefficients because they are the coefficients of the predetermined variables in the reduced-form equations. Note that each equation includes only one endogenous variable, the dependent variable, and that each equation has ex- actly the same set of predetermined variables. The reduced-form coefficients, such as are known as impact multipliers because they measure the impact on the endogenous variable of a one-unit increase in the value of the predetermined variable, after allowing for the feedback effects from the entire simultaneous system. There are at least three reasons for using reduced-form equations: 1. Since the reduced-form equations have no inherent simultaneity, they do not violate Classical Assumption III. Therefore, they can be estimated with OLS without encountering the problems discussed in this chapter. 2. The interpretation of the reduced-form coefficients as impact multipli- ers means that they have economic meaning and useful applications of their own. For example, if you wanted to compare a government spend- ing increase with a tax cut in terms of the per-dollar impact in the first year, estimates of the impact multipliers (reduced-form coefficients or ) would allow such a comparison. 3. Perhaps most importantly, reduced-form equations play a crucial role in the estimation technique most frequently used for simultaneous equations. This technique, Two-Stage Least Squares, will be explained in Section 3. To conclude, let’s return to the cola supply and demand model and specify the reduced-form equations for that model. (To test yourself, flip back to Equations 4 and 5 and see if you can get the right answer before going on.) Since the equilibrium condition forces QD to be equal to QS, we need only two reduced-form equations: (8) (9)Pt 5 �4 1 �5X1t 1 �6X2t 1 �7X3t 1 v2t Qt 5 �0 1 �1X1t 1 �2X2t 1 �3X3t 1 v1t �s �1 and �5, �s Y2t 5 �4 1 �5X1t 1 �6X2t 1 �7X3t 1 v2t Y1t 5 �0 1 �1X1t 1 �2X2t 1 �3X3t 1 v1t 450 SIMULTANEOUS EQUATIONS Even though P never appears on the left side of a structural equation, it’s an endogenous variable and should be treated as such. The Bias of Ordinary Least Squares (OLS) All the Classical Assumptions must be met for OLS estimates to be BLUE; when an assumption is violated, we must determine which of the proper- ties no longer holds. It turns out that applying OLS directly to the struc- tural equations of a simultaneous system produces biased estimates of the coefficients. Such bias is called simultaneous equations bias or simultane- ity bias. Understanding Simultaneity Bias Simultaneity bias refers to the fact that in a simultaneous system, the ex- pected values of the OLS-estimated structural coefficients are not equal to the true We are therefore faced with the problem that in a simultane- ous system: (10) Why does this simultaneity bias exist? Recall from Section 1 that in simulta- neous equations systems, the error terms (the tend to be correlated with the endogenous variables (the Ys) whenever the Ys appear as explanatory variables. Let’s follow through what this correlation means (assuming posi- tive coefficients for simplicity) in typical structural equations like 11 and 12: (11) (12) Since we cannot observe the error term and don’t know when is above average, it will appear as if every time Y1 is above average, so too is Y2. As a re- sult, the OLS estimation program will tend to attribute increases in Y1 caused by the error term to Y2, thus typically overestimating This overestimation is simultaneity bias. If the error term is abnormally negative, Y1t is less than it would have been otherwise, causing Y2t to be less than it would have been oth- erwise, and the computer program will attribute the decrease in Y1 to Y2, once again causing us to overestimate (that is, induce upward bias).�1 �1.�1 �1t(�1) Y2t 5 �0 1 �1Y1t 1 �2Zt 1 �2t Y1t 5 �0 1 �1Y2t 1 �2Xt 1 �1t �s) E(�̂) 2 � �s. (�̂s) 2 451 SIMULTANEOUS EQUATIONS 3. Monte Carlo experiments are computer-generated simulations that typically follow seven steps: 1. Assume a “true” model with specific coefficient values and an error term distribution. 2. Select values for the independent variables. 3. Select an estimating technique (usually OLS). 4. Create various samples of the dependent variable, using the assumed model, by randomly generating error terms from the assumed distribution; often, the number of samples created runs into the thousands. 5. Compute the estimates of the s from the various samples using the estimating technique. 6. Summarize and evaluate the results. 7. Consider sensitivity analyses using different values, distributions, or estimating techniques. � Recall that the causation between Y1 and Y2 runs in both directions be- cause the two variables are interdependent. As a result, when estimated by OLS, can no longer be interpreted as the impact of Y2 on Y1, holding X constant. Instead, now measures some mix of the effects of the two en- dogenous variables on each other! In addition, consider It’s supposed to be the effect of X on Y1 holding Y2 constant, but how can we expect Y2 to be held constant when a change in Y1 takes place? As a result, there is potential bias in all the estimated coefficients in a simultaneous system. What does this bias look like? It’s possible to derive an equation for the expected value of the regression coefficients in a simultaneous system that is estimated by OLS. This equation shows that as long as the error term and any of the explanatory variables in the equation are correlated, then the co- efficient estimates will be biased. In addition, it also shows that the bias will have the same sign as the correlation between the error term and the en- dogenous variable that appears as an explanatory variable in that error term’s equation. Since that correlation is usually positive in economic and business examples, the bias usually will be positive, although the direction of the bias in any given situation will depend on the specific details of the structural equations and the model’s underlying theory. This does not mean that every coefficient from a simultaneous system estimated with OLS will be a bad approximation of the true population coefficient. However, it’s vital to consider an alternative to OLS whenever si- multaneous equations systems are being estimated. Before we investigate the alternative estimation technique most frequently used (Two-Stage Least Squares), let’s look at an example of simultaneity bias. An Example of Simultaneity Bias To show how the application of OLS to simultaneous equations estima- tion causes bias, we used a Monte Carlo experiment3 to generate an exam- ple of such biased estimates. Since it’s impossible to know whether any bias exists unless you also know the true we arbitrarily picked a set of�s, �2. �̂1 �1, 452 SIMULTANEOUS EQUATIONS 4. Other assumptions included a normal distribution for the error term, , , , , and In addition, we assumed that the error terms of the two equations were not correlated. N 5 20.�2D 5 2, r 2 xz 5 0.4� 2 S 5 3 �0 5 0�0 5 0 coefficients to be considered “true.” We then stochastically generated data sets based on these “true” coefficients, and obtained repeated OLS esti- mates of these coefficients from the generated data sets. The expected value of these estimates turned out to be quite different from the true coef- ficient values, thus exemplifying the bias in OLS estimates of coefficients in simultaneous systems. We used a supply and demand model as the basis for our example: (13) (14) where: Qt 5 the quantity demanded and supplied in time period t Pt 5 the price in time period t Xt 5 a “demand-side” exogenous variable, such as income Zt 5 a “supply-side” exogenous variable, such as weather 5 classical error terms (different for each equation) The first step was to choose a set of true coefficient values that corre- sponded to our expectations for this model: In other words, we have a negative relationship between price and quantity demanded, a positive relationship between price and quantity supplied, and positive relationships between the exogenous variables and their respective dependent variables. The next step was to randomly generate a number of data sets based on the true values. This also meant specifying some other characteristics of the data4 before generating the different data sets (5,000 in this case). The final step was to apply OLS to the generated data sets and to calculate the estimated coefficients of the demand equation (13). (Similar results were obtained for the supply equation.) The arithmetic means of the results for the 5,000 regressions were: (15)Q̂Dt 5 �̂0 2 0.37Pt 1 1.84Xt �1 5 21 �2 5 11 �1 5 11 �2 5 11 �t Qt 5 �0 1 �1Pt 1 �2Zt 1 �St Qt 5 �0 1 �1Pt 1 �2Xt 1 �Dt 453 SIMULTANEOUS EQUATIONS �3 �2 �1 True �1 True �2 E(�2) = 1.84 E(�1) = – 0.37 Sampling Distribution of �2 0 1 2 3 Sampling Distribution of �1 Figure 2 Sampling Distributions Showing Simultaneity Bias of OLS Estimates In the experiment in Section 2, simultaneity bias is evident in the distribution of the estimates of which had a mean value of compared with a true value of , and in the estimates of which had a mean value of 1.84 compared with a true value of 1.00. �2,21.00 20.37�1, In other words, the expected value of should have been , but instead it was the expected value of should have been but instead it was 1.84: This is simultaneity bias! As the diagram of the sampling distributions of the in Figure 2 shows, the OLS estimates of were almost never very close to and the OLS estimates of were distributed over a wide range of values. Two-Stage Least Squares (2SLS) How can we get rid of (or at least reduce) simultaneity bias? There are a num- ber of estimation techniques that help mitigate simultaneity bias, but the most frequently used alternative to OLS is called Two-Stage Least Squares (2SLS). 