Chapitre07_Solutions.pdf

Stock/Watson - Introduction to Econometrics 4th
Edition - Answers to Exercises: Chapter 7
_____________________________________________________________________________________________________
1
7.1
(1) t = 10.47/0.29 = 36.1; p-value = 2(-36.1) ≈ 0
(2) t = 10.44/0.29 = 36.0; p-value = 2(-36.0) ≈ 0
(3) t = 10.42/0.29 = 35.9; p-value = 2(-35.9) ≈ 0
7.2
(a) Yes, the college-high school earnings difference is statistically significant at the 5%
level. The t-statistic is 10.47/0.29 = 36.1 which is larger in absolute value than 1.96, the
5% critical value. The 95% confidence interval is 10.47 ± 1.96×0.29 = [9.90, 11.04]
(b) Yes, the female-male earnings difference is statistically significant at the 5% level.
The t-statistic is -4.69/0.29 = -16.2 which is larger in absolute value than 1.96, the 5%
critical value. The 95% confidence interval is -4.69 ± 1.96×0.29 = [-5.26, -4.12]
7.3.
(a) Yes, age is an important determinant of earnings. The t-statistic is 0.61/0.04 = 15.3,
with a p-value less than .01; this implies that the coefficient on age is statistically
significant at the 1% level. The 95% confidence interval is 0.61 (1.96  0.04) = [0.53,
0.69].
(b) Age  [$0.53, $0.69] 5  [$0.53, $0.69] = [$2.65, $3.45].

_____________________________________________________________________________________________________
2
7.4.
(a) The F-statistic testing the coefficients on the regional regressors are zero is 9.32. The
1% critical value (from the 3,
F  distribution) is 3.78. Because 9.32 > 3.78, the regional
effects are significant at the 1% level.
(b) The expected difference between Juanita and Molly is (X6,Juanita X6,Molly)  6 6.
Thus a 95% confidence interval is 0.44 (1.96  0.37) = [-$1.17, $0.29].
(c) The expected difference between Juanita and Jennifer is
(X5,Juanita X5,Jennifer)  5 (X6,Juanita X6,Jennifer)  6 5 6.
A 95% confidence interval could be constructed using the general methods discussed in
Section 7.3. In this case, an easy way to do this is to omit Midwest from the regression and
replace it with X5 West. In this new regression the coefficient on South measures the
difference in wages between the South and the Midwest, and a 95% confidence interval can
be computed directly.
7.5. The t-statistic for the difference in the college coefficients is
t = (b̂college,2015
- b̂college,1992
)/SE(b̂college,2015
- b̂college,1992
).
Because b̂college,2015 and ,1992
ˆ
college
 are computed from independent samples, they are
independent, which means that cov(b̂college,2015
, b̂college,1992
) = 0 .
Thus, var(b̂college,2015
- b̂college,1992
) = var(b̂college,2015
)+ var(b̂college,1998
).
This implies that SE(b̂college,2014
- b̂college,1992
) = (0.292
+0.342
)
1
2
= 0.45.
Thus, the t-statistic is (10.44 − 8.94)/0.40 = 3.33. The estimated change is
statistically significant at the 5% significance level (3.33 >1.96).

_____________________________________________________________________________________________________
3
7.6. In isolation, these results do imply gender discrimination. Sex discrimination means
that two workers, identical in every way but gender, are paid different wages. Thus, it is
also important to control for characteristics of the workers that may affect their
productivity (education, years of experience, etc.) If these characteristics are
systematically different between men and women, then they may be responsible for the
difference in mean wages. (If this were true, it would raise an interesting and important
question of why women tend to have less education or less experience than men, but that
is a question about something other than sex discrimination.) These are potentially
important omitted variables in the regression that will lead to bias in the OLS coefficient
estimator for Female. Since these characteristics were not controlled for in the statistical
analysis, it is premature to reach a conclusion about sex discrimination.

_____________________________________________________________________________________________________
4
7.7. (a) The t-statistic is 0.485
2.61
0.186 1.96.
=  Therefore, the coefficient on BDR is not
statistically significantly different from zero.
(b) The coefficient on BDR measures the partial effect of the number of bedrooms
holding house size (Hsize) constant. Yet, the typical 5-bedroom house is much
larger than the typical 2-bedroom house. Thus, the results in (a) says little about
the conventional wisdom.
(c) The 99% confidence interval for effect of lot size on price is 2000  [.002
2.58  .00048] or 1.52 to 6.48 (in thousands of dollars).
(d) Choosing the scale of the variables should be done to make the regression
results easy to read and to interpret. If the lot size were measured in thousands
of square feet, the estimate coefficient would be 2 instead of 0.002.
This would make the results easier to read an interpret: on average, a one
thousand increase in lot size is associated with a two thousand dollar increase in
the price of a house.
(e) The 10% critical value from the 2,
F  distribution is 2.30. Because 0.08 < 2.30,
the coefficients are not jointly significant at the 10% level.

