Essentials of Econometrics
Tutorial Exercises SW 14
14.1 Consider the AR(1) model Yt = βo + β1 Yt−1 + ut . Suppose that the process is stationary.
(a) Show that E(Yt ) = E(Yt−1 ). (Hint: Read Key Concept 14.5 in the textbook.)
(b) Show that E(Yt ) = β0 / (1 − β1 )
14.2 The index of industrial production (IPt ) is a monthly time series that measures the quantity
of industrial commodities produced in a given month. This problem uses data on this index
for the United States. All regressions are estimated over the sample period 1960:1 to 2000:12
(that is, January 1960 through December 2000). Let Yt = 1200 × ln(IPt /IPt−1 ).
(a) The forecaster states that Yt shows the monthly percentage change in IP , measured in
percentage points per annum. Is this correct? Why?
(b) Suppose a forecaster estimates the following AR(4) model for Yt :
Ybt = 1.377 + 0.318Yt−1 + 0.123Yt−2 + 0.068Yt−3 + 0.001Yt−4.
(0.062)
(0.078)
(0.055)
(0.068)
(0.056)
Use this AR(4) to forecast the value of Yt in January 2001 using the following values of
IP for August 2000 through December 2000:
Date 2000:7 2000:8 2000:9 2000:10 2000:11 2000:12
IP
147.595 148.650 148.973 148.660 148.206 147.3
(c) Worried about potential seasonal fluctuations in production, the forecaster adds Yt−12 to
the autoregression. The estimated coefficient on Yt−12 is -0.054 with a standard error of
0.053. Is this coefficient statistically significant?
14.3 Using the same data as in Exercise 14.2, a researcher tests for a stochastic trend in ln(IPt )
using the following regression:
∆\
ln(IPt ) = 0.061 + 0.00004t − 0.018 ln(IPt−1 ) + 0.333∆ ln(IPt−1 ) + 0.162∆ ln(IPt−2 ).
(0.024)
(0.00001)
(0.007)
(0.075)
(0.055)
where the standard errors shown in parentheses are computed using the homoskedasticity-only
formula and the regressor ”t” is a linear time trend.
(a) Use the ADF statistic to test for a stochastic trend (unit root) in ln(IP ).
(b) Do these results support the specification used in Exercise 14.2? Explain.
14.4 The forecaster in Exercise 14.2 augments her AR(4) model for IP growth to include four lagged
values of ∆Rt where Rt is the interest rate on three-month U.S. Treasury bills (measured in
percentage points at an annual rate).
(a) The Granger-causality F -statistic on the four lags of ∆Rt is 2.35. Do interest rates help
to predict IP growth? Explain.
1
(b) The researcher also regresses ∆Rt on a constant, four lags of ∆Rt and four lags of IP
growth. The resulting Granger-causality F -statistic on the four lags of IP growth is
2.87. Does IP growth help to predict interest rates? Explain.
14.8 Suppose that Yt is the monthly value of the number of new home construction projects started
in the United States. Because of the weather, Yt has a pronounced seasonal pattern; for
example, housing starts are low in January and high in June. Let µJan denote the average
value of housing starts in January and µF eb , µM ar , …, µDec denote the average values in the
other months. Show that the values of µjan , µF eb , …, µDec can be estimated from the OLS
regression Yt = β0 + β1 F ebt + β2 M art + … + β11 Dect + ut , where F ebt is a binary variable
equal to 1 if t is February, M art is a binary variable equal to 1 if t is March, and so forth.
Show that β0 = µJan , β0 + β1 = µF eb , β0 + β2 = µM ar , and so forth.
14.11 Suppose that ∆Yt follows the AR(1) model ∆Yt = β0 + β1 ∆Yt−1 + ut .
(a) Show that Yt follows an AR(2) model.
(b) Derive the AR(2) coefficients for Yt as a function of β0 and β1 .
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6.1
Definition: The adjusted coefficient of determination (also known as adjusted R² or R²).s
represents the proportion of variation in the dependent variable that can be explained by the
estimated regression-line.
The range of R² is between 0 and 1. Higher the value of R2, better is the fit for regression
model.
Note: Another similar quantity R2 is also used to comment on the fit of the regression model..
However, R² increases whenever a new independent variable is added in the regression. Hence,
you want to use adjusted R² with multiple regression analysis. Adjusted R2 increases only when
you add new independent variables that do increase the explanatory power of the regression
equation, making it a much more useful measure of how well a multiple regression equation fits
the sample data than R2-1
Formula:
Ŕ² = 1 −
n-1
n-k-1
(1 – R²)¶
Where n sample size, k = number of independent variables in the regression equation and R²
is the coefficient of determination.
• → Regression (1)¶
n =4000, k =2 (college, female), R2 = 0.176 |
1
Ŕ² = 1 −
4000 – 1
4000 2 1
(1 -0.176) =
0.17559
་
• → Regression (2)¶
n=4000, k =3 (college, female, age), R2 = 0.190|||
1
Ŕ² = 1 −
4000-1
4000 3 1
(1 -0.190) = 0.18939
• → Regression (3)¶
n =4000, k=6 (college, female, age, Northeast, Midwest, South), R2 = 0.194¶|
