Introduction to Econometrics Answers Chapter 8 Exercises

Stock/Watson - Introduction to Econometrics - 3rd
Updated Edition - Answers to Exercises: Chapter 8
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1
8.1. (a) The percentage increase in sales is 198 196
196
100 1.0204%.
−
 = The approximation
is 100 × [ln (198) − ln (196)] = 1.0152%.
(b) When Sales2014 = 205, the percentage increase is 205 196
196
100 4.5918%
−
 = and
the approximation is 100 × [ln (205) − ln (196)] = 4.4895%. When Sales2014 =
250, the percentage increase is 250 196
196
100 27.551%
−
 = and the approximation is
100 × [ln (250) − ln (196)] = 24.335%. When Sales2014 = 500, the percentage
increase is 500 196
196
100 155.1%
−
 = and the approximation is
100×[ln(500)−ln(196)] = 93.649%.
(c) The approximation works well when the change is small. The quality of the
approximation deteriorates as the percentage change increases.

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2
8.2. (a) According to the regression results in column (1), the house price is expected to
increase by 21% (= 100% × 0.00042 × 500 ) with an additional 500 square feet
and other factors held constant. The 95% confidence interval for the percentage
change is 100% × 500 × (0.00042 ± 1.96 × 0.000038) = [17.276% to 24.724%].
(b) Because the regressions in columns (1) and (2) have the same dependent
variable, 2
R can be used to compare the fit of these two regressions. The log-log
regression in column (2) has the higher 2
,
R so it is better so use ln(Size) to
explain house prices.
(c) The house price is expected to increase by 7.1% ( = 100% × 0.071 × 1). The 95%
confidence interval for this effect is 100% × (0.071 ± 1.96 × 0.034) = [0.436% to
13.764%].
(d) The house price is expected to increase by 0.36% (100% × 0.0036 × 1 = 0.36%)
with an additional bedroom while other factors are held constant. The effect is
not statistically significant at a 5% significance level:
0.0036
0.037
| | 0.09730 1.96.
t = =  Note that this coefficient measures the effect of an
additional bedroom holding the size of the house constant. Thus, it measures
the effect of converting existing space (from, say a family room) into a
bedroom.
(e) The quadratic term ln(Size)2
is not important. The coefficient estimate is not
statistically significant at a 5% significance level: 0.0078
0.14
| | 0.05571 1.96.
t = = 
(f) The house price is expected to increase by 7.1% ( = 100% × 0.071 × 1) when a
swimming pool is added to a house without a view and other factors are held
constant. The house price is expected to increase by 7.32% ( = 100% × (0.071 ×
1 + 0.0022 × 1) ) when a swimming pool is added to a house with a view and
other factors are held constant. The difference in the expected percentage change
in price is 0.22%. The difference is not statistically significant at a 5%
significance level: 0.0022
0.10
| | 0.022 1.96.
t = = 

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3
8.3. (a) The regression functions for hypothetical values of the regression coefficients
that are consistent with the educator’s statement are: 1 > 0 and 2 < 0. When
TestScore is plotted against STR the regression will show three horizontal
segments. The first segment will be for values of STR < 20; the next segment for
20 ≤ STR ≤ 25; the final segment for STR > 25. The first segment will be higher
than the second, and the second segment will be higher than the third.
(b) It happens because of perfect multicollinearity. With all three class size binary
variables included in the regression, it is impossible to compute the OLS
estimates because the intercept is a perfect linear function of the three class size
regressors.

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4
8.4. (a) With 2 years of experience, the man’s expected AHE is
With 3 years of experience, the man’s expected AHE is
Difference = 0.0147-0.0009 = 0.014 (or 1.4%).
(b) Region affects the level of ln(AHE), but not the change associated with years of
experience.
(c) With 10 years of experience, the man’s expected AHE is
With 11 years of experience, the man’s expected AHE is
Difference =0.0147-0.0038 = 0.011 (or 1.1%).
(d) The regression in nonlinear in experience (it includes Potential experience2
).

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5
(e) Let 1 denote the coefficient on Potential Experience and 2 denote the coefficient on
(Potential Experience)2
. Then, following the discussion in the paragraphs just about
equation (8.7) in the text, the expected change in part(a) is given by 1 + 52 and the
expected change in (b) is given by 1 + 212. The difference between these, say
(b)−(a), is 162. Because the estimated value of 2 is significant at the 5% level (the
t-statistic for 2
ˆ
 is tstat = −.000183/.000024 = −7.6), the difference between the
effects in (a) and (b) (=162) is significant at the 5% level.
(f) No. These would affect the level of ln(AHE), but not the change associated with
another year of experience.
(g) In this case, you would estimate a new regression that includes the interaction terms
Female × Potential experience and Female × (Potential experience)2
.