_____________________________________________________________________________________________________
5
7.8. (a) Using the expressions for R2
and 2
,
R algebra shows that
2 2 2 2
1 1
1 (1 ), so 1 (1 ).
1 1
n n k
R R R R
n k n
− − −
= − − = − −
− − −
2 420 1 1
Column 1: 1 (1 0.049) 0.051
420 1
R
− −
= − − =
−
2 420 2 1
Column 2: 1 (1 0.424) 0.427
420 1
R
− −
= − − =
−
2 420 3 1
Column 3: 1 (1 0.773) 0.775
420 1
R
− −
= − − =
−
2 420 3 1
Column 4: 1 (1 0.626) 0.629
420 1
R
− −
= − − =
−
2 420 4 1
Column 5: 1 (1 0.773) 0.775
420 1
R
− −
= − − =
−
(b) 0 3 4
1 3 4
: 0
: , 0
H
H
 
 
= =
 
Unrestricted regression (Column 5):
2
0 1 1 2 2 3 3 4 4 unrestricted
, 0.775
Y X X X X R
    
= + + + + =
Restricted regression (Column 2):
2
0 1 1 2 2 restricted
, 0.427
Y X X R
  
= + + =
2 2
unrestricted restricted
unrestricted
2
unrestricted unrestricted
( )/
, 420, 4, 2
(1 )/( 1)
(0.775 0.427)/2 0.348/2 0.174
322.22
(1 0.775)/(420 4 1) (0.225)/415 0.00054
HomoskedasticityOnly
R R q
F n k q
R n k
−
= = = =
− − −
−
= = = =
− − −
5% Critical value form F2,00 4.61; FHomoskedasticityOnly F2,00 so Ho is rejected at
the 5% level.

_____________________________________________________________________________________________________
6
(c) t3 13.921 and t4 0.814, q 2; |t3| > c (Where c 2.807, the 1%
Benferroni critical value from Table 7.3). Thus the null hypothesis is rejected at
the 1% level.
(d) 1.01 2.58  0.27
7.9. (a) Estimate
0 1 2 1 2
( )
i i i i i
Y X X X u
  
= + + + +
and test whether  0.
(b) Estimate
Yi
= b0
+g X1i
+ b2
(X2i
- 2X1i
)+ui
(c) Estimate
1 0 1 2 2 1
( )
i i i i i i
Y X X X X u
  
− = + + − +

_____________________________________________________________________________________________________
7
7.10. Because 2 2 2
1 , restricted unrestricted
SSR SSR
SSR
TSS TSS
R R R
−
= − − = and
2
1 .
unrestricted
SSR
unrestricted TSS
R
− = Thus
2 2
2
( )/
(1 )/( 1)
/
/( 1)
( )/
/ (
restricted unrestricted
unrestricted
unrestricted unrestricted
SSR SSR
TSS
SSR
unrestricted
TSS
restricted unrestricted
unrestricted un
R R q
F
R n k
q
n k
SSR SSR q
SSR n k
−
−
=
− − −
=
− −
−
=
− 1)
restricted −

_____________________________________________________________________________________________________
8
Empirical Exercise 7.1
Calculations for this exercise are carried out in the STATA file EE_7_1.do.
The following table summarizes some regressions
Dependent variable is Birthweight
Regressor (1) (2) (3) (4)
Smoker -253.2
(26.8)
[-305.8, -
200.7]
-217.6
(26.1)
[-268.8, -
166.4]
-175.4
(26.8)
[-228.0, -
122.8]
-177.0
(27.3)
[-230.5, -
123.4]
Alcohol -30.5
(72.6)
-21.1
(73.0)
-14.8
(72.9)
Nprevist 34.1
(3.6)
29.6
(3.6)
29.8
(3.6)
Unmarried -187.1**
(27.7)
-199.3
(31.0)
Age -2.5
(2.4)
Years of
education
-0.238
(5.53)
Intercept 3432.1
(11.9)
3051.2
(43.7)
3134.4
(44.1)
3199.4
(90.6)
SER 583.7 570.5 565.7 565.8
R2
0.028 0.072 0.087 0.087
n 3000 3000 3000 3000
Standard errors are shown in parentheses and 95% confidence interval for Smoker is shown in
brackets
(a) See the table
(b) see table
(c) Yes it seems so. The coefficient changes falls by roughly 30% in magnitude when
additional regressors are added to (1). This change is substantively large and large
relative to the standard error in (1).
(d) Yes it seems so. The coefficient changes falls by roughly 20% in magnitude when
unmarried is added as an additional regression. This change is substantively large and
large relative to the standard error in (2).
(e) (i) -241.4 to -132.9

