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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

determined entirely by your justification.

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

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

determined entirely by your justification.

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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