# Practice Final—Spring 2021 Prof. Beheshti April 27, 2021 Define and Discuss Problem 1. For each of the following terms provide the mathematical definition (if appli- cable). Then, explain the terms...

Practice Final—Spring 2021
Prof. Beheshti
April 27, 2021
Define and Discuss
Problem 1. For each of the following terms provide the mathematical definition (if appli-
cable). Then, explain the terms like you were talking to a smart non-economist. For the
explanation, use no more than two sentences. (2 points each).
1. Exclusion restriction (in context of IV)
2. Instrumental variable
3. Pre-trend
4. First stage
5. Covariance
1
True/False/Uncertain For each of the following statements, decide whether
it is true, false, or uncertain. Justify your answer. (2 points each). Note: Your score will be
Problem 2. For the next several problems, consider the following regression model
Y = β0 + β1X + � (1)
where we are concerned that E[�|X] 6= 0, and Z is an instrument for X.
1. If Corr(X,Z) < 0, then Z is an invalid instrument.
2. For Z to be a valid instrument, we must assume that Z only affects Y through its
impact on X.
3. The IV estimator for β1 is obtained by regressing Y on �̂.
For the next several problems, consider a difference-in-differences model of the form
Y = β0 + β1 · Post+ β2 · Treated+ β3 · Post · Treated+ � (2)
4. We can evaluate the parallel trends assumption by checking whether β̂0 = 0.
5. β̂3 will measure the causal effect of treatment even if the parallel trends assumption is
not satisfied.
6. The parallel trends assumption is that, before the treatment, the treated and control
groups were trending in parallel.
2
Application and Interpretation
Figure 1 on the following page presents a screenshot from RStudio. The dataset ”USSeat-
Belts” contains panel data on each state from 1983 to 1997. The variable fatalities is number
of traffic fatalities per million miles driven. The variable drinkage is a binary variable that
equals 1 if the minimum drinking age is 21 and 0 if the minimum drinking age is 18. The
variable state is a factor variable that reports the state.
Problem 7. Write down the regression being estimated in line 8.
Problem 8. Interpret the value XXXXXXXXXXIs this necessarily the causal effect of increasing
the minimum drinking age from 18 to 21? Why or why not?
Problem 9. How does the regression in line 11 differ from the regression in line 8? What
problems does this solve relative to the first regression? Is the coefficient on drinkage from
this regression necessarily the causal effect of increasing the minimum drinking age from 18
to 21? Why or why not?
For the next several problems consider the following scenario. You want to estimate how
receiving an opioid prescription (think OxyContin, Vicodin, Percocet) affects the probability
that someone uses heroin in the future. Since opioid consumption is likely endogenous, you
instrument for receiving an opioid prescription with an indicator variable for whether your
primary care physician is a high prescriber (some doctors prescribe far more opioids than
others).
Background note: opioids are powerful painkillers that are commonly misused for their
recreational effects (i.e., to get high).
Problem 10. Why might it be a bad idea to just regress heroin use on past prescription
opioid consumption?
Problem 11. Explain what the relevance condition means in this context. (2 points)
Problem 12. Explain what the exclusion restriction means in this context. (2 points)
Problem 13. Can you think of any potential violations of the exclusion restriction? (2
points)
3
Figure 1: RStudio Screenshot
4
Derivations
Problem 14. For the following question, you may assume SLR.1-SLR.4. Suppose the true
regression model is
Y = β0 + β1X + �
Derive the OLS estimator for β1.
Problem 15. Consider the following 2SLS regression model
Y = β0 + β1X + � (3)
and we instrument for X with Z. That is, the first-stage regression is
X = α0 + α1Z + η (4)
and the reduced form regression is
Y = δ0 + δ1Z + µ (5)
Assume Z is a valid instrument. Show that β1 =
δ1
α1
. Hint: start by expanding the covariance
of Y and Z.
5
May 10, 2021

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