3 �221.00, �1�̂s E(�̂2) 5 1.84 2 1.00 E(�̂1) 5 20.37 2 21.00 11.00,�̂220.37; 21.00�̂1 454 SIMULTANEOUS EQUATIONS What Is Two-Stage Least Squares? OLS encounters bias in the estimation of simultaneous equations mainly be- cause such equations violate Classical Assumption III, so one solution to the problem is to explore ways to avoid violating that assumption. We could do this if we could find a variable that is: 1. a good proxy for the endogenous variable, and 2. uncorrelated with the error term. If we then substitute this new variable for the endogenous variable where it appears as an explanatory variable, our new explanatory variable will be uncorrelated with the error term, and Classical Assumption III will be met. That is, consider Equation 16 in the following system: (16) (17) If we could find a variable that was highly correlated with Y2 but that was uncorrelated with then we could substitute this new variable for Y2 on the right side of Equation 16, and we’d conform to Classical Assumption III. This new variable is called an instrumental variable. An instrumental variable replaces an endogenous variable (when it is an explanatory vari- able); it is a good substitute for the endogenous variable and is independent of the error term. Since there is no joint causality between the instrumental variable and any endogenous variable, the use of the instrumental variable avoids the viola- tion of Classical Assumption III. The job of finding such a variable is another story, though. How do we go about finding variables with these qualifica- tions? For simultaneous equations systems, it turns out that finding instru- mental variables is straightforward. We use 2SLS. Two-Stage Least Squares (2SLS) is a method of systematically creating instrumental variables to replace the endogenous variables where they ap- pear as explanatory variables in simultaneous equations systems. 2SLS does this by running a regression on the reduced form of the right-side endoge- nous variables in need of replacement and then using the (or fitted val- ues) from those reduced-form regressions as the instrumental variables. Why do we do this? Every predetermined variable in the simultaneous sys- tem is a candidate to be an instrumental variable for every endogenous vari- able, but if we choose only one, we’re throwing away information. To avoid this, we use a linear combination of all the predetermined variables. We form this linear combination by running a regression for a given endogenous Ŷs �1, Y2t 5 �0 1 �1Y1t 1 �2X2t 1 �2t Y1t 5 �0 1 �1Y2t 1 �2X1t 1 �1t 455 SIMULTANEOUS EQUATIONS variable as a function of all the predetermined variables—the predicted value of the endogenous variable is the instrument we want. Thus, the 2SLS two-step procedure is: Since the predetermined (exogenous plus lagged endogenous) variables are uncorrelated with the reduced-form error term, the OLS estimates of the reduced-form coefficients (the can then be used to calculate estimates of the endogenous variables: (18) (19) These then are used as instruments in the structural equations.Ŷs Ŷ2t 5 �̂3 1 �̂4X1t 1 �̂5X2t Ŷ1t 5 �̂0 1 �̂1X1t 1 �̂2X2t �̂s) are unbiased. These �̂s STAGE ONE: Run OLS on the reduced-form equations for each of the endoge- nous variables that appear as explanatory variables in the structural equations in the system. STAGE TWO: Substitute the reduced form for the Ys that appear on the right side (only) of the structural equations, and then estimate these revised structural equations with OLS. Ŷs That is, stage two consists of estimating the following equations with OLS: (20) (21) Note that the dependent variables are still the original endogenous variables and that the substitutions are only for the endogenous variables where they appear on the right-hand side of the structural equations. This procedure produces consistent (for large samples), but biased (for small samples), esti- mates of the coefficients of the structural equations. Y2t 5 �0 1 �1Ŷ1t 1 �2X2t 1 u2t Y1t 5 �0 1 �1Ŷ2t 1 �2X1t 1 u1t 456 SIMULTANEOUS EQUATIONS 5. Most econometric software packages, including EViews and Stata, offer such a 2SLS option. For more on this issue, see Exercise 9 and footnote 9 of this chapter. 6. This bias is caused by remaining correlation between the produced by the first-stage reduced-form regressions and the . The effect of the correlation tends to decrease as the sam- ple size increases. Even for small samples, though, it’s worth noting that the expected bias due to 2SLS usually is smaller than the expected bias due to OLS. �s Ŷs If second-stage equations such as Equations 20 and 21 are estimated with OLS, the will be incorrect, so be sure to use your computer’s 2SLS es- timation procedure.5 This description of 2SLS can be generalized to m different simultaneous structural equations. Each reduced-form equation has as explanatory variables every predetermined variable in the entire system of equations. The OLS estimates of the reduced-form equations are used to compute the estimated values of all the endogenous variables that appear as ex- planatory variables in the m structural equations. After substituting these fitted values for the original values of the endogenous independent vari- ables, OLS is applied to each stochastic equation in the set of structural equations. The Properties of Two-Stage Least Squares 1. 2SLS estimates are still biased in small samples. For small samples, the ex- pected value of a produced by 2SLS is still not equal to the true ,6 but as the sample size gets larger, the expected value of the ap- proaches the true As the sample size gets bigger, the variances of both the OLS and the 2SLS estimates decrease. OLS estimates become very precise estimates of the wrong number, and 2SLS estimates be- come very precise estimates of the correct number. As a result, the larger the sample size, the better a technique 2SLS is. To illustrate, let’s look again at the example of Section 2. The 2SLS es- timate of was . This estimate is biased, but it’s much closer to the truth than is the OLS estimate of . We then re- turned to that example and expanded the data set from 5,000 different samples of size 20 each to 5,000 different samples of 50 observations each. As expected, the average for 2SLS moved from compared to the true value of By contrast, the OLS average esti- mate went from Such results are typical; large sample20.37 to 20.44. 21.00. 21.25 to 21.06�̂1 2 0.37(�1 5 21.00) 2 1.25�1 �. �̂ ��̂ SE(�̂)s 457 SIMULTANEOUS EQUATIONS sizes will allow 2SLS to produce unbiased estimates, but OLS still will produce biased estimates. 2. The bias in 2SLS for small samples typically is of the opposite sign of the bias in OLS. Recall that the bias in OLS typically was positive, indicating that a produced by OLS for a simultaneous system is likely to be greater than the true For 2SLS, the expected bias is negative, and thus a produced by 2SLS is likely to be less than the true For any given set of data, the 2SLS estimate can be larger than the OLS estimate, but it can be shown that the majority of 2SLS estimates are likely to be less than the corresponding OLS estimates. For large samples, there is little bias in 2SLS. Return to the example of Section 2. Compared to the true value of the small sample 2SLS average estimate was as mentioned earlier. This means that the 2SLS estimates showed negative bias. The OLS estimates, on the other hand, averaged ; since is more positive than the OLS estimates exhibited posi- tive bias. Thus, the observed bias due to OLS was opposite the observed bias due to 2SLS, as is generally the case. 3. If the fit of the reduced-form equation is quite poor, then 2SLS will not rid the equation of bias even in a large sample. Recall that the instrumental vari- able is supposed to be a good substitute for the endogenous variable. To the extent that the fit (as measured by of the reduced-form equation is poor, then the instrumental variable isn’t highly correlated with the original endogenous variable, and there is no reason to expect 2SLS to be effective. As the of the reduced-form equation increases, the usefulness of 2SLS will increase. 