Essentials of EconometricsTutorial Exercises SW 10

10.1 This exercise refers to the drunk driving panel data regression summarized in Table 1 (see

Table 1 at the last page).

(a) New Jersey has a population of 8.1 million people. Suppose that New Jersey increased

the tax on a case of beer by $1 (in 1988 dollars). Use the results in column (4) to predict

the number of lives that would be saved over the next year. Construct a 95% confidence

interval for your answer.

(b) The drinking age in New Jersey is 21. Suppose that New Jersey lowered its drinking

age to 18. Use the results in column (4) to predict the change in the number of traﬃc

fatalities in the next year. Construct a 95% confidence interval for your answer.

(c) Suppose that real income per capita in New Jersey increases by 1% in the next year. Use

the results in column (4) to predict the change in the number of traﬃc fatalities in the

next year. Construct a 90% confidence interval for your answer.

(d) Should time eﬀects be included in the regression? Why or why not?

(e) A researcher conjectures that the unemployment rate has a diﬀerent eﬀect on traﬃc fatalities in the western states than in the other states. How would you test this hypothesis?

(Be specific about the specification of the regression and the statistical test you would

use.)

10.2 Consider the following binary variable version of the fixed eﬀects model:

= 0 + 1 + 1 1 + 2 2 + + +

where and are observed regressors. In particular, let 1 be a binary variable that

equals 1 when = 1 and equals 0 otherwise, let 2 equal 1 when = 2 and equal 0 otherwise,

and so on.

(a) Suppose that = 3. Show that the binary regressors and the “constant” regressor are

perfectly multicollinear; that is, express one of the variables 1 , 2 , 3 , and 0 as

a perfect linear function of others, where 0 = 1 for all , .

(b) Show the result in (a) for general .

(c) What will happen if you try to estimate the coeﬃcients of the regression by OLS?

10.3 Recall the list of five potential threats to the internal validity of a regression study. Apply

this list to the empirical analysis of the Eﬀect of Drunk Driving Laws on Traﬃc Deaths in

textbook Section 10.6 and thereby draw conclusions about its internal validity.

10.4 Using the regression in the following equation

= 0 + 1 + 2 2 + 3 3 + + +

where and are observed regressors. In particular, let 2 be a binary variable that

equals 1 when = 2 and equals 0 otherwise, let 3 equal 1 when = 3 and equal 0 otherwise,

and so on. What is the slope and intercept for

1

(a) Entity 1 in time period 1?

(b) Entity 1 in time period 3?

(c) Entity 3 in time period 1?

(d) Entity 3 in time period 3?

10.5 Consider the model with a single regressor = 1 1 + + + . This model also can

be written as

= 0 + 1 1 + 2 2 + + + 2 2 + + +

where 2 = 1 if = 2 and 0 otherwise, 2 = 1 if = 2 and 0 otherwise, and so forth. How

are the coeﬃcients ( 0 , 2 ,…, , 2 ,…, ) related to the coeﬃcients (1 ,…, , 1 ,…, )?

10.6 Do the fixed eﬀects regression assumptions in Key Concept 10.3 in textbook imply that

(˜ ˜ ) = 0 for 6= in Equation (10.28) (page 421 in the textbook)? Explain.

10.7 A reseracher believes that traﬃc fatalities increase when roads are icy and so states with more

snow will have more facilities than other states. Comment on the following methods designed

to estimate the eﬀect of snow on fatalities:

(a) The researcher collects data on the average snowfall for each state and adds this regressor

( ) to the regressions given in Table 10.1.

(b) The researcher collects data on the snowfall in each state for each year in the sample

( ) and adds this regressor to the regressions.

10.8 Consider observations ( ) from the linear panel data model

= 1 + + +

where = 1 ; = 1 ; and + is an unobserved individual-specific time trend.

How would you estimate 1 ?

