End term exam for EC9A3
This test has FOUR exercises:
Exercise 1 (25 marks) Reprove bounds for ATE
Exercise 2 (30 marks) Small questions on LATE
Exercise 3 (35 marks) Measurement error and 2SLS
Exercise 4 (10 marks) Asymptotic distribution of a ratio
Exercise 1 (25 marks). Reprove the following theorem we saw together in the notes:
Theorem 0.0.1 Bounds for the ATE with binary outcome
If Yi is binary, and Zi satises random instrument and exclusion restriction, then
B− ≤ AT E ≤ B+ ,
where
B− = P (Di = 1|Zi = 1)E(Yi |Di = 1, Zi = 1)
− (P (Di = 0|Zi = 0)E(Yi |Di = 0, Zi = 0) + P (Di = 1|Zi = 0))
and
B+ = P (Di = 1|Zi = 1)E(Yi |Di = 1, Zi = 1) + P (Di = 0|Zi = 1)
− P (Di = 0|Zi = 0)E(Yi |Di = 0, Zi = 0).
Exercise 2 (30 marks): Short questions on LATE
Assume you are writing a paper trying to measure the eect of some treatment Di on an
outcome Yi . On that purpose you have found a binary instrument Zi .
a) (5 marks) For the LATE theorem to apply, Assumption 14 in the notes needs to hold.
This assumption requires that the instrument Zi satisfy 4 conditions. Give the mathematical
expression of these 4 conditions.
b) (5 marks) One of these 4 conditions is that the instrument should be as good as randomly
assigned. What should you do to convince yourself and the reader of the paper that the
instrument is indeed as good as randomly assigned? Your answer should be less than
ve lines long.
c) (5 marks) Assume that the test you suggested in b) conrmed your intuition that indeed
the instrument is as good as randomly assigned, and you have good reasons to believe that
the instrument also satises the three other conditions listed in a). Which regression should
you run to estimate the eect of Di on Yi ? Specically, which treatment eect parameter will
this regression estimate?
d) (10 marks) In this question, let's assume that you are interested in measuring the eect of
a job training program on the probability that unemployed people will nd a job in less than
1
6 months. You do not nd it ethical to run a randomized experiment in which you prevent
the control group from following the training. Therefore, you come up with the following idea:
you will randomly select a group of unemployed to which you will send a letter letting them
know about this training (content of the training, how to enrol, etc.) and telling them that
if they attend the training, they will receive 200 GBP. Unemployed not randomly selected to
be part of this group can attend the training program, but they will not receive the letter,
and they will not receive 200 GBP if they follow the training. The rst group of unemployed
is called the encouragement group, and the second group is called the control group.1 In this
context, Yi is a dummy for whether unemployed i nds a job in less than 6 months, Di is a
dummy for whether she follows the training, and Zi is a dummy equal to 1 if she is in the
encouragement group. Do you think Zi will satisfy the four requirements of Assumption 14?
Your answer should be less than ten lines long.
e) (5 marks) During the lectures, we said that in general, we cannot say which units in the
sample are compliers. When you have a binary instrument such that P (Di = 1|Zi = 0) = 0,
for which units in the population can you say with certainty that they are compliers? Your
answer should be less than ve lines long.
Exercise 3 (35 marks). Measurement error, and the eect of police on crime (inspired from
Chaln and McCrary, 2013)
Assume you want to measure the eect of police on crime. On that purpose, you are going
to run a regression in which your dependent variable will be Yit , the logarithm of the number
of crimes in city i at time t, and your main explanatory variable will be Pit , the logarithm
of the number of police forces in city i at time t. For now, we do not discuss the causal
interpretation of that regression. Let Yit = α + βPit + eit denote the regression decomposition
of Yit with respect to Pit . eit is a regression residual, so by construction it is uncorrelated with
it ,Pit )
Pit . Moroever, we have β = cov(Y
V (Pit ) . Do not forget this regression decomposition,
you will need to use it in the next questions!
β is what you would like to be able to measure. Unfortunately, police forces are not well
measured, and you do not observe Pit but you have two imperfect measures of it, Pit1 and
Pit2 .
Assumption 1 The random measurement error model
1
Randomized experiments where the treatment group is only encouraged to receive the treatment are called
encouragement designs.