4. 2SLS estimates have increased variances and SE( )s. While 2SLS does an excellent job of reducing the amount of bias in the s, there’s a price to pay for this reduced bias. This price is that 2SLS estimates tend to have higher variances and SE( )s than do OLS estimates of the same equations. On balance, then, 2SLS will almost always be a better estimator of the co- efficients of a simultaneous system than OLS will be. The major exception to this general rule is when the fit of the reduced-form equation in question is quite poor for a small sample. �̂ �̂ �̂ R2 R2) 21.00,20.37 20.37 21.25,21.00 for �1, �.�̂ �. �̂ 458 SIMULTANEOUS EQUATIONS An Example of Two-Stage Least Squares Let’s work through an example of 2SLS, a naive linear Keynesian macroeco- nomic model of the U.S. economy. We’ll specify the following system: (22) (23) (24) (25) where: Yt 5 Gross Domestic Product (GDP) in year t COt 5 total personal consumption in year t It 5 total gross private domestic investment in year t Gt 5 government purchases of goods and services in year t NXt 5 net exports of goods and services (exports minus imports) in year t Tt 5 taxes (actually equal to taxes, depreciation, corporate prof- its, government transfers, and other adjustments necessary to convert GDP to disposable income) in year t rt 5 the interest rate in year t YDt 5 disposable income in year t All variables are in real terms (measured in billions of 2000 dollars) except the interest rate variable, which is measured in nominal percent. The data for this example are from 1976 through 2007 and are presented in Table 1. Equations 22 through 25 are the structural equations of the system, but only Equations 23 and 25 are stochastic (behavioral) and need to be esti- mated. The other two are identities, as can be determined by the lack of coef- ficients. Stop for a second and look at the system; which variables are endogenous? Which are predetermined? The endogenous variables are those that are jointly determined by the system, namely, Yt, COt, YDt, and It. To see why these four variables are simultaneously determined, note that if you change one of them and follow this change through the system, the change will get back to the original causal variable. For instance, if It goes up for some rea- son, that will cause Yt to go up, which will feed right back into It again. They’re simultaneously determined. It 5 �3 1 �4Yt 1 �5rt21 1 �2t YDt 5 Yt 2 Tt COt 5 �0 1 �1YDt 1 �2COt21 1 �1t Yt 5 COt 1 It 1 Gt 1 NXt 459 SIMULTANEOUS EQUATIONS Table 1 Data for the Small Macromodel YEAR Y CO I G YD r 1975 NA 2876.9 NA NA NA 8.83 1976 4540.9 3035.5 544.7 1031.9 3432.2 8.43 1977 4750.5 3164.1 627.0 1043.3 3552.9 8.02 1978 5015.0 3303.1 702.6 1074.0 3718.8 8.73 1979 5173.4 3383.4 725.0 1094.1 3811.2 9.63 1980 5161.7 3374.1 645.3 1115.4 3857.7 11.94 1981 5291.7 3422.2 704.9 1125.6 3960.0 14.17 1982 5189.3 3470.3 606.0 1145.4 4044.9 13.79 1983 5423.8 3668.6 662.5 1187.3 4177.7 12.04 1984 5813.6 3863.3 857.7 1227.0 4494.1 12.71 1985 6053.7 4064.0 849.7 1312.5 4645.2 11.37 1986 6263.6 4228.9 843.9 1392.5 4791.0 9.02 1987 6475.1 4369.8 870.0 1426.7 4874.5 9.38 1988 6742.7 4546.9 890.5 1445.1 5082.6 9.71 1989 6981.4 4675.0 926.2 1482.5 5224.8 9.26 1990 7112.5 4770.3 895.1 1530.0 5324.2 9.32 1991 7100.5 4778.4 822.2 1547.2 5351.7 8.77 1992 7336.6 4934.8 889.0 1555.3 5536.3 8.14 1993 7532.7 5099.8 968.3 1541.1 5594.2 7.22 1994 7835.5 5290.7 1099.6 1541.3 5746.4 7.96 1995 8031.7 5433.5 1134.0 1549.7 5905.7 7.59 1996 8328.9 5619.4 1234.3 1564.9 6080.9 7.37 1997 8703.5 5831.8 1387.7 1594.0 6295.8 7.26 1998 9066.9 6125.8 1524.1 1624.4 6663.9 6.53 1999 9470.3 6438.6 1642.6 1686.9 6861.3 7.04 2000 9817.0 6739.4 1735.5 1721.6 7194.0 7.62 2001 9890.7 6910.4 1598.4 1780.3 7333.3 7.08 2002 10048.8 7099.3 1557.1 1858.8 7562.2 6.49 2003 10301.0 7295.3 1613.1 1904.8 7729.9 5.67 2004 10675.8 7561.4 1770.2 1931.8 8008.9 5.63 2005 10989.5 7791.7 1873.5 1939.0 8121.4 5.24 2006 11294.8 8029.0 1912.5 1971.2 8407.0 5.59 2007 11523.9 8252.8 1809.7 2012.1 8644.0 5.56 Source: The Economic Report of the President, 2009. Note that T and NX can be calculated using Equations 22 and 24. Datafile = MACRO14 460 SIMULTANEOUS EQUATIONS 7. Although this sentence is technically correct, it overstates the case. In particular, there are a couple of circumstances in which an econometrician might want to consider to be part of the simultaneous system for theoretical reasons. For our naive Keynesian model with a lagged interest rate effect, however, the equation is not in the simultaneous system. 8. This investment equation is a simplified mix of the accelerator and the neoclassical theo- ries of the investment function. The former emphasizes that changes in the level of output are the key determinant of investment, and the latter emphasizes that user cost of capital (the opportunity cost that the firm incurs as a consequence of owning an asset) is the key. For an introduction to the determinants of consumption and investment, see any intermediate macroeconomics textbook. rt 2 1 What about interest rates? Is rt an endogenous variable? The surprising an- swer is that, strictly speaking, rt is not endogenous in this system because (not rt) appears in the investment equation. Thus, there is no simultaneous feedback through the interest rate in this simple model.7 Given this answer, which are the predetermined variables? The predeter- mined variables are . To sum, the simultaneous system has four structural equations, four endogenous variables, and five pre- determined variables. What is the economic content of the stochastic structural equations? The consumption function, Equation 23, is a dynamic model distributed lag con- sumption function. The investment function, Equation 25, includes simplified multiplier and cost of capital components. The multiplier term measures the stimulus to investment that is generated by an increase in GDP. In a Keynesian model, thus would be expected to be positive. On the other hand, the higher the cost of capital, the less investment we’d expect to be undertaken (holding multiplier effects constant), mainly because the expected rate of return on marginal capital investments is no longer sufficient to cover the higher cost of capital. Thus is expected to be negative. It takes time to plan and start up investment projects, though, so the interest rate is lagged one year.8 Stage One: Even though there are four endogenous variables, only two of them appear on the right-hand side of stochastic equations, so only two reduced- form equations need to be estimated to apply 2SLS. These reduced-form �5 �4 �4 Gt, NXt, Tt, COt21, and rt21 rt21 461 SIMULTANEOUS EQUATIONS 9. A few notes about 2SLS estimation and this model are in order. The 2SLS estimates in Equations 28 and 29 are correct, but if you were to estimate those equations with OLS (using as instruments generated as in Equation 26) you would obtain the same coefficient estimates but a different set of estimates of the standard errors (and t-scores). This difference comes about because running OLS on the second stage alone ignores the fact that the first stage was run at all. To get accurate estimated standard errors and t-scores, the estimation should be done with a 2SLS program. Ŷs and YDs equations are estimated automatically by all 2SLS computer estimation pro- grams, but it’s instructive to take a look at one anyway: (26) This reduced form has an excellent overall fit but is almost surely suffering from severe multicollinearity. Note that we don’t test any hypotheses on reduced forms, nor do we consider dropping a variable that is statistically and theoretically irrelevant. The whole purpose of stage one of 2SLS is not to generate meaningful reduced-form estimated equations but rather to gener- ate useful instruments to use as substitutes for endogenous variables in the second stage. To do that, we calculate the for all 32 observa- tions by plugging the actual values of all 5 predetermined variables into reduced-form equations like Equation 26. Stage Two: We then substitute these and for the endogenous vari- ables where they appear on the right sides of Equations 23 and 25. For ex- ample, the from Equation 26 would be substituted into Equation 23, re- sulting in: (27) If we estimate Equation 27 and the other second-stage equation given the data in Table 1, we obtain the following 2SLS9 results: (28) N 5 32 R2 5 .999 DW 5 0.83 2.73 4.84 (0.13) (0.14) COt 5 2 209.06 1 0.37YDt 1 0.66COt21 COt 5 �0 1 �1YDt 1 �2COt21 1 �1t YDt YDts,Ŷts, Ŷts and YDts (Ŷs) DW 5 2.21R2 5 .998N 5 32 t 5 3.49 2 2.30 3.68 7.60 4.12 (0.22) (0.16) (0.14) (0.09) (9.14) YDt 5 2288.55 1 0.78Gt 2 0.37NXt 1 0.52Tt 1 0.67COt21 1 37.63rt 2 1 462 SIMULTANEOUS EQUATIONS (29) If we had estimated these equations with OLS alone instead of with 2SLS, we would have obtained: (30) (31) Let’s compare the OLS and 2SLS results. First, there doesn’t seem to be much difference between them. If OLS is biased, how could this occur? When the