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6
8.5. (a) (1) The demand for older journals is less elastic than for younger journals
because the interaction term between the log of journal age and price per citation
is positive. (2) There is a linear relationship between log price and log of quantity
follows because the estimated coefficients on log price squared and log price
cubed are both insignificant. (3) The demand is greater for journals with more
characters follows from the positive and statistically significant coefficient
estimate on the log of characters.
(b) (i) The effect of ln(Price per citation) is given by [ −0.899 + 0.141 × ln(Age)] 
ln(Price per citation). Using Age = 80, the elasticity is [−0.899 + 0.141 
ln(80)] = −0.28.
(ii) As described in equation (8.8) and the footnote on page 263, the standard
error can be found by dividing 0.28, the absolute value of the estimate, by the
square root of the F-statistic testing
ln(Price per citation) + ln(80)  ln(Age)×ln(Price per citation) = 0.
(c) ( )
ln ln( ) ln( )
Characters
a
Characters a
= − for any constant a. Thus, estimated
parameter on Characters will not change and the constant (intercept) will
change.

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7
8.6. (a) (i) There are several ways to do this. Here is one. Create an indicator variable,
say DV1, that equals one if %Eligible is greater than 20% and less than 50%.
Create another indicator, say DV2, that equals one if %Eligible is greater
than 50%. Run the regression:
0 1 2 3
% 1 % 2 % other regressors
TestScore Eligible DV Eligible DV Eligible
   
= + +  +  +
The coefficient 1 shows the marginal effect of %Eligible on TestScores for
values of %Eligible < 20%, 1 + 2 shows the marginal effect for values of
%Eligible between 20% and 50% and 1 + 3 shows the marginal effect for
values of %Eligible greater than 50%.
(ii) The linear model implies that 2 = 3 = 0, which can be tested using an F-
test.
(b) (i) There are several ways to do this, perhaps the easiest is to include an
interaction term STR  ln(Income) to the regression in column (7).
(ii) Estimate the regression in part (b.i) and test the null hypothesis that the
coefficient on the interaction term is equal to zero.

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8
8.7. (a) (i) ln(Earnings) for females are, on average, 0.44 lower for men than
for women.
(ii) The error term has a standard deviation of 2.65 (measured in log-points).
(iii)Yes. But the regression does not control for many factors (size of firm,
industry, profitability, experience and so forth).
(iv)No. In isolation, these results do not imply gender discrimination. Gender
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 gender
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 gender discrimination.
(b) (i) If MarketValue increases by 1%, earnings increase by 0.37%
(ii) Female is correlated with the two new included variables and at least one of
the variables is important for explaining ln(Earnings). Thus the regression in
part (a) suffered from omitted variable bias.
(c) Forgetting about the effect or Return, whose effects are small and statistically
insignificant, the omitted variable bias formula (see equation (6.1)) suggests that
Female is negatively correlated with ln(MarketValue).

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9
8.8 (a) and (b)
(c)

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10
(d)
(e)

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11
8.9. Note that
2
0 1 2
2
0 1 2 2
( 21 ) ( 21 ).
Y X X
X X X
  
   
= + +
= + + + −
Define a new independent variable 2
21 ,
Z X X
= − and estimate
0 2 i
Y X Z u
  
= + + + 
The confidence interval is ( )
ˆ ˆ
1 96 SE .
 
  
8.10. (a) 1 1 2 1 2 1 1 3 1 2
( , ) ( , ) ,
Y f X X X f X X X X X
 
 = +  − =  +   so 1 1 3 2.
Y
X
X
 


= +
(b) 1 2 2 1 2 2 3 1 2
( , ) ( , ) ,
Y f X X X f X X X X X
 
 = +  − =  + 
2 so
2 2 3 1.
Y
X
X
 


= +
(c)
1 1 2 2 1 2
0 1 1 1 2 2 2 3 1 1 2 2
0 1 1 2 2 3 1 2
1 3 2 1 2 3 1 2 3 1 2
( , ) ( , )
( ) ( ) ( )( )
( )
( ) ( ) .
Y f X X X X f X X
X X X X X X X X
X X X X
X X X X X X
   
   
    
 = +  +  −
= + +  + +  + +  + 
− + + +
= +  + +  +  

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12
8.11. Linear model: E(Y | X) = 0 + 1X, so that 1
( | )
dE Y X
dX

= and the elasticity is
1
1
0 1
( | )
X
X
E Y X X


 
=
+
Log-Log Model: E(Y | X) = ( )
0 1 0 1 0 1
ln( ) ln( ) ln( )
| ( | )
X u X X
u
E e X e E e X ce
     
+ + + +
= = ,
where c = E(eu
| X), which does not depend on X because u and X are assumed to be
independent.
Thus 0 1 ln( )
1
1
( | ) ( | )
X
dE Y X E Y X
ce
dX X X
 


+
= = and the elasticity is 1.

