Week 1: OLS & Experiments

Solutions to exercises from Stock & Watson 4e

(not discussed in tutorial)

Question 7.7
a) The t-statistic is 0.567−0
1.23
= 0.461 < 1.96. Therefore, the coefficient on BDR is not statis-
tically significantly different from zero. [KC7.1]

b) The coefficient on BDR measures the partial effect of the number of bedrooms holding
house size (Hsize) constant. Yet, the typical 4-bedroom house is larger than the typical
3-bedroom house. Thus, the results in (a) says little about the conventional wisdom.

c) The 95% confidence interval for the effect of lot size on price is 2500 × [.005 ± 1.96 ×
.00072] or 8.97 to 16.03 (in thousands of dollars). [KC7.2]

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 5 instead of 0.005.

e) The 10% critical value from the F(2,∞) distribution is 2.30. Because 2.38 > 2.30, the
coefficients are jointly significant at the 10% level. [CH7.2]

Question 8.7
a)

i. ln(Earnings) for females are, on average, 0.44 lower for women than for men, so the
wage gap is 44%.
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 discrimi-
nation 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 char-
acteristics 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

1
 that will lead to bias in the OLS coefficient estimator for Female. Since these char-
acteristics 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
iii. Forgetting about the effect of Return, whose effect seems small and statistically
insignificant, the omitted variable bias formula (see equation (6.1)) suggests that
Female is negatively correlated with ln(MarketValue)

Question 13.2
a) On average, a student in class A (the “small class”) is expected to score higher than a
student in class B (the “regular class”) by 15.93 points with a standard error 4.08. The
90% confidence interval for the predicted difference in average test scores is 15.93 ± 1.64
× 4.08 = [9.24, 22.62].

b) On average, a student in class A taught by a teacher with 6 years of experience is expected

to score lower than a student in class B taught by a teacher with 12 years of experience
by 0.74 × 6 = 4.44 points. The standard error for the score difference is 0.35 × 6 = 2.1.
The 95% confidence interval for the predicted lower score for students in classroom A is
4.44 ± 1.96 × 2.1 = [0.32, 8.56]. [Covariance rules KC 2.4]

c) The expected difference in average test scores in 15.93 + 0.74 × (-6) = 11.49. Because
of random assignment, the estimators of the small class effect and the teacher experience
effect are uncorreleated. Thus, the standard error for the difference in average test scores is
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[4.082 + (−6)2 × 0.352 ] 2 = 4.59. The 95% confidence interval for the predicted difference
in average test scores in classrooms A and B is 11.49 ± 1.96 × 4.59 = [2.50, 20.48].
[Covariance rules KC 2.4]

d) This question made sense in the 3e/u3e versions of the book because the intercept is
present there in cloumns (1) and (2). In the 4e of the book, the intercept in Table 13.2 is
surpressed, so the question doesnt make any sense there. Still one can pose the question
if the intercept be included in col (3).
The intercept is not included in the regression to avoid the perfect multicollinearity prob-
lem that exists among the intercept and school indicator variables. In fact this experiment
randomizes students to classes within school, not between schools. Therefore all regres-
sions should include school dummies, an example of CMI. Because each school (roughly)
has as many Regular, Reg+TA, and Small classes (because each school must have at least
one class of each type), the estimated treatment effects are not sensitive to the inclusion
of the school dummies. The TeacherExperience (TYE) variable, however, is, and drops
from 1.47→0.74. Teachers are not randomized over schools, the teachers are randomized
within schools. A raw TYE comparison compares kids from good neighborhood schools
(where experieced teachers go) to teachers from bad neighborhood school (where the less
experienced teachers go). Including school FE removes this selection bias.
Question 13.3
a) The estimated average treatment effect is 1348 - 1395 = -47 points.
 
b) For the treatment group we have M ale = 0.6 with standard error SE M ale = s.d.(M √ ale) =
n
√
q
M ale(1−M ale) 0.6(1−0.6)
√
100
= 10
= 0.049 (using KC3.4 at the first step and eq2.7 at the second,
the standard deviation of a binary variable). For q the control group we have M ale = 0.4
M ale(1−M ale)
√
0.4(1−0.4)
 
SD(M ale)
with standard error SE M ale = √
n
= √
100
= 10
= 0.049. To
T C
M ale −M ale = 0.6−0.4 = 0.2 is significant we cal-
test whether the difference in means r
T C T 2 C 2 T C
       
culate the SE M ale − M ale = SE M ale + SE M ale − 2cov M ale , M ale =
√  T C

0.0492 + 0.0492 = 0.069 (assuming independent samples so cov M ale , M ale = 0).
The t-test then gives t = | 0.20−0
0.069
| = 2.89 > 1.96, so we reject the hypothesis that there is
no significant difference in the population, indication that the assignment was non ran-
dom (at least with regard to gender).

Question: How could we get this estimate and test by regression?

T C
 on the treatment dummy will give the β̂ = M ale −M ale =
A simple regression of Male
0.6 − 0.4 = 0.2 with SE β̂ = 0.069

Question 13.5
a) This is an example of attrition, which poses a threat to internal validity. After the male
athletes leave the experiment, the remaining subjects are representative of a population
that excludes male athletes. If the average causal effect for this population is the same
as the average causal effect for the population that includes the male athletes, then the
attrition does not affect the internal validity of the experiment. On the other hand, if the
average causal effect for male athletes differs from the rest of population, internal validity
has been compromised.

b) This is an example of partial compliance which is a threat to internal validity. The local
area network is a failure to follow treatment protocol, and this leads to bias in the OLS
estimator of the average causal effect.

c) This poses no threat to internal validity. As stated, the study is focused on the effect of
dorm room Internet connections. The treatment is making the connections available in
the room; the treatment is not the use of the Internet. Thus, the art majors received the
treatment (although they chose not to use the Internet).

d) As in part (b) this is an example of partial compliance. Failure to follow treatment

protocol leads to bias in the OLS estimator.