fit of the stage-one reduced-form equations is excellent, as in Equation 26, then Y and are virtually identical, and the second stage of 2SLS is quite sim- ilar to the OLS estimate. Second, we’d expect positive bias in the OLS estima- tion and smaller negative bias in the 2SLS estimation, but the differences be- tween OLS and 2SLS appear to be in the expected direction only about half the time. This might have been caused by the extreme multicollinearity in the 2SLS estimations as well as by the superb fit of the reduced forms mentioned previously. Also, take a look at the Durbin–Watson statistics. DW is well below the dL of 1.31 (one-sided 5-percent significance, N 5 32, in all the equa- tions despite DW’s bias toward 2 in the consumption equation (because it’s a dynamic model). Consequently, positive serial correlation is likely to exist in the residuals of both equations. Applying GLS to the two 2SLS-estimated equations is tricky, however, especially because, as mentioned, serial correla- tion causes bias in an equation with a lagged dependent variable, as in the consumption function. One solution to this problem, running GLS and 2SLS, is discussed in Exercise 12. Finally, what about nonstationarity? Time-series models like these have the potential to be spurious in the face of nonstationarity. Are any of these regres- sions spurious? Well, as you can guess from looking at the data, quite a few K 5 2) Ŷ N 5 32 R2 5 .956 DW 5 0.47 15.87 2 0.83 (0.01) (11.19) Ît 5 2267.16 1 0.19Yt 2 9.26rt21 N 5 32 (annual 1976–2007) R2 5 .999 DW 5 0.77 4.70 5.66 (0.10) (0.10) COt 5 2 266.65 1 0.46YDt 1 0.56COt21 N 5 32 R2 5 .956 DW 5 0.47 15.82 2 0.85 (0.01) (11.20) Ît 5 2 261.48 1 0.19Ŷt 2 9.55rt21 463 SIMULTANEOUS EQUATIONS of the series in this model are, indeed, nonstationary. Luckily, the interest rate is stationary. In addition, it turns out that the consumption function is reason- ably cointegrated (see Exercise 15 of this chapter), so Equations 28 and 30 probably can stand as estimated. Unfortunately, the investment equation suf- fers from nonstationarity that almost surely results in an inflated t-score for GDP and a low t-score for (because is stationary when all the other variables in the equation are nonstationary). In fact, most macromodels en- counter similar problems with the significance (and sometimes the sign) of the interest rate variable in investment equations, at least partially because of the nonstationarity of the other variables in the equation. Given the tools covered so far in this text, however, there is little we can do to improve the situation. These caveats aside, this model has provided us with a complete example of the use of 2SLS to estimate a simultaneous system. However, the applica- tion of 2SLS requires that the equation being estimated be “identified,” so before we can conclude our study of simultaneous equations, we need to ad- dress the problem of identification. The Identification Problem Two-Stage Least Squares cannot be applied to an equation unless that equa- tion is identified. Before estimating any equation in a simultaneous system, you therefore must address the identification problem. Once an equation is found to be identified, then it can be estimated with 2SLS, but if an equation is not identified (underidentified), then 2SLS cannot be used no matter how large the sample. Such underidentified equations can be estimated with OLS, but OLS estimates of underidentified equations are difficult to interpret be- cause the estimates don’t necessarily match the coefficients we want to esti- mate. It’s important to point out that an equation being identified (and therefore capable of being estimated with 2SLS) does not ensure that the re- sulting 2SLS estimates will be good ones. The question being asked is not how good the 2SLS estimates will be but whether the 2SLS estimates can be obtained at all. What Is the Identification Problem? Identification is a precondition for the application of 2SLS to equations in simultaneous systems; a structural equation is identified only when enough of the system’s predetermined variables are omitted from the equation in question to allow that equation to be distinguished from all the others in the 4 rt21rt21 464 SIMULTANEOUS EQUATIONS system. Note that one equation in a simultaneous system might be identified and another might not. How could we have equations that we could not identify? To see how, let’s consider a supply and demand simultaneous system in which only price and quantity are specified: (32) (33) where: Although we’ve labeled one equation as the demand equation and the other as the supply equation, the computer will not be able to identify them from the data because the right-side and the left-side variables are exactly the same in both equations; without some predetermined variables included to distin- guish between these two equations, it would be impossible to distinguish supply from demand. What if we added a predetermined variable like weather (W) to the supply equation for an agricultural product? Then, Equation 33 would become: (34) In such a circumstance, every time W changed, the supply curve would shift, but the demand curve would not, so that eventually we would be able to col- lect a good picture of what the demand curve looked like. Figure 3 demonstrates this. Given four different values of W, we get four different supply curves, each of which intersects with the constant demand curve at a different equilibrium price and quantity (intersections 1–4). These equilibria are the data that we would be able to observe in the real world and are all that we could feed into the computer. As a result, we would be able to identify the demand curve because we left out at least one predetermined variable; when this predetermined variable changed, but the demand curve didn’t, the supply curve shifted so that quantity demanded moved along the demand curve and we gathered enough information to estimate the coeffi- cients of the demand curve. The supply curve, on the other hand, remains as much a mystery as ever because its shifts give us no clue whatsoever about its shape. In essence, the demand curve was identified by the predetermined variable that was included in the system but excluded from the demand equation. The supply curve is not identified because there is no such ex- cluded predetermined variable for it. QSt 5 �0 1 �1Pt 1 �2Wt 1 �St QDt 5 QSt QSt 5 �0 1 �1Pt 1 �St (supply) QDt 5 �0 1 �1Pt 1 �Dt (demand) 465 SIMULTANEOUS EQUATIONS Even if we added W to the demand curve as well, that would not identify the supply curve. In fact, if we had W in both equations, the two would be identical again, and although both would shift when W changed, those shifts would give us no information about either curve! As illustrated in Figure 4, the observed equilibrium prices and quantities would be almost random in- tersections describing neither the demand nor the supply curve. That is, the shifts in the supply curve are the same as before, but now the demand curve also shifts with W. In this case, it’s not possible to identify either the demand curve or the supply curve.10 The way to identify both curves is to have at least one predetermined vari- able in each equation that is not in the other, as in: (35) (36) Now when W changes, the supply curve shifts, and we can identify the demand curves from the data on equilibrium prices and quantities. When X changes, the demand curve shifts, and we can identify the supply curve from the data. To sum, identification is a precondition for the application of 2SLS to equations in simultaneous systems. A structural equation is identified only QSt 5 �0 1 �1Pt 1 �2Wt 1 �St QDt 5 �0 1 �1Pt 1 �2Xt 1 �Dt 10. An exception would be if you knew the relative magnitudes of the true coefficients of W in the two equations, but such knowledge is unlikely. 0 1 3 2 4 Q D S1P S3 S2 S4 Figure 3 A Shifting Supply Curve Allows the Identification of the Demand Curve If the supply curve shifts but the demand curve does not, then we move along the demand curve, allowing us to identify and estimate the demand curve (but not the supply curve). 