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13
8.12. (a) Because of random assignment within the group of returning students E(X1i | ui)
= 0 in “ - regression,” so that 1
ˆ
 is an unbiased estimator of 1.
(b) Because of random assignment within the group of returning students E(X1i | ui) =
0 in “- regression,” so that 1
ˆ
 is an unbiased estimator of 1.
(c) Write E(ui | X1i, X2i) = E(ui | X2i) = 0 + 1X2i, where linearity is assumed for the
conditional expected value. Thus,
E(Y |X1, X2) = 0 + 1X1 + 2X2 + 3X1X2 + E(ui | X1i, X2i)
= (0 + 0) + 1X1 + (2 + 1) X2 + 3X1X2.
Using this expression, E(Y | X1 = 1, X2 = 0) − E(Y | X1 = 0, X2 = 0) = 1, which
is equal to 1 from (a).
Also, E(Y| X1 = 1, X2 = 1) − E(Y| X1 = 0, X2 = 1) = 1 + 3, which is equal to 1
from (b). Together, these results imply that 3 = 1 − 1.
(d) Defining vi = ui – E(ui | X1i, X2i) = ui – E(ui | X2i), then
Yi = (0 + 0) + 1X1 + (2 + 1) X2 + 3X1X2 + vi,
where E(vi | X1i, X2i) = 0. Thus, applying OLS to the equation will yield a biased
estimate of the constant term [ 0
ˆ
( )
E  = 0 + 0], an unbiased estimate of 1
[E(b̂1
) = b1
], a biased estimate of 2 [ 2
ˆ
( )
E  = 2 + 1], and an unbiased estimate
of 3 [ 3
ˆ
( )
E  = 3].

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14
Empirical Exercise 8.1
Calculations for this exercise are carried out in the STATA file EE_8_1.do.
(a) The table shows the sample mean (Y ) and its standard error for lead and no-lead
cities. The difference in the sample means is 0.022 with a standard error of 0.024. The
estimate implies that cities with lead pipes have a larger infant mortality rate (by 0.02
deaths per 100 people in the population), but the standard error is large (0.024) and the
difference is not statistically significant (t = 0.022/0.024 = 0.090).
n Y SE(Y )
Lead 117 0.403 0.014
No Lead 55 0.381 0.020
Difference 0.022 0.024
(b) The regression is
= 0.919 + 0.462×lead − 0.075×pH − 0.057×lead×pH
(0.150) (0.208) (0.021) (0.028)
(i) The first coefficient is the intercept, which shows the level of Infrate when lead = 0
and pH = 0. It dictates the level of the regression line.
The second coefficient and fourth coefficients measure the effect of lead on the infant
mortality rate. Comparing 2 cities, one with lead pipes (lead = 1) and one without lead
pipes (lead = 0), but the same of pH, the difference in predicted infant mortality rate is
The third and fourth coefficients measure the effect of pH on the infant mortality rate.
Comparing 2 cities, one with a pH = 6 and the other with pH = 5, but the same of lead,
the difference in predicted infant mortality rate is
so the difference is -0.075 for cities without lead pipes and −0.132 for cities with lead
pipes.

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15
(ii)
Solid: Cities with lead pipes
Dashed: Cities without lead pipes
The infant mortality rate is higher for cities with lead pipes, but the difference declines as
the pH level increases. For example:
The 10th
percentile of pH is 6.4. At this level, the difference in infant mortality rates is
The 50th
The 90th
(iii) The F-statistic for the coefficient on lead and the interaction term is F = 3.94, which
has a p-value of 0.02, so the coefficients are jointly statistically significantly different
from zero at the 5% but not the 1% significance level.
(iv) The interaction term has a t statistic of t = −2.02, so the coefficient is significant at
the 5% but not the 1% significance level.

