Instrumental Variables continued

Instrumental Variables Continued

• Review
• Estimation and Inference
• IV in Potential outcomes framework: LATE

Endogeneity Problem

• Suppose effect of \(X\) on \(Y\) can be represented as \[Y=\beta_0+\beta_1X + u\]

• But with endogeneity \[E[Xu]\neq 0\]

• Can arise from omitted confounders
• OLS inconsistent for \(\beta_1\)
  • Finds slope of (Best Linear Predictor of) Conditional Expectation Function
  • But this isn’t what we want

A Possible Solution: Instrumental Variables

• Find instrument \(Z\) such that \[Cov(Z,u)=0\] \[Cov(Z,X)\neq 0\] \[E[u]=0\]
• \(Z\) correlated with \(X\) but not residual \(u\)
• \(Z\) affects treatment only
  • No direct effect on \(Y\)
  • No correlation with omitted confounders
• Then \(\beta_1=\frac{Cov(Z,Y)}{Cov(Z,X)}\)
• Estimate by Instrumental Variables Estimator \[\hat{\beta}_1^{IV}=\frac{\frac{1}{n}\sum_{i=1}^{n}(Z_i-\bar{Z})(Y_i-\bar{Y})}{\frac{1}{n}\sum_{i=1}^{n}(Z_i-\bar{Z})(X_i-\bar{X})}\]

IV assumptions

• Assume the following
• (IV1) Linear Model \[Y=\beta_0+\beta_1X + u\]
• (IV2) Random sampling: \((Y_i,X_i,Z_i)\) drawn i.i.d. from population satisfying linear model assumptions
• (IV3) Relevance \[Cov(Z,X)\neq 0\]
• (IV4(i)) Exogeneity \[Cov(Z,u)=0\]
• (IV4(ii)) Mean 0 errors \[E[u]=0\]
• Sometimes also assume (IV5) Homoskedasticity \[E[u^2|Z]=\sigma^2\] (\(\sigma^2\) a finite nonzero constant)

IV asymptotics

• IV estimator \(\hat{\beta}_1^{IV}\) is consistent for \(\beta_1\) under (IV1)-(IV4(i))
• IV estimator of constant term \(\hat{\beta}_0^{IV}\) consistent under (IV1)-(IV4)
• IV estimator \(\hat{\beta}^{IV}\) asymptotically normal under (IV1)-(IV4)
• Standard errors of IV estimator derived in usual way using CLT under (IV1)-(IV5)
• Formula is in book, and applied by software
  • ivreg in AER library
• Different formula if residuals are heteroskedastic
  • (IV1)-(IV4) hold, but (IV5) doesn’t
  • Obtain from vcovHC command in sandwich library

Example application: Estimating a demand curve

• Quantity demanded \(Y\) is causally affected by price \(X\)
  • But also other things, including shifts in demand curve
• Want estimate of effect of price on demand
  • Sometimes firms experiment with prices, but rarely
• Instead, can find instrument \(Z\) for price
  • \(Z\) relevant if it affects price
  • \(Z\) exogenous if independent of any shifts in demand curve
• Need something which independently shifts supply curve
• Idea: instrument \(Z\) using
  • Random change in tax on item
  • Random change in cost of inputs to production

Example: Cigarette demand

• Let’s estimate a demand curve for cigarettes across states (1995 data)
• Use cigarette tax increases as instrument
  • Works if tax changes determined by state fiscal needs unrelated to cigarette demand
• Price elasticity estimated by IV regression of log sales on log price, using tax difference as instrument

ivreg(log(packs) ~ log(rprice) | tdiff, data = c1995)

• Decompose into reduced form: effect of tax on quantity demanded, divided by first stage: effect of taxes on prices
• OLS version of estimate gives correlation of price and quantity, but could come from supply or demand

Example: Cigarette demand (Code)

#Load library containing IV command 'ivreg'
library(AER)
# Load data on cigarette prices and quantities
data("CigarettesSW")
#Use real prices as X
CigarettesSW$rprice <- with(CigarettesSW, price/cpi)
#Use changes in cigarette tax as
# instrument which shifts supply curve
CigarettesSW$tdiff <- with(CigarettesSW, (taxs - tax)/cpi)
#data from different states in 1995
c1995 <- subset(CigarettesSW, year == "1995")

IV estimate of demand elasticity for cigarettes (code)

#To get IV estimate of effect of x on y using
#  z as instrument syntax is ivreg(y ~ x | z)
# Effect of log(price) on log(quantity)
# Elasticity
fm_ivreg <- ivreg(log(packs) ~ log(rprice)
                  | tdiff, data = c1995)
#Obtain (robust) standard errors
fm_ivreg.results<-coeftest(fm_ivreg,
         vcov = vcovHC(fm_ivreg, type = "HC0"))