466 SIMULTANEOUS EQUATIONS 0 1 3 2 4 Q S1 D1 P S3 S2 S4 D3 D2 D4 Figure 4 If Both the Supply Curve and the Demand Curve Shift, Neither Curve Is Identified If both the supply curve and the demand curve shift in response to the same variable, then we move from one equilibrium to another, and the resulting data points identify neither curve. To allow such an identification, at least one exogenous factor must cause one curve to shift while allowing the other to remain constant. when the predetermined variables are arranged within the system so as to allow us to use the observed equilibrium points to distinguish the shape of the equation in question. Most systems are quite a bit more complicated than the previous ones, however, so econometricians need a general method by which to determine whether equations are identified. The method typi- cally used is the order condition of identification. The Order Condition of Identification The order condition is a systematic method of determining whether a partic- ular equation in a simultaneous system has the potential to be identified. If an equation can meet the order condition, then it is identified in all but a very small number of cases. We thus say that the order condition is a neces- sary but not sufficient condition of identification.11 11. A sufficient condition for an equation to be identified is called the rank condition, but most researchers examine just the order condition before estimating an equation with 2SLS. These re- searchers let the computer estimation procedure tell them whether the rank condition has been met (by its ability to apply 2SLS to the equation). Those interested in the rank condition are en- couraged to consult an advanced econometrics text. 467 SIMULTANEOUS EQUATIONS What is the order condition? Recall that we have used the phrases endoge- nous and predetermined to refer to the two kinds of variables in a simultane- ous system. Endogenous variables are those that are jointly determined in the system in the current time period. Predetermined variables are exogenous variables plus any lagged endogenous variables that might be in the model. For each equation in the system, we need to determine: 1. The number of predetermined (exogenous plus lagged endogenous) variables in the entire simultaneous system. 2. The number of slope coefficients estimated in the equation in question. In equation form, a structural equation meets the order condition if: The number of predetermined variables The number of slope coefficients (in the simultaneous system) (in the equation) Two Examples of the Application of the Order Condition Let’s apply the order condition to some of the simultaneous equations sys- tems encountered in this chapter. For example, consider once again the cola supply and demand model of Section 1: (37) (38) (39) Equation 37 is identified by the order condition because the number of pre- determined variables in the system (three, X1, X2, and X3) is equal to the number of slope coefficients in the equation (three: This particular result (equality) implies that Equation 37 is exactly identified by the order condition. Equation 38 is also identified by the order condition because there still are three predetermined variables in the system, but there �1, �2, and �3). QSt 5 QDt QSt 5 �0 1 �1Pt 1 �2X3t 1 �St QDt 5 �0 1 �1Pt 1 �2X1t 1 �3X2t 1 �Dt $ THE ORDER CONDITION: A necessary condition for an equation to be identified is that the number of predetermined (exogenous plus lagged endoge- nous) variables in the system be greater than or equal to the number of slope coefficients in the equation of interest. 468 SIMULTANEOUS EQUATIONS are only two slope coefficients in the equation; this condition implies that Equation 38 is overidentified. 2SLS can be applied to equations that are identi- fied (which includes exactly identified and overidentified), but not to equa- tions that are underidentified. A more complicated example is the small macroeconomic model of Section 3: (22) (23) (24) (25) As we’ve noted, there are five predetermined variables (exogenous plus lagged endogenous) in this system Equation 23 has two slope coefficients so this equation is overidentified and meets the order condition of identification. As the reader can verify, Equa- tion 25 also turns out to be overidentified. Since the 2SLS computer program did indeed come up with estimates of the in the model, we knew this al- ready. Note that Equations 22 and 24 are identities and are not estimated, so we’re not concerned with their identification properties. Summary 1. Most economic and business models are inherently simultaneous be- cause of the dual causality, feedback loops, or joint determination of particular variables. These simultaneously determined variables are called endogenous, and nonsimultaneously determined variables are called exogenous. 2. A structural equation characterizes the theory underlying a particular variable and is the kind of equation we have used to date in this text. A reduced-form equation expresses a particular endogenous variable solely in terms of an error term and all the predetermined (exogenous and lagged endogenous) variables in the simultaneous system. 3. Simultaneous equations models violate the Classical Assumption of in- dependence between the error term and the explanatory variables be- cause of the feedback effects of the endogenous variables. For example, an unusually high observation of an equation’s error term works 5 �s (5 . 2)(�1 and �2), (Gt, NXt, Tt, COt21, and rt21). It 5 �3 1 �4Yt 1 �5rt21 1 �2t YDt 5 Yt 2 Tt COt 5 �0 1 �1YDt 1 �2COt21 1 �1t Yt 5 COt 1 It 1 Gt 1 NXt 469 SIMULTANEOUS EQUATIONS through the simultaneous system and eventually causes a high value for the endogenous variables that appear as explanatory variables in the equation in question, thus violating the assumption of no correla- tion (Classical Assumption III). 4. If OLS is applied to the coefficients of a simultaneous system, the re- sulting estimates are biased and inconsistent. This occurs mainly be- cause of the violation of Classical Assumption III; the OLS regression package attributes to explanatory variables changes in the dependent variable actually caused by the error term (with which the explanatory variables are correlated). 5. Two-Stage Least Squares is a method of decreasing the amount of bias in the estimation of simultaneous equations systems. It works by systemat- ically using the reduced-form equations of the system to create substi- tutes for the endogenous variables that are independent of the error terms (called instrumental variables). It then runs OLS on the structural equations of the system with the instrumental variables replacing the endogenous variables where they appear as explanatory variables. 6. Two-Stage Least Squares estimates are biased (with a sign opposite that of the OLS bias) but consistent (becoming more unbiased with closer to zero variance as the sample size gets larger). If the fit of the reduced-form equations is poor, then 2SLS will not work very well. The larger the sample size, the better it is to use 2SLS. 7. 2SLS cannot be applied to an equation that’s not identified. A neces- sary (but not sufficient) requirement for identification is the order condition, which requires that the number of predetermined vari- ables in the system be greater than or equal to the number of slope coefficients in the equation of interest. Sufficiency is usually deter- mined by the ability of 2SLS to estimate the coefficients. EXERCISES (The answer to Exercise 2 is at the end of the chapter.) 1. Write the meaning of each of the following terms without referring to the book (or your notes), and compare your definition with the ver- sion in the text for each: a. endogenous variable b. predetermined variable 470 SIMULTANEOUS EQUATIONS c. structural equation d. reduced-form equation e. simultaneity bias f. Two-Stage Least Squares g. identification h. order condition for identification 2. Damodar Gujarati12 estimated the following two money supply equations on U.S. annual data. The first was estimated with OLS, and the second was estimated with 2SLS (with Investment and Govern- ment Expenditure as predetermined variables in the reduced form equation). OLS: 2SLS: where: M2t � the M2 money stock in year t, in billions of dollars GDPt � Gross Domestic Product in year t, in billions of dollars a. What, exactly, does the caret (hat) over in the 2SLS equation mean? b. Which equation makes more sense on theoretical grounds? Ex- plain. c. Which equation is more likely to have biased coefficients? Explain. d. If you had to choose one equation, which would you prefer? Why? (Hint: Assume that the residuals are cointegrated.) e. If your friend claims that “it doesn’t matter which equation you use because they’re virtually identical,” how would you respond? 