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16
(v) The mean of pH is 7.5. At this level, the difference in infant mortality rates is
The standard deviation of pH is 0.69, so that the mean plus 1 standard devation is 8.19
and the mean minus 1 standard deviation is 6.81. The infant mortality rates at the pH
levels are:
(vi) Write the regression as
Infrate = 0 + llead + 2pH + 3lead×pH + u
so the effect of lead on Infrate is 1 + 3pH. Thus, we want to construct a 95%
confidence interval for 1 + 6.53. Using method 2 of Section 7.3, add and subtract
6.53lead to the regression to obtain:
Infrate = 0 + (l + 6.53)lead + 2pH + 3(lead×pH −0.65lead) + u
or
The estimated regression is
= 0.919 + 0.092×lead − 0.075×pH − 0.057×lead×(pH−6.5)
(0.150) (0.033) (0.021) (0.028)
and the 95% confidence interval for the coefficient on lead (which is l + 6.53) is 0.027
to 0.157.
(c) There are several demographic variables in the dataset. You should add these and see
if the conclusions from (b) change in an important way.

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17
Empirical Exercise 8.2
Calculations for this exercise are carried out in the STATA file EE_8_2.do.
This table contains the results from seven regressions that are referenced in these answers.
Data from 2015
(1) (2) (3) (4) (5) (6) (7) (8)
Dependent Variable
AHE ln(AHE) ln(AHE) ln(AHE) ln(AHE) ln(AHE) ln(AHE) ln(AHE)
Age 0.531
(0.045)
0.024
(0.002)
0.134
(0.046)
0.135
(0.046)
0.139
(0.06)
0.160
(0.063)
0.160
(0.072)
Age2
−0.0019
(0.00077)
−0.0019
(0.00077)
−0.0018
(0.0010)
−0.0023
(0.0011)
-0.0023
(0.0012)
ln(Age) 0.72
(0.06)
Female  Age −
(0.091)

()
Female  Age2 -00001
(0.0015)
-0.0002
(0.0016)
Bachelor  Age −
(0.091)
−
(0.093)
Bachelor  Age2 0.0009
(0.0015)
0.0008
(0.0016)
Female −4.14
(0.26)
−0.18
(0.01)
−0.18
(0.01)
−0.18
(0.01)
−0.19
(0.02)
-0.03
(1.33)
−0.19
(0.02)
-0.04
(1.36)
Bachelor 9.85
(0.26)
0.46
(0.01)
0.46
(0.01)
0.46
(0.01)
0.45
(0.02)
0.45
(0.02)
1.09
(1.34)
0.94
(1.36)
Female  Bachelor 0.023
(0.023)
0.023
(0.023)
0.024
(0.023)
0.024
(0.023)
F-statistic and p-values on joint hypotheses
F-stat. on terms
involving Age
76.6
(0.00)
76.8
(0.00)
38.8
(0.00)
38.52
(0.00
26.0
(0.00)
Interaction terms of
Female
with Age and Age2
2.64
(0.07)
3.18
(0.04)
Interaction of Bachelor
with Age and Age2
1.03
(0.36)
1.57
(0.21)
SER 10.92 0.48 0.48 0.48 0.48 0.48 0.48 0.48
2
R 0.19 0.21 0.21 0.21 0.21 0.21 0.21 0.21
Note: intercept is included in all regressions. Sample size is n = 7098

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18
(a) The regression results for this question are shown in column (1) of the table. If Age
increases from 25 to 26, earnings are predicted to increase by $0.531 per hour. If Age
increases from 33 to 34, earnings are predicted to increase by $0.531 per hour. These
values are the same because the regression is a linear function relating AHE and Age.
(b) The regression results for this question are shown in column (2) of the table. If Age
increases from 25 to 26, ln(AHE) is predicted to increase by 0.024, so earnings are
predicted to increase by 2.4%. If Age increases from 34 to 35, ln(AHE) is predicted to
increase by 0.024, o earnings are predicted to increase by 2.4%. These values, in
percentage terms, are the same because the regression is a linear function relating
ln(AHE) and Age.
(c) The regression results for this question are shown in column (3) of the table. If Age
increases from 25 to 26, then ln(Age) has increased by ln(26) − ln(25) = 0.0392 (or
3.92%). The predicted increase in ln(AHE) is 0.72  (.0392) = 0.028. This means that
earnings are predicted to increase by 2.8%. If Age increases from 34 to 35, then ln(Age)
has increased by ln(35) − ln(34) = .0290 (or 2.90%). The predicted increase in ln(AHE)
is 0.72  (0.0290) = 0.021. This means that earnings are predicted to increase by 2.1%.
(d) The regression results for this question are shown in column (4) of the table. When
Age increases from 25 to 26, the predicted change in ln(AHE) is
(0.134  26 − 0.0019  262
) − (0.134  25 − 0.0019  252
) = 0.037.
This means that earnings are predicted to increase by 3.7%.
When Age increases from 34 to 35, the predicted change in ln(AHE) is
(0.134  34 − 0.0019  342
) − (0.134  33 − 0.0019  332
) = 0.007.
This means that earnings are predicted to increase by 0.7%.
(e) The regressions differ in their choice of one of the regressors. They can be compared
on the basis of the 2
.
R The regression in (3) has a (marginally) higher 2
,
R so it is
preferred.
(f) The regression in (4) adds the variable Age2
to regression (2). The coefficient on Age2
is statistically significant (t = −2.41). This suggests that (4) is preferred to (2).
(g) The regressions differ in their choice of the regressors (ln(Age) in (3) and Age and
Age2
in (4)). They can be compared on the basis of the R2
. The regression in (4) has
a (marginally) higher R2
, so it is preferred.

