I would need help with pages 2-5 tomorrow from 1pm - 2:30pm central time, we are allowed to be in groups but I do not know anyone in my class. If anyone could help me that would be much appreciated!!
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 -0.0056781. is 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 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="" -0.0056781.="" is="" 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.="">