Econometrics

Assignment 3:
Be sure that you explicitly answer each question and explain each step, as if you were writing
solutions so that another student in the class would be able to follow your thoughts. Part of
your grade will depend on explaining each step of your answers.
1. You are interested in the effect of attending college on income. For n people, you are
able to observe:
Yi
: person i’s income
Di
: a dummy variable equal to 1 if person i attended college, and zero otherwise.
(a) (1 point) You run an OLS regression of Yi
. on a constant and Di
. Write the formula
for βˆ
1, the estimated coefficient on Di
, in terms of Di and Yi
.
(b) (2 points) Explain the formula in terms of variance and covariance, specifically
describing it in terms of income and college.
(c) (2 points) You run the regression and find that βˆ
1=$40,000. Interpret this coefficient.
(d) (3 points) Your estimate of βˆ
1 is significant at the 5% level. Can you conclude that
attending college increases income? Why or why not?
(e) (3 points) Now suppose you can observe fi
, an indicator for whether person i is
female. For the purpose of the question, make two assumptions:
1) Women are more likely than men to attend college
2) Women tend to earn lower incomes than men
How will including fi
in your regression affect your estimate of βˆ
1?
(f) (2 points) What is the attribution problem in the context of this question?
(g) (2 points) Suppose again that you observe both fi
, and indicator for whether person
i is female. You regress Yi on a constant, Di
, fi
, and the interaction term Di ∗ fi
.
Explain what information you can learn from the coefficient on the interaction
term.
2. You are teaching 140A, and you receive a grant that allows you to provide 50% of your
students with a tutor for one hour per week. You want to know how tutoring affects
students’ grades in the course.(a) (2 points) Suppose you assign tutors to the students will the lowest GPAs. Will
you be able to determine the effect of a tutor on students’ grades?
(b) (1 point) How should you assign tutors so you can find the effect of a tutor on
course performance?
(c) (2 points) With your treatment assignment mechanism from part b, will you encounter the attribution problem? Explain.
(d) (3 points) You regress Yi (student i’s grade in 140A) on Di (a dummy variable equal
to 1 if student i received tutoring, and 0 otherwise). Under what circumstances
would you expect βˆ
1, the coefficient on Di
, to be close to the average treatment
effect on the treated? Explain.
(e) (5 points) Show that with random assignment, βˆ
1 is an unbiased estimator of the
average treatment effect on the treated. You must justify every step to receive full
credit.
(f) (2 points) Suppose there are 400 students in 140A. What is the approximate distribution of βˆ√1−AT T
V (βˆ
1)
? What theorem tells us the approximate distribution?
(g) (2 points) Under random assignment, is βˆ
1 an unbiased estimator of the average
treatment effect (ATE)?
AT E =
1
nΣ
n
i=1E(Y (1) − Y (0))
3. Newspaper articles, Twitter posts, and politician interviews have all given statistics to
back their claims. However, after taking 140A, you should be able to spot when these
claims and statistics are extremely unreliable. The following is an excerpt from an article
from cnbc.com titled “Here’s how much more money you could make just from joining
a frat”:
Joining a fraternity can cost you over $1,000 extra dollars a year in college, but odds
are that you could end up seeing a return on that investment.In a paper titled “Social
Animal House: The Economic and Academic Consequences of Fraternity Membership,”
researchers from Union College found that going Greek raises your income by 36 percent
down the line. The academic cost, meanwhile, is a small one: a 0.25 point drop in GPA on
the traditional 4-point-scale. In other words, you might graduate with a 2.75 GPA while
your classmate who didn’t pledge ends up with a 3.00. But if he makes $75,000 a year,
you could expect to make $102,000. These results suggest that fraternity membership
causally produces large gains in social capital, which more than outweigh its negative
effects on human capital,” the researchers write.
To those that are unfamiliar, a fraternity is an organization traditionally of men associated together for various interests. These organizations typically require an extensive
process to be accepted into the fraternity and membership is completely voluntary. The
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