OLS estimate of demand elasticity for cigarettes (code)

#OLS estimate of effect of log(price)
 #on log(quantity)  with no instrument
fm_ols <- lm(formula= log(packs) ~
               log(rprice), data = c1995)
#Obtain (robust) standard errors
fm_ols.results<-coeftest(fm_ols,
    vcov = vcovHC(fm_ols, type = "HC0"))

Components of IV estimates of demand elasticity for cigarettes (code)

# First stage: z on x
firststage<-lm(formula= log(rprice) ~
                 tdiff, data = c1995)
firststage.results<-coeftest(firststage,
    vcov = vcovHC(firststage, type = "HC0"))
# Reduced form
reducedform<-lm(formula= log(packs) ~
                tdiff, data = c1995)
reducedform.results<-coeftest(reducedform,
         vcov = vcovHC(reducedform, type = "HC0"))

Results of IV estimates of demand elasticity for cigarettes (code)

#Display results in big fancy table
library(stargazer)
stargazer(fm_ols.results,fm_ivreg.results,
          reducedform.results,firststage.results,
    type="html", 
    title="Cigarette Demand Elasticity: OLS, 
      IV, Reduced Form, First Stage",
    header=FALSE,
    column.labels=c("log(packs)","log(packs)",
                    "log(packs)","log(price)"),
    omit.table.layout="n")

Results of IV estimates of demand elasticity for cigarettes

Cigarette Demand Elasticity: OLS, IV, Reduced Form, First Stage

	Dependent variable:
	log(packs)	log(packs)	log(packs)	log(price)
	(1)	(2)	(3)	(4)
log(rprice)	-1.213***	-1.084***
	(0.190)	(0.312)
tdiff			-0.033***	0.031***
			(0.010)	(0.005)
Constant	10.339***	9.720***	4.717***	4.617***
	(0.915)	(1.496)	(0.064)	(0.028)

Results

• OLS says 1% difference in prices associated with 1.2% lower cigarette consumption
  • Not causal effect of changing prices
• IV says 1% increase in prices (due to change in tax) leads to 1% lower cigarette consumption
• Justification: states that saw 1 % point increase in tax saw
  • 3% increase in prices (First stage)
  • 3% decrease in sales (Reduced form)
• If tax increases effectively random, can interpret both as causal
• If tax increases affect demand only through prices, ratio is effect of price on quantity
• IV elasticity \(\neq\) OLS estimate: states with higher cig prices also had other characteristics which reduced cigarette sales

Potential Outcomes Again

• Let’s extend heterogeneous treatment effects model to IV case
• Recall potential outcomes model of causal effect of binary variable \(X\in\{0,1\}\) on outcome \(Y\) \[Y_i=Y^0_i(1-X_i)+Y^1_iX_i\]
• \(Y^0_i\) is value of \(Y_i\) given do \(X_i=0\)
• \(Y^1_i\) is value of \(Y_i\) given do \(X_i=1\)
• E.g. \(X_i\) is a drug, \(Y_i\) is a health outcome

• Difference of conditional averages converges to difference in conditional expectations \[\frac{1}{n_1}\sum_{i=1}^{n}Y_iX_i-\frac{1}{n_0}\sum_{i=1}^{n}Y_i(1-X_i)\overset{p}{\to} E[Y_i|X_i=1]-E[Y_i|X_i=0]\]

  • Same as slope in regression of \(X\) on \(Y\)
• If \(X_i\) independent of \((Y^0_i,Y^1_i)\), e.g. because \(X_i\) assigned randomly difference becomes \(E[Y^1_i-Y^0_i]\)
  • Average Treatment Effect (ATE)

Potential Outcomes in IV

• In IV case, \(X_i\) not random, but \(Z_i\) is
  • Both binary \(\{0,1\}\)
• E.g., maybe you ran a controlled experiment, giving drug to one group \(Z_i=1\), not to other \(Z_i=0\)
  • But some people forget to take the drug
• Likely that people who forget not random
  • Associated with stress, memory, physical ability, etc
• Goal is causal effect of \(X_i\) on \(Y_i\)
• Experiment assigns \(Z_i\): told to take the drug,
  • Not \(X_i\): actually take the drug
• Use \(Z_i\) as instrument for \(X_i\)
• Random assignment of IV means \(Z_i\) independent of \((Y^0_i,Y^1_i,X^0_i,X^1_i)\)

IV Estimate in Binary Z, Binary X case

• First Stage: Regression of \(X\) on \(Z\) gives \(E[X_i|Z_i=1]-E[X_i|Z_i=0]\)
• Since \(Z\) random, this is ATE of Z on X
• Reduced form: Regression of \(Y\) on \(Z\) gives \(E[Y_i|Z_i=1]-E[Y_i|Z_i=0]\)
• Since \(Z\) random, this is ATE of Z on Y
• IV is reduced form over first stage \[\hat{\beta}_1^{IV}= \frac{\frac{1}{n_1}\sum_{i=1}^{n}Z_iY_i-\frac{1}{n_0}\sum_{i=1}^{n}(1-Z_i)Y_i}{\frac{1}{n_1}\sum_{i=1}^{n}Z_iX_i-\frac{1}{n_0}\sum_{i=1}^{n}(1-Z_i)X_i}\]
• Converges to \[\frac{E[Y_i|Z_i=1]-E[Y_i|Z_i=0]}{E[X_i|Z_i=1]-E[X_i|Z_i=0]}\]
• In constant effects case, this is \(\beta\)
  • But what if effects heterogeneous?

