Answer each of the following questions with True, False, or
Uncertain and justify your answers with a concise argument or a proof if necessary. You
may cite theorems proved in class but you must show precisely how you are applying
the theorem and check that the assumptions required for the theorem hold. Correct
answers without justification will receive minimal credit.
(a) (5 points) Let X ∼ N(0, 1) and suppose Y is another random variable equal to
X
2 − 1. Then Y ∼ N(−1,
1
2
).
(b) (5 points) In a univariate regression, Y = β0 + β1X + U, a violation of the (A2)0
assumption that cov(X, U) = 0 will lead to inconsistent estimation of β1 unless
we have a large sample (i.e., n → ∞)
(c) (5 points) The residuals from an OLS regression will sum exactly to zero.
(d) (5 points) A one-sided hypothesis test of H0 : β1 = 0 in a univariate regression
will have a different critical value than if it were a two-sided hypothesis test.
(e) (5 points) A regression of Y on X will deliver the same slope estimate as a
regression of X on Y .
(f) (5 points) In a multivariate regression, Y = β0+β1X1+β2X2+U, multicollinearity will arise if cov(X1, X2) > 0.
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2. (35 POINTS) In the United States, approximately half a million individuals are
detained prior to trial each year. The high rate of pre-trial detention, particularly for
poor and minority defendants, has contributed to a debate about the effectiveness of
the bail system. Critics argue, for example, that excessive bail conditions and pretrial detention puts pressure on defendants to plead guilty. Others claim that the bail
system is functioning properly and that releasing more defendants will increase pretrial flight and crime rates. A recent paper by Dobbie, Goldin, and Yang explores this
question with data on individual cases, linked to individuals’ subsequent trial, crime,
and employment outcomes. Consider the following regression:
Pleadi = β0 + β1Releasedi + Ui (1)
where Pleadi
is a dummy variable taking a value of 1 if individual i pleads guilty at
her eventual trial and 0 otherwise, and Releasedi takes a value of 1 if individual i is
released by the judge at the pre-trial hearing and 0 otherwise.
(a) (5 points) Do you think cov(Ui
, Releasedi) = 0? Why or why not?
Because of the concern for omitted variables, the researchers adopt an instrumental variables strategy, which exploits the random assignment of cases to judges
in bail hearings. For simplicity, suppose there are two judges and that the researchers have data on these judges’ release rates during a period of time that
precedes the sample period. Define an instrument Zi
, which is the pre-period
release rate of individual i’s judge, so Zi =
z if assigned to judge 1
z¯ if assigned to judge 2
(b) (10 points) What are the two conditions are required to hold if Zi
is a valid
instrument. (Please state both in math and in words what the conditions mean
in this context.)
(c) (5 points) Do you think the exogeneity condition is likely to hold in this context?
Why or why not?
(d) (5 points) Suppose z = ¯z, so that the two judges are no different in their release
rates. Which of the two IV conditions would fail to hold?
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(e) (5 points) Suppose high-income defendants are able to game the system and
bribe the clerk so that they are assigned the judge with the higher release rate.
Which of the two IV conditions would fail to hold?
(f) (5 points) Suppose the two conditions for a valid instrument are satisfied. Now,
the researchers’ carry out their instrumental variables regression and obtain an
estimate: βˆIV
1 = −.09. How would you interpret this result and the magnitude
of the estimate. (For context, the fraction of defendants who plead guilty in the
researchers’ sample is .44.)
3. (30 POINTS TOTAL) Many states permit employers to ask job applicants whether
they have been convicted of a crime in the past, while other states have enacted “banthe-box” legislation, which forbids employers from asking job candidates whether they
have a criminal conviction. “Ban-the-box”legislation is intended, in part, to reduce discrimination against African-American job applicants, who are more likely to have prior
conviction compared to white applicants. Economists have pointed out that “ban-thebox”legislation may actually lead to more discrimination against African-American job
applicants. The argument is that removing information about job applicants’ criminal
histories could lead employers who don’t want to hire ex-offenders to guess who the
ex-offenders are, and avoid interviewing them.
In this problem we will consider empirical work by economists Jennifer Doleac and
Benjamin Hansen on this question. They collect data on several states, some of which
have adopted “ban-the-box” policies, and they collect demographic and employment
data for young men with less than a college degree. They estimates the following
regression:
Employedi = α + β1BT Bi + β2AfAmi + β3(BT Bi × AfAmi) + i
where Employedi takes a value of 1 if individual i is employed and 0 otherwise; BT Bi
takes a value of 1 if individual i lives in a “ban-the-box” state and 0 otherwise; and
AfAmi takes a value of 1 if an individual is African-American, and 0 otherwise.
(a) (5 points) What null hypothesis would you test if you wished to see if AfricanAmerican men have lower employment rates in states that do not have “ban-thebox” policies in place, compared to non-African-American men?
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(b) (5 points) What null hypothesis would you test if you wished to see if the “banthe-box” policy appears to negatively affect the employment of African-American
men, compared to the effect of the policy on non-African-American men?
(c) (5 points) The researchers estimate βˆ
3 = −.07 with SE(βˆ
3) = .01. Interpret the
result and the magnitude of the estimate. The sample size is n = 503, 419.
(d) (5 points) Perform a two-sided test that the “ban-the-box” policy has a negative effect on employment for African-American men compared to non-AfricanAmerican men. State your test statistic and its distribution. Do you reject your
null hypothesis at the 5% significance level? (Hint: if W ∼ t, then P r{W ≤
1.96} = .975 and P r{W ≤ 1.645} = .95).
(e) (5 points) Suppose that there was actually a mistake in the researchers’ code,
and the standard error is actually SE(βˆ
3) = .07. Would we be able to reject the
hypothesis of (d)?
(f) (5 points) Suppose we wish to test the null hypothesis that the “ban-the-box”
policy has no effect on employment both for African-American and non-African
American men? State the null hypothesis and discuss the procedure for constructing your test statistic.