10.10 In a study of the eﬀect on earnings of education using panel data on annual earnings for a large

number of workers, a researcher regresses earnings in a given year on age, education, union

status, and the worker’s earnings in the previous year using fixed eﬀects regression. Will this

regression give reliable estimates of the eﬀects of the regressors (age, education, union status,

and previous year’s earnings) on earnings? Explain. (Hint: Check the fixed eﬀects regression

assumptions in Section 10.5.)

2

Table 1 Regression Analysis of the Eﬀect of Drunk Driving Laws on Traﬃc Deaths

D ep en d ent va ria b le: tra ﬃ c fata lity rate (d ea th s p er 1 0,0 0 0 )

R eg resso r

(1 )

036∗∗

B e er ta x

(005)

(2 )

(3 )

−066∗

−064+

(029)

(036)

D rin k in g a g e 1 8

(4 )

−045

(030)

0028

(0070)

−0018

D rin k in g a g e 1 9

(0050)

0032

D rin k in g a g e 2 0

(0051)

(5 )

−069∗

(035)

−0010

(6 )

−046

(031)

(7 )

−093∗∗

(034)

0037

(0083)

(0102)

−0076

−0065

(0056)

−0113

(0068)

−0100+

D rin k in g a g e

(0099)

(0125)

−0002

(0021)

0038

0085

A vera g e veh ic le m ile s p er d river

0008

0017

U n e m p loy m ent ra te

−0063∗∗

M a n d a to ry ja il o r co m m u n ity serv ic e?

(0103)

(0007)

(0112)

(0011)

(0013)

0089

0009

0124

−0063∗∗

−0091∗∗

(0103)

(0007)

(0013)

182∗∗

R ea l in co m e p e r ca p ita (lo g a rith m )

0039

179∗∗

(0164)

(0049)

(0021)

100

(064)

(068)

Ye ars

1 98 2 -8 8

1 9 8 2 -8 8

1 9 8 2 -8 8

1 9 8 2 -8 8

1 9 8 2 -8 8

1 9 8 2 -8 8

S ta te eﬀ e cts?

T im e eﬀ ects?

C lu ste red sta n d a rd erro rs?

no

no

no

ye s

no

ye s

yes

yes

yes

ye s

ye s

ye s

ye s

ye s

ye s

yes

yes

yes

1982

& 8 8 o n ly

yes

yes

yes

(064)

−

−

S ta tistic a n d

Va lu e s

Testin g E x clu sio n o f G ro u p s o f Va ria b le s

422

T im e eﬀ ects= 0

(0002)

1012

(0001)

035

D rin k in g a g e co e ﬃ cie nts= 0

(0786)

348

(0006)

1028

(0001)

141

3749

(0001)

042

(0253)

(0738)

U n e m p loy m ent ra te,

2962

in co m e p e r c ap ita = 0

3196

(0001)

̄2

0 .09 1

0 .8 8 9

0 .8 9 1

0 .9 2 6

0 .8 9 3

2520

(0001)

(0001)

0 .9 2 6

0 .8 9 9

T h e se re g ressio n s w ere e stim a ted u sin g p a n e l d a ta fo r 4 8 U .S . sta tes. R e g ressio n s (1) th ro u g h (6 )u se d a ta fo r a ll ye ars 1 9 8 2 to 1 98 8 , a n d reg ressio n (7 ) u ses

d a ta fro m 1 9 82 a n d 1 9 8 8 o n ly. T h e d a ta set is d e sc rib e d in A p p en d ix 1 0 .1 . S ta n d a rd erro rs a re g iven in p a renth esis u n d er th e co e ﬃ cie nts, a n d −

a re g ive n in p a renth e se s u n d er th e −sta tistic s. T h e in d iv id u a l co e ﬃ cie nt is sta tistic a lly sig n ifi c a nt a t th e + 10%, ∗ 5%, o r ∗ 1% sig n ifi c a n ce le vel.

3

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