Goal of potential outcome interpretation

• Under constant treatment effects, a valid IV gives ATE, independent of characteristics
• Issue is subjects induced into treatment by IV may respond differently than other subjects when made to take treatment
  • Drug more/less effective among people already healthier, and so less likely to forget
  • Students who attend a school after winning a voucher are those who expect it will help them out the most
• It turns out that, if effects differ, IV can only give you Average Treatment Effect for subpopulation
• Call this the Local Average Treatment Effect, LATE
• Still interesting if we want to give treatment just to this group
  • Next, will show who this group is

Causal effect of Z on X

• In potential outcomes notation, \(X\) given \(Z\) takes value \[X_i=X^0_i(1-Z_i)+X^1_iZ_i\]
• Can distinguish 4 groups
  • Always Takers: Take drug whether assigned or not \[X^0_i=X^1_i=1\]
  • Never Takers: Don’t take drug whether assigned to or not \[X^0_i=X^1_i=0\]
  • Compliers: Take drug if assigned, Don’t if not \[X^0_i=0,\ X^1_i=1\]
  • Defiers: Take drug if not assigned, don’t take if assigned \[X^0_i=1,\ X^1_i=0\]

Reduced form in potential outcomes framework

• Rewrite reduced form in terms of potential outcomes \[E[Y_i|Z_i=1]-E[Y_i|Z_i=0]=\] \[E[X^1_iY^1_i+(1-X^1_i)Y^0_i|Z_i=1]-E[X^0_iY^1_i+(1-X^0_i)Y^0_i|Z_i=0]\]
• Use independence: \(Z_i\) independent of \((Y^0_i,Y^1_i,X^0_i,X^1_i)\) \[= E[(X^1_i-X^0_i)(Y^1_i-Y^0_i)]\]
• Decompose reduced form into effects on the 4 groups \[= P(X_i^1=1,X_i^0=0)E[1*(Y^1_i-Y^0_i)|X_i^1=1,X_i^0=0]\] \[+P(X_i^1=1,X_i^0=1)E[0*(Y^1_i-Y^0_i)|X_i^1=1,X_i^0=1]\] \[+P(X_i^1=0,X_i^0=0)E[0*(Y^1_i-Y^0_i)|X_i^1=0,X_i^0=0]\] \[+P(X_i^1=0,X_i^0=1)E[-1*(Y^1_i-Y^0_i)|X_i^1=0,X_i^0=1]\]

Reduced form in potential outcomes framework, ctd

• Always takers and never takers unaffected by instrument
• Reduced form is weighted difference in causal effects on compliers and defiers \[E[Y_i|Z_i=1]-E[Y_i|Z_i=0]=\] \[= P(X_i^1=1,X_i^0=0)E[1*(Y^1_i-Y^0_i)|X_i^1=1,X_i^0=0]\] \[+P(X_i^1=0,X_i^0=1)E[-1*(Y^1_i-Y^0_i)|X_i^1=0,X_i^0=1]\]

Monotonicity

• In many cases, reasonable to assume no defiers
  • E.g. drug only given to treatment group, so no way for control to take it, even though treatment group may sometimes forget
  • If IV Z is something like “I give you money to do X”, rational people will always be no less likely to do X when it is more profitable
• Can fail if instrument can move outcome in either direction
  • Judge A gives longer sentences to one group of criminals than Judge B, but shorter to a different group
    
• If monotonicity holds, \(P(X_i^1=0,X_i^0=1)=0\) and \[E[Y_i|Z_i=1]-E[Y_i|Z_i=0]=\] \[P(X_i^1=1,X_i^0=0)E[Y^1_i-Y^0_i|X_i^1=1,X_i^0=0]\]
• Reduced form is causal effect of \(X\) for the compliers, multiplied by probability that the instrument induces compliance