2
We say that two proxies Pit1 and Pit2 for Pit satisfy a random measurement error model if:
Pit1 = Pit + uit1
Pit2 = Pit + uit2
E(uit1 ) = E(uit2 ) = 0
V (uit1 ) > 0, V (uit2 ) > 0
(uit1 , uit2 ) ⊥⊥ (Yit , Pit )
uit1 ⊥⊥ uit2 .
uit1 is the dierence between Pit1 and Pit . Similarly, uit2 is the dierence between Pit2 and
Pit . Therefore, uit1 and uit2 are measurement errors. For the random measurement error
assumption to be satised, uit1 and uit2 should have mean 0 and strictly positive variances,
they should be independent of Pit and Yit , and they should be independent of each other.
a) (5 marks) Show that if the random measurement error assumption is satised, the population coecient of Pit1 in an OLS regression of Yit on Pit1 is not equal to β . Show that the
same is true for the population coecient of Pit2 in an OLS regression of Yit on Pit2 .
b) (7 marks) Show that if the random measurement error assumption is satised, the population coecient of Pit1 in a 2SLS regression of Yit on Pit1 using Pit2 as an instrument for Pit1
is equal to β . Show that under those assumptions, the population coecient of Pit2 in a 2SLS
regression of Yit on Pit2 using Pit1 as an instrument for Pit2 is also equal to β .
c) (3 marks) Explain the intuition of the result in question b).
d)
d.1) (5 marks) Show that if the random measurement error assumption is satised, then the
following 4 equations must hold:
E [Yit − α − βPit1 ] = 0
E [Yit − α − βPit2 ] = 0
E [Pit2 (Yit − α − βPit1 )] = 0
E [Pit1 (Yit − α − βPit2 )] = 0
d.2) (3 marks) Based on what you have shown in question d.1), give the name of an estimation
method dierent from 2SLS which you could use to estimate (α, β). You just need to give
the name of that estimation method.
d.3) (2 marks) Based on what you have shown in question d.1), give the name of a statistical
test you could use to test the random measurement error assumption. You just need to
give the name of this statistical test.
Introduction to the next questions
3
Chaln and McCrary include city xed eects and year xed eects in their regressions. If
police forces were properly measured, this type of regression would estimate the causal eect
of police on crime under reasonable assumptions which you do not need to know to be able
to answer the next questions. Let's just assume henceforth that the coecient of Pit in a
regression of Yit on Pit and city and year xed eects captures the causal eect of police on
crime. Surprisingly, papers using these regressions have consistently found very low eects of
police on crime. McCrary argue that this is because police forces are measured with error,
so instead of regressing Yit on Pit and city and year xed eects, these papers are actually
regressing Yit on Pit1 and city and year xed eects, where Pit1 is a noisy measure of Pit .
To substantiate their claim, they use a data set with two noisy measures of police forces,
and follow the approach we described in the previous questions. In questions a) and b), we
have considered simple univariate regressions of Yit on Pit , Pit1 , or Pit2 , but you can take for
granted without reproving it that the results of these questions still apply to more complicated
regressions with city and year xed eects.
Chaln and McCrary start estimating an OLS regression of Yit on Pit1 (plus city and year
xed eects). The coecient of Pit1 is −0.117, with a standard error of 0.037. Then, Chaln
and McCrary estimate a 2SLS regression of Yit on Pit1 (plus city and year xed eects), using
Pit2 (plus city and year xed eects) as an instrument for Pit1 . The coecient of Pit1 is now
equal to −0.361, with a standard error of 0.143. They nally estimate a 2SLS regression of
Yit on Pit2 (plus city and year xed eects), using Pit1 (plus city and year xed eects) as an
instrument for Pit2 . This gives them a coecient of −0.336, with a standard error of 0.106.
e) (4 marks) Why is the absolute value of the 2SLS coecient of Pit1 (0.361) much greater
than that of the OLS coecient of Pit1 (0.117)?
f) (4 marks) Given the value of the 2 2SLS coecients, do you think that the test mentioned
in question d.3) will be rejected?
g) (2 marks) Based on the 2SLS coecient of Pit1 , what is the eect of police on crime? (you
should reply with a sentence like: increasing policy forces by xxx increases/reduces crime by
xxx).
Exercise 4 (10 marks). Asymptotic distribution of a ratio.
Assume you are interested in some parameter θ = αβ , with α 6= 0. Assume also that you have
been able to nd an estimator βb for β and an estimator αb for α such that
√
n(b
α − α, βb − β)0 ,→ N (0, V ),
where V is a 2 × 2 matrix. Let θb = αβb .
√
a) (5 marks) Use a theorem we saw in the notes to show that n(θb− θ) also converges towards
a normal distribution with mean 0. Justify carefully why you can use this theorem here.
√
b) (5 marks) write the asymptotic variance of n(θb − θ) as a function of α, β and V .
b
4