3. Section 1 works through Equations 2 and 3 to show the violation of Classical Assumption III by an unexpected increase in �1. GDP R2 5 .987t 5 41.24 (0.013) M2t 5 146.8 1 0.551GDPt R2 5 .986 t 5 40.97 (0.013) M2t 5 115.0 1 0.561GDPt 12. Damodar Gujarati, Essentials of Econometrics (Boston: Irwin McGraw-Hill, 1999), p. 492, with special thanks to Bill Wood. 471 SIMULTANEOUS EQUATIONS Show the violation of Classical Assumption III by working through the following examples: a. a decrease in in Equation 3 b. an increase in in Equation 4 c. an increase in in Equation 23 4. The word recursive is used to describe an equation that has an impact on a simultaneous system without any feedback from the system to the equation. Which of the equations in the following systems are si- multaneous, and which are recursive? Be sure to specify which vari- ables are endogenous and which are predetermined: a. b. c. 5. Section 2 makes the statement that the correlation between the and the Ys (where they appear as explanatory variables) usually is positive in economics. To see if this is true, investigate the sign of the error term/explanatory variable correlation in the following cases: a. the three examples in Exercise 3 b. the more general case of all the equations in a typical supply and demand model (for instance, the model for cola in Section 1) c. the more general case of all the equations in a simple macroeco- nomic model (for instance, the small macroeconomic model in Section 3) 6. Determine the identification properties of the following equations. In particular, be sure to note the number of predetermined variables in the system, the number of slope coefficients in the equation, and whether the equation is underidentified, overidentified, or exactly identified. a. Equations 2–3 b. Equations 13–14 c. part a of Exercise 4 (assume all equations are stochastic) d. part b of Exercise 4 (assume all equations are stochastic) �s Y2t 5 f(Y3t, X5t) Yt 5 f(Y2t, X1t, X2t) Ht 5 g(Zt, Bt, CSt, Dt) Xt 5 g(Zt, Pt 2 1) Zt 5 g(Xt, Yt, Ht) Y3t 5 f(X2t, X1t21, X4t21) Y2t 5 f(Y3t, Y1t, X4t) Y1t 5 f(Y2t, X1t, X2t21) �1 �D �2 472 SIMULTANEOUS EQUATIONS 7. Determine the identification properties of the following equations. In particular, be sure to note the number of predetermined variables in the system, the number of slope coefficients in the equation, and whether the equation is underidentified, overidentified, or exactly identified. (Assume that all equations are stochastic unless specified otherwise.) a. At 5 f(Bt, Ct, Dt) Bt 5 f(At, Ct) b. Y1t 5 f(Y2t, X1t, X2t, X3t) Y2t 5 f(X2t) X2t 5 f(Y1t, X4t, X3t) c. Ct 5 f(Yt) It 5 f(Yt, Rt, Et, Dt) Rt 5 Yt 5 Ct � It � Gt (nonstochastic) 8. Return to the supply and demand example for cola in Section 1 and explain exactly how 2SLS would estimate the of Equations 4 and 5. Write out the equations to be estimated in both stages, and indicate precisely what, if any, substitutions would be made in the second stage. 9. As an exercise to gain familiarity with the 2SLS program on your com- puter, take the data provided for the simple Keynesian model in Sec- tion 3, and: a. Estimate the investment function with OLS. b. Estimate the reduced form for Y with OLS. c. Substitute the from your reduced form into the investment func- tion and run the second stage yourself with OLS. d. Estimate the investment function with your computer’s 2SLS pro- gram (if there is one) and compare the results with those obtained in part c. 10. Suppose that one of your friends recently estimated a simultaneous equation research project and found the OLS results to be virtually iden- tical to the 2SLS results. How would you respond if he or she said “What a waste of time! I shouldn’t have bothered with 2SLS in the first place! Besides, this proves that there wasn’t any bias in my model anyway.” a. What is the value of 2SLS in such a case? b. Does the similarity between the 2SLS and OLS estimates indicate a lack of bias? Ŷ �s and �s f(Mt, Rt21, Yt 2 Yt21) 473 SIMULTANEOUS EQUATIONS 11. Think over the problem of building a model for the supply of and de- mand for labor (measured in hours worked) as a function of the wage and other variables. a. Completely specify labor supply and labor demand equations and hypothesize the expected signs of the coefficients of your variables. b. Is this system simultaneous? That is, is there likely to be feedback between the wage and hours demanded and supplied? Why or why not? c. Is your system likely to encounter biased estimates? Why? d. What sort of estimation procedure would you use to obtain your coefficient estimates? (Hint: Be sure to determine the identification properties of your equations.) 12. Let’s analyze the problem of serial correlation in simultaneous mod- els. For instance, recall that in our small macroeconomic model, the 2SLS version of the consumption function, Equation 28, was: (28) where CO is consumption and YD is disposable income. a. Test Equation 28 to confirm that we do indeed have a serial correla- tion problem. (Hint: This should seem familiar.) b. Equation 28 will encounter both simultaneity bias and bias due to serial correlation with a lagged endogenous variable. If you could solve only one of these two problems, which would you choose? Why? (Hint: Compare Equation 28 with the OLS version of the consumption function, Equation 30.) c. Suppose you wanted to solve both problems? Can you think of a way to adjust for both serial correlation and simultaneity bias at the same time? Would it make more sense to run GLS first and then 2SLS, or would you rather run 2SLS first and then GLS? Could they be run simultaneously? 13. Suppose that a fad for oats (resulting from the announcement of the health benefits of oat bran) has made you toy with the idea of becom- ing a broker in the oat market. Before spending your money, you de- cide to build a simple model of supply and demand (identical to those in Sections 1 and 2) of the market for oats: QDt 5 QSt QSt 5 �0 1 �1Pt 1 �2Wt 1 �St QDt 5 �0 1 �1Pt 1 �2YDt 1 �Dt N 5 32 R2 5 .999 DW 5 0.83 2.73 4.84 (0.13) (0.14) COt 5 2 209.06 1 0.37YDt 1 0.66COt21 474 SIMULTANEOUS EQUATIONS 13. These data are from the excellent course materials that Professors Bruce Gensemer and James Keeler prepared to supplement the use of this text at Kenyon College. where: QDt 5 the quantity of oats demanded in time period t QSt 5 the quantity of oats supplied in time period t Pt 5 the price of oats in time period t Wt 5 average oat-farmer wages in time period t YDt 5 disposable income in time period t a. You notice that no left-hand-side variable appears on the right side of either of your stochastic simultaneous equations. Does this mean that OLS estimation will encounter no simultaneity bias? Why or why not? b. You expect that when Pt goes up, QDt will fall. Does this mean that if you encounter simultaneity bias in the demand equation, it will be negative instead of the positive bias we typically associ- ate with OLS estimation of simultaneous equations? Explain your answer. c. Carefully outline how you would apply 2SLS to this system. How many equations (including reduced forms) would you have to estimate? Specify precisely which variables would be in each equation. d. Given the following hypothetical data,13 estimate OLS and 2SLS versions of your oat supply and demand equations. e. Compare your OLS and 2SLS estimates. How do they compare with your prior expectations? Which equation do you prefer? Why? Year Q P W YD 1 50 10 100 15 2 54 12 102 12 3 65 9 105 11 4 84 15 107 17 5 75 14 110 19 6 85 15 111 30 7 90 16 111 28 8 60 14 113 25 9 40 17 117 23 10 70 19 120 35 Datafile � OATS14 475 SIMULTANEOUS EQUATIONS 14. James F. Ragan, Jr., “The Voluntary Leaver Provisions of Unemployment Insurance and Their Effect on Quit and Unemployment Rates,” Southern Economic Journal, Vol. 15, No. 1, pp. 135–146. 14. Simultaneous equations make sense in cross-sectional as well as time- series applications. For example, James Ragan14 examined the effects of unemployment insurance (hereafter UI) eligibility standards on unemployment rates and the rate at which workers quit their jobs. Ragan used a pooled data set that contained observations from a number of different states from four different years (requirements for UI eligibility differ by state). His results are as follows (t-scores in parentheses): where: QUi 5 the quit rate (quits per 100 employees) in the ith state URi 5 the unemployment rate in the ith state UNi 5 union membership as a percentage of nonagricul- tural employment in the ith state REi 5 average hourly earnings in the ith state relative to the average hourly earnings for the United States ILi 5 dummy variable equal to 1 if workers in the ith state are eligible for UI if they are forced to quit a job because of illness, 0 otherwise QMi 5 dummy variable equal to 1 if the ith state main- tains full UI benefits for the quitter (rather than lowering benefits), 0 otherwise MXi 5 maximum weekly UI benefits relative to average hourly earnings in the ith state a. Hypothesize the expected signs for the coefficients of each of the explanatory variables in the system. Use economic theory to justify (2.03) (2.05) 1 0.56ILi 1 0.63QMi 1 c (1.01) (3.29) (1.71) URi 5 2 0.54 1 0.44QUi 1 0.13UNi 1 0.049MXi (0.01) (2 0.52) 1 0.003ILi 2 0.25QMi 1 c (0.10) (2 0.63) (2 1.98) (2 0.73) QUi 5 7.00 1 0.089URi 2 0.063UNi 2 2.83REi 2 0.032MXi 476 SIMULTANEOUS EQUATIONS your answers. Which estimated coefficients are different from your expectations? b. Ragan felt that these two equations would encounter simultaneity bias if they were estimated with OLS. Do you agree? Explain your answer. (Hint: Start by deciding which variables are endogenous and why.) c. The actual equations included a number of variables not docu- mented earlier, but the only predetermined variable in the system that was included in the QU equation but not the UR equation was RE. What does this information tell you about the identification properties of the QU equation? The UR equation? d. What are the implications of the lack of significance of the endoge- nous variables where they appear on the right-hand side of the equations? e. What, if any, policy recommendations do these results suggest? 