First stage in potential outcomes framework

• First stage is \(E[X_i|Z=1]-E[X_i|Z_i=0]\)
• Independence means this equals \(E[X^1_i]-E[X^0_i]\) \[= P(X^1_i=1)-P(X^0_i=1)\]
• Apply law of total probability \[= (P(X^1_i=1 \cap X^0_i=1)+P(X^1_i=1 \cap X^0_i=0))\] \[-(P(X^0_i=1 \cap X^1_i=1)+P(X^0_i=1 \cap X^1_i=0))\] \[= P(X^1_i=1 \cap X^0_i=0)-P(X^0_i=1 \cap X^1_i=0)\]
• No defiers implies \[E[X_i|Z=1]-E[X_i|Z_i=0] = P(X^1_i=1 \cap X^0_i=0)\]
• First stage estimates proportion of compliers

LATE

• \(E[Y^1_i-Y^0_i|X_i^1=1,X_i^0=0]\) is called the LATE
  • Local Average Treatment Effect
• Recovered by \[LATE=\frac{E[Y_i|Z_i=1]-E[Y_i|Z_i=0]}{P(X_i^1=1,X_i^0=0)}\]
• Probability of compliance \(P(X_i^1=1,X_i^0=0)=E[X_i|Z=1]-E[X_i|Z_i=0]\) \[LATE=\frac{E[Y_i|Z_i=1]-E[Y_i|Z_i=0]}{E[X_i|Z=1]-E[X_i|Z_i=0]}\]
• This is \(\beta^{IV}\)
• Under independence and monotonicity, IV estimates LATE
• With constant treatment effects, do not need monotonicity
  • IV measures ATE

Interpretation

\[\hat{\beta}^{IV}_1\overset{p}{\rightarrow}E[Y^1_i-Y^0_i|X_i^1=1,X_i^0=0]\]

• IV gives causal effect of treatment on compliers
• This is average treatment effect over a subpopulation
  • People induced to change their behavior by instrument
• Maybe not the same as population as a whole
  • People who remember to take medicine probably healthier than those who forget
  • Then LATE gives effect of drug on healthier people, not always the same as on all people
• If policy is to change \(X\) for all people, IV only measures the effect if effects are the same on everyone
• If policy is to induce people to do \(X\), by changing \(Z\), then IV gives what we want

Extensions

• In non-binary instrument case, IV gives weighted average of conditional causal effects over different subgroups
• See Angrist & Pischke book for many details

Example: Charter School Effectiveness

• Policy question
  • Many states now have privately run but publicly funded schools.
  • Do these schools improve learning outcomes?
• In certain school districts, admission to school for some students is by lottery
  • \(Z_i=1\) if admitted, \(Z_i=0\) if not
  • Independent of potential learning outcomes since random
• Not all who are admitted attend
  • \(X_i=1\) if attending, \(0\) otherwise
• If people who decide to attend differ from those who don’t, difference in scores may provide biased measure of causal effects
• Use IV to estimate causal effect of attending a charter school on Math test scores \[\hat{\beta}^{IV}=\frac{\text{Average score, admitted - Average score, not admitted}}{\%\text{ of admitted who attend - }\%\text{ of non-admitted who attend}}\]

Results (from Angrist and Pischke, Table 3.1)

• Data from KIPP school in Lynn, Massachussetts, 2005-2008
• Sample of 371 students apply to school via lottery
• 253 admitted, of whom 199 attend
• 118 not admitted, of whom 5 attend (got in some other way)
• First stage: admitted students \(199/253 - 5/118= 0.74\) percent increased likelihood of attending
• Average Math score of lottery winners: 0.00 standard deviations below state mean
• Average score for non-winners: 0.36 standard deviations below state mean
• Reduced form:
  • Lottery winners have math scores 0.36 standard deviations higher than nonwinners.
  • ATE says winning boosts math scores
• IV estimate: Reduced form/first stage = 0.36/0.74=0.48 standard deviations higher

Interpretation

• Lottery boosts math scores by \(.36\sigma\), and since this can only be coming from going to the school, which only 3/4 of winners did, effect of going to school is \(\approx .5\sigma\) units on math tests
• Constant effects model says \(0.48\sigma\) is “the” causal effect of going to charter school for this population
• If school more or less effective for some students, interpretation more limited
• Always takers: students who would go even if not admitted
  • Probably these are people with involved parents
  • Likely to differ on education-related characteristics
• Never takers: students who won’t attend even if admitted
  • Again could differ on education-related characteristics
• Defiers: students who would only go if not admitted
  • Seems reasonable to rule this out
• LATE: Causal effect of charter school for students in group who, if admitted, will choose to attend

Conclusions

• When instrument available, IV estimator handles endogeneity
• Linear constant coefficients case:
  • Requires Exogeneity and Relevance
  • Estimates (constant) effect
• Heterogeneous treatment effects case
  • Requires Independence and Monotonicity (No Defiers)
  • Estimates Local Average Treatment Effect
    • Average of Treatment Effects for compliers only

Next Class or Two

• Multivariate Instrumental Variables
  • 2SLS Estimator
• Testing IV assumptions
• More IV applications