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19
(h) Regression functions
(2) Black line
(3) Blue dashed
(4) Red dots
The regression functions from (2) and (3) are similar. The quadratic regression (4)
shows more curvature. The regression functions for a female with a high school
diploma will look just like these, but they will be shifted by the amount of the
coefficient on the binary regressor Female. The regression functions for workers with
a bachelor’s degree will also look just like these, but they would be shifted by the
amount of the coefficient on the binary variable Bachelor.
(i) This regression is shown in column (5). The coefficient on the interaction term
Female  Bachelor shows the “extra effect” of Bachelor on ln(AHE) for women
relative the effect for men.
Predicted values of ln(AHE):
Alexis: 0.135 30 −0.0019×302
− 0.191 + 0.451+ 0.023×1 + 0.41 = 
Jane: 0.135 30 −0.0019×302
− 0.191 + 0.450+ +0.023×0 + 0.41 = 2.56
Bob: 0.135 30 −0.0019×302
− 0.190 + 0.451+ 0.023×0 + 0.41 = 3.20
Jim: 0.135 30 −0.0019×302
− 0.190 + 0.450+ 0.023×1 + 0.41 = 2.75
Difference in ln(AHE): Alexis − Jane = 3.03 − 2.56 = 0.47
Difference in ln(AHE): Bob − Jim = 3.20 − 2.75 = 0.45

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20
Notice that the difference in the difference predicted effects is 0.47 − 0.45 = 0.02,
which is the value of the coefficient on the interaction term.
(j) This regression is shown in (6), which includes two additional regressors: the
interactions of Female and the age variables, Age and Age2
. The F-statistic testing the
restriction that the coefficients on these interaction terms is equal to zero is F = 2.64
with a p-value of 0.07. This implies that there is statistically significant evidence at
the 10% but not 5% level that there is a different effect of Age on ln(AHE) for men
and women.
(k) This regression is shown in (7), which includes two additional regressors that are
interactions of Bachelor and the age variables, Age and Age2
. The F-statistic testing
the restriction that the coefficients on these interaction terms is zero is 1.03 with a p-
value of 0.36. This implies that there not is statistically significant evidence (at the
10% level) that there is a different effect of Age on ln(AHE) for high school and
college graduates.
(l) The estimated regressions suggest that earnings increase as workers age from 25–34,
the range of age studied in this sample. Education and sex are significant predictors
of earnings, and there are statistically significant interaction effects between age and
sex. The figure below shows the regressions predicted value of ln(AHE) for male and
females with high school and college degrees from (6)
Green (line): male with college degree
Red (dots): female with college degree
Blue (dashed): male without college degree
Black (line): female without college degree

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21
The table below summarizes the regressions predictions for increases in earnings as a
person ages from 25 to 32 and 32 to 34.
Gender, Education
Predicted ln
(AHE) at Age
Predicted Increase
in ln(AHE)
(In percentage
points per year)
25 32 34 25 to 32 32 to 34
Females, High School 2.46 2.60 2.59 2.0 -0.2
Males, High School 2.60 2.81 2.83 3.0 0.9
Females, BA 2.91 3.09 3.11 2.5 1.0
Males, BA 3.03 3.28 3.32 3.5 2.2
Earnings for those with a college education are higher than those with a high school degree, and
earnings of the college educated increase more rapidly early in their careers (age 25–34).
Earnings for men are higher than those of women, and earnings of men increase more rapidly
early in their careers (age 25–34). For all categories of workers (men/women, high
school/college) earnings increase more rapidly from age 25–32 than from 32–34.

Introduction to Econometrics Answers Chapter 8 Exercises

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