15. Return to the consumption function of the small macromodel of Sec- tion 3 and consider again the issue of cointegration as a possible solu- tion to the problem of nonstationarity. a. Which of the variables in the equation are nonstationary? (Hint: See Exercises 10 and 11 in Chapter 12.) b. Test the possibility that Equation 30 is cointegrated. That is, test the hypothesis that the residuals of Equation 30 are stationary. (Hint: Use the Dickey–Fuller test.) c. Equation 30 is a dynamic model distributed lag equation. Do you think that this makes it more or less likely that the equation is cointegrated? d. Equation 30 is the OLS estimate of the consumption function. Would your approach be any different if you were going to test the 2SLS estimate for cointegration? How? Why? Appendix: Errors in the Variables Until now, we have implicitly assumed that our data were measured accu- rately. That is, although the stochastic error term was defined as including measurement error, we never explicitly discussed what the existence of such measurement error did to the coefficient estimates. Unfortunately, in the real world, errors of measurement are common. Mismeasurement might result from the data being based on a sample, as are almost all na- tional aggregate statistics, or simply because the data were reported incor- rectly. Whatever the cause, these errors in the variables are mistakes in the 6 477 SIMULTANEOUS EQUATIONS measurement of the dependent and/or one or more of the independent variables that are large enough to have potential impacts on the estimation of the coefficients. Such errors in the variables might be better called “measurement errors in the data.” We will tackle this subject by first exam- ining errors in the dependent variable and then moving on to look at the more serious problem of errors in an independent variable. We assume a single equation model. The reason we have included this topic here is that errors in explanatory variables give rise to biased OLS estimates very simi- lar to simultaneity bias. Measurement Errors in the Data for the Dependent Variable Suppose that the true regression model is (40) and further suppose that the dependent variable, Yi, is measured incorrectly, so that is observed instead of Yi, where (41) and where vi is an error of measurement that has all the properties of a classi- cal error term. What does this mismeasurement do to the estimation of Equa- tion 40? To see what happens when let’s add vi to both sides of Equation 40, obtaining (42) which is the same as (43) where That is, we estimate Equation 43 when in reality we want to estimate Equation 40. Take another look at Equation 43. When vi changes, both the dependent variable and the error term move together. This is no cause for alarm, however, since the dependent variable is always correlated with the error term. Although the extra movement will increase the variability of Y and therefore be likely to decrease the overall statistical fit of the equation, an error of measurement in the dependent variable does not cause any bias in the estimates of the �s. �i* �i* 5 (�i 1 vi). Yi* 5 �0 1 �1Xi 1 �i* Yi 1 vi 5 �0 1 �1Xi 1 �i 1 vi Yi* 5 Yi 1 vi, Yi* 5 Yi 1 vi Yi* Yi 5 �0 1 �1Xi 1 �i 478 SIMULTANEOUS EQUATIONS Measurement Errors in the Data for an Independent Variable This is not the case when the mismeasurement is in the data for one or more of the independent variables. Unfortunately, such errors in the independent variables cause bias that is quite similar in nature (and in remedy) to simul- taneity bias. To see this, once again suppose that the true regression model is Equation 40: (40) But now suppose that the independent variable, Xi, is measured incorrectly, so that is observed instead of Xi, where (44) but where ui is an error of measurement like vi in Equation 41. To see what this mismeasurement does to the estimation of Equation 40, let’s add the term to Equation 40, obtaining (45) which can be rewritten as (46) or (47) where In this case, we estimate Equation 47 when we should be trying to estimate Equation 40. Notice what happens to Equation 47 when ui changes, however. When ui changes, the stochastic error term and the independent variable move in opposite directions; they are correlated! Such a correlation is a direct violation of Classical Assumption III in a way that is remarkably similar to the violation (described in Section 1) of the same assumption in simultaneous equations. Not surprisingly, this vio- lation causes the same problem, bias, for errors-in-the-variables models that it causes for simultaneous equations. That is, because of the measurement error in the independent variable, the OLS estimates of the coefficients of Equation 47 are biased. A frequently used technique to rid an equation of the bias caused by mea- surement errors in the data for one or more of the independent variables is to Xi*�i** �i** 5 (�i 2 �1ui). Yi 5 �0 1 �1Xi* 1 �i** Yi 5 �0 1 �1(Xi 1 ui) 1 (�i 2 �1ui) Yi 5 �0 1 �1Xi 1 �i 1 (�1ui 2 �1ui) 0 5 (�1ui 2 �1ui) Xi* 5 Xi 1 ui Xi* Yi 5 �0 1 �1Xi 1 �i 479 SIMULTANEOUS EQUATIONS 15. If errors exist in the data for the dependent variable and one or more of the independent variables, then both decreased overall statistical fit and bias in the estimated coefficients will re- sult. Indeed, a famous econometrician, Zvi Griliches, warned that errors in the data coming from their measurement, usually computed from samples or estimates, imply that the fancier estimating techniques should be avoided because they are more sensitive to data errors than is OLS. See Zvi Griliches, “Data and Econometricians—the Uneasy Alliance,” American Economic Review, Vol. 75, No. 2, p. 199. See also, B. D. McCullough and H. D. Vinod, “The Numerical Re- liability of Econometric Software,” Journal of Economic Literature, Vol. 37, pp. 633–665. use an instrumental variable, the same technique used to alleviate simultaneity bias. A substitute for X is chosen that is highly correlated with X but is uncor- related with Recall that 2SLS is an instrumental variables technique. Such techniques are applied only rarely to errors in the variables problems, how- ever, because although we may suspect that there are errors in the variables, it’s unusual to know positively that they exist, and it’s difficult to find an in- strumental variable that satisfies both conditions. As a result, is about as good a proxy for X as we usually can find, and no action is taken. If the mis- measurement in X were known to be large, however, some remedy would be required. To sum, an error of measurement in one or more of the independent vari- ables will cause the error term of Equation 47 to be correlated with the inde- pendent variable, causing bias analogous to simultaneity bias.15 X* �. 480 482 Answers Exercise 2 a. The caret over GDP is an indication that two-stage least squares was used. A reduced-form equation was run with GDP as a func- tion of investment and government expenditure. The estimated GDPs from the reduced form were then substituted for GDP where it appears on the right-hand side of the money supply equation in order to act as a proxy (an instrumental variable) for GDP. b. The 2SLS equation makes significantly more sense from a theo- retical point of view. Most economists agree that GDP has an im- pact on the money supply and that the money supply also has an impact on GDP, leading to a simultaneous model being the model of choice. c. The OLS equation is more likely to have biased coefficients, but the 2SLS model also will face potential bias in small samples. The bias in the OLS model is likely to be positive, while the bias in the 2SLS model is likely be negative (and smaller in absolute value). d. We prefer the 2SLS model by a wide margin, because it is theoret- ically more compelling, and because it has less expected bias. e. It’s true that in this case the 2SLS and OLS estimates are virtually identical, but that doesn’t change the fact that 2SLS is preferable from both a theoretical and econometric point of view. SIMULTANEOUS EQUATIONS 481 Accurate forecasting is vital to successful planning, so it’s the primary goal of many business and governmental uses of econometrics. For example, manu- facturing firms need sales forecasts, banks need interest rate forecasts, and governments need unemployment and inflation rate forecasts. To many business and government leaders, the words econometrics and forecasting mean the same thing. Such a simplification gives econometrics a bad name because many econometricians overestimate their ability to produce ac- curate forecasts, resulting in unrealistic claims and unhappy clients. Some of their clients would probably applaud the nineteenth century New York law (luckily unenforced but apparently also unrepealed) that provides that persons “pretending to forecast the future” shall be liable to a $250 fine and/or six months in prison.1 Although many econometricians might wish that such con- sultants would call themselves “futurists” or “soothsayers,” it’s impossible to ignore the importance of econometrics in forecasting in today’s world. The ways in which the prediction of future events is accomplished are quite varied. At one extreme, some forecasters use models with hundreds of equations.2 At the other extreme, quite accurate forecasts can be created with nothing more than a good imagination and a healthy dose of self-confidence. 