_____________________________________________________________________________________________________
9
(ii) Yes. The 95% confidence interval does not include zero. Alternatively, the t-
statistics Is -6.76 which is large in absolute value than the 5% crtical value of 1.96.
(iii) Yes. On average, birthweight is 187 grams lower for unmarried mothers.
(iv) As the question suggests, unmarried is a control variable that captures the effects of
several factors that differ between married and unmarried mothers such as age, education,
income, diet and other health factors, and so forth.
f. I have added on additional regression in the table that includes Age and Educ (years of
education) in regression (4). The coefficient on Smoker is very similar to its value in
regression (3).

_____________________________________________________________________________________________________
10
Empirical Exercise 7.2
Calculations for this exercise are carried out in the STATA file EE_7_2.do.
(a) From Key Concept 6.1, omitted variable bias arises if
1. X is correlated with the omitted variable.
and
2. The omitted variable is a determinant of the dependent variable, Y.
The mechanism described in the problem explains why height (X) and cognitive ability
(the omitted variable) are correlated and why cognitive ability is a determinant of earning
(Y). The mechanism suggests that height and cognitive ability are positively correlated
and that cognitive ability has a positive effect on earnings. Thus, X will be positively
correlated with the error in (2) leading to a positive bias in the estimated coefficient.
The following table summarizes some regressions
Dependent variable is Earnings
Women Men
Regressor (1) (2) (3) (4)
Height 511.2
(97.6)
135.1
(92.3)
1307
(98.9)
744.7
(92.3)
LT_HS -31858
(835)
-31400
(870)
HS -20418
(638)
-20346
(702)
Some_Col -12649
(717)
-12611
(798)
Intercept 12650
(6299)
50750
(6004)
-43130
(6925)
9863
(6541)
SER 26801 24917 26671 24623
R2
0.003 0.138 0.021 0.165
n 9974 9974 7896 7896
(b) (i) The estimated coefficient on height falls by approximately 75%, from 511 to 135
when the education variables are added as control variables in the regression. This is
consistent with positive omitted bias in (1).
(ii) College is perfectly collinear with other eductation regressors and the constant
regressor.
(iii) The F-statistic is 578, which is larger than the 1% critical value of 3.78 (from the
F3, distribution), and therefore the null hypothesis that the coefficients on he education
variables are jointly equal to zero is rejected at the 1% significance level.

_____________________________________________________________________________________________________
11
(iv) The coefficients measure the effect of education on earnings relative to the omitted
category, which is College. Thus, the estimated coefficient on the “Less than High
School” regressor implies that workers with less than a high school education on average
earn $31,858 less per year than a college graduate; a worker with a high school education
on average earns $20,418 less per year than a college graduate; a worker with a some
college on average earns $12,649 less per year than a college graduate.
(c) (i) The estimated coefficient on height falls by approximately 50%, from 1307 to 745.
This is consistent with positive omitted bias in the simple regression (1).
(ii) Same answer as (b)
(iii) The F-statistic is 500, which is larger than the 1% critical value of 3.78 (from the
F3, distribution), and therefore the null hypothesis that the coefficients on he education
variables are jointly equal to zero is rejected at the 1% significance level.
(iv) The coefficients measure the effect of education on earnings relative to the omitted
category, which is College. Thus, the estimated coefficient on the “Less than High
School” regressor implies that workers with less than a high school education on average
earn $31,400 less per year than a college graduate; a worker with a high school education
on average earns $20,346 less per year than a college graduate; a worker with a some
college on average earns $12,611 less per year than a college graduate.

Chapitre07_Solutions.pdf

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Chapitre07_Solutions.pdf

Similar to Chapitre07_Solutions.pdf (20)

Recently uploaded

Recently uploaded (20)

Chapitre07_Solutions.pdf