1. Section 899 of the N.Y. State Criminal Code: the law does not apply to “ecclesiastical bodies acting in good faith and without personal fees.” 2. For an interesting comparison of such models, see Ray C. Fair and Robert J. Shiller, “Comparing Information in Forecasts from Econometric Models,” American Economic Review, Vol. 80, No. 3, pp. 375–389. 1 What Is Forecasting? 2 More Complex Forecasting Problems 3 ARIMA Models 4 Summary and Exercises Forecasting From Chapter 15 of Using Econometrics: A Practical Guide, 6/e. A. H. Studenmund. Copyright © 2011 by Pearson Education. Published by Addison-Wesley. All rights reserved. 483 FORECASTING 3. See, for example, G. Elliott, C. W. J. Granger, and A. G. Timmermann, Handbook of Economic Forecasting (Oxford, UK: North-Holland Elsevier, 2006), and N. Carnot, V. Koen, and B. Tissot, Economic Forecasting (Basingstoke, UK: Palgrave MacMillan, 2005). Unfortunately, it’s unrealistic to think we can cover even a small portion of the topic of forecasting in one short chapter. Indeed, there are a number of excellent books and journals on this subject alone.3 Instead, this chapter is meant to be a brief introduction to the use of econometrics in forecasting. We will begin by using simple linear equations and then move on to investi- gate a few more complex forecasting situations. The chapter concludes with an introduction to a technique, called ARIMA, that calculates forecasts en- tirely from past movements of the dependent variable without the use of any independent variables at all. ARIMA is almost universally used as a bench- mark forecast, so it’s important to understand even though it’s not based on economic theory. What Is Forecasting? In general, forecasting is the act of predicting the future; in econometrics, forecasting is the estimation of the expected value of a dependent variable for observations that are not part of the same data set. In most forecasts, the values being predicted are for time periods in the future, but cross-sectional predictions of values for countries or people not in the sample are also com- mon. To simplify terminology, the words prediction and forecast will be used interchangeably in this chapter. (Some authors limit the use of the word fore- cast to out-of-sample prediction for a time series.) We’ve already encountered an example of a forecasting equation. Think back to the weight/height example of Section 4 from Chapter 1 and recall that the purpose of that model was to guess the weight of a male customer based on his height. In that example, the first step in building a forecast was to estimate Equation 21 from Chapter 1: (A) That is, we estimated that a customer’s weight on average equaled a base of 103.4 pounds plus 6.38 pounds for each inch over 5 feet. To actually make the prediction, all we had to do was to substitute the height of the individual whose weight we were trying to predict into the estimated equation. For a male who is tall, for example, we’d calculate: (1)Predicted weight 5 103.4 1 6.38 ? (13 inches over five feet) 6r1s (inches over five feet)5 103.4 1 6.38 ? HeightiEstimated weighti 1 484 FORECASTING or The weight-guessing equation is a specific example of using a single linear equation to predict or forecast. Our use of such an equation to make a fore- cast can be summarized into two steps: 1. Specify and estimate an equation that has as its dependent variable the item that we wish to forecast. We obtain a forecasting equation by specifying and estimating an equation for the variable we want to predict: (2) The use of to denote the sample size is fairly stan- dard for time-series forecasts (t stands for “time”). 2. Obtain values for each of the independent variables for the observations for which we want a forecast and substitute them into our forecasting equation. To calculate a forecast with Equation 2, this would mean finding values for period for and and substituting them into the equation: (3) What is the meaning of this It is a prediction of the value that Y will take in observation (outside the sample) based upon our values of and based upon the particular specification and estimation that produced Equation 2. To understand these steps more clearly, let’s look at two applications of this forecasting approach: Forecasting Chicken Consumption: Let’s return to the chicken demand model, Equation 8 from Chapter 6, to see how well that equation forecasts aggregate per capita chicken consumption: (B) (0.03) (0.02) (0.01) t � � 3.38 � 1.86 � 15.7 R2 5 .9904 N 5 29 (annual 197422002) DW d 5 0.99 Ŷt 5 27.7 2 0.11PCt 1 0.03PBt 1 0.23YDt X1T11 and X2T11 T 1 1 ŶT11? ŶT11 5 �̂0 1 �̂1X1T11 1 �̂2X2T11 X2X1T 1 1 (t 5 1, 2, . . . , T) Ŷt 5 �̂0 1 �̂1X1t 1 �̂2X2t (t 5 1, 2, . . . , T) 103.4 1 82.9 5 186.3 pounds 485 FORECASTING 4. The rest of the actual values are PC: 2004 � 24.8, 2005 � 26.8; PB: 2004 � 406.5, 2005 � 409.1; YD: 2004 � 295.17, 2005 � 306.16. Many software packages, including EViews and Stata, have forecasting modules that will allow you to calculate forecasts using equations like Equation 4 automatically. If you use that module, you’ll note that the forecasts differ slightly because we rounded the coefficient estimates. 5. For a summary of seven different methods of measuring forecasting accuracy, see Peter Kennedy, A Guide to Econometrics (Malden, MA: Blackwell, 2008), pp. 334–335. where: Y � pounds of chicken consumption per capita PC and PB � the prices of chicken and beef, respectively, per pound YD � per capita U.S. disposable income To make these forecasts as realistic as possible, we held out the last three available years from the data set used to estimate Equation 8 from Chapter 6. We’ll thus be able to compare the equation’s forecasts with what actually happened. To forecast with the model, we first obtain values for the three in- dependent variables and then substitute them into Equation 8 from Chapter 6. For 2003, PB � 374.6, and YD � 280.2 giving us: (4) Continuing on through 2005, we end up with4: Year Forecast Actual Percent Error 2003 99.63 95.63 4.2 2004 105.06 98.58 6.6 2005 107.44 100.60 6.8 How does the model do? Well, forecasting accuracy, like beauty, is in the eye of the beholder, and there are many ways to answer the question.5 The sim- plest method is to take the mean of the percentage errors (in absolute value), an approach called, not surprisingly, the mean absolute percentage error (MAPE) method. The MAPE for our forecast is 6.2 percent. The most popular alternative method of evaluating forecast accuracy is the root mean square error criterion (RMSE), which is calculated by squaring the forecasting error for each time period, averaging these squared amounts, and then taking the square root of this average. One advantage of the RMSE is that it penalizes large errors because the errors are squared before they’re added together. For the chicken demand forecasts, the RMSE of our forecast is 5.97 pounds (or 6 percent). Ŷ2003 5 27.7 2 0.11(34.1) 1 0.03(374.6) 1 0.23(280.2) 5 99.63 PC 5 34.1, 486 FORECASTING As you can see in Figure 1, it really doesn’t matter which method you use, because the unconditional forecasts generated by Equation 8 from Chapter 6 track quite well with reality. We missed by around 6 percent. Forecasting Stock Prices: Some students react to the previous example by wanting to build a model to forecast stock prices and make a killing on the stock market. “If we could predict the price of a stock three years from now to 0 2003 2004 2005 Time Forecasted Actual C o n su m p ti o n o f C h ic k e n Pounds per Capita 100 75 50 25 0 1 32 4 Time Forecasted Actual K e ll o g g ’s S to c k P ri c e $ 30 25 20 15 10 5 Figure 1 Forecasting Examples In the chicken consumption example, the equation’s forecast errors averaged around 6 percent. For the stock price model, even actual values for the independent variables and an excellent fit within the sample could not produce an accurate forecast. 487 FORECASTING stock based on our forecast, we’d have lost money! Since other attempts to forecast stock prices have also encountered difficulties, this doesn’t seem like a reasonable use for econometric forecasting. Individual stock prices (and many other items) are simply too variable and depend on too many nonquantif

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