Instrumental Variables: Conclusion

Today

• Conclude Instrumental Variables
  • Simultaneous Equations
  • Measurement error

Instrumental Variables: Model and Estimator

• Interested in coefficients of linear model \(Y_{1i}=X_i^{\prime}\beta+u_i\) \[=\beta_0+\beta_1Y_{2i} + \beta_2Y_{3i} + \ldots + \beta_{\ell}Y_{\ell i} + \beta_{\ell+1}Z_{1i} + ... + \beta_{k+\ell}Z_{ki} + u_{i}\]
• But regressors are endogenous \[E[Y_{-1i}u_i]\neq 0\]
• Solution: Use instrumental variables \(Z_j\) \(j=0\ldots m>k+1\) that are
  • Exogenous \(E[Z_{ij}u_i]=0\ \forall j=0\ldots m\)
  • Relevant: no exact linear relationships among variables in \(\hat{X}\)
• Apply Two Stage Least Squares (2SLS) estimator
  • 1. Regress \(X\) on \(Z\) to get predicted \(\hat{X}\)
  • 2. Regress \(Y\) on \(\hat{X}\) to get \(\hat{\beta}^{2SLS}\)

Sources of endogeneity: Why Use 2SLS?

\[E[Y_{-1i}u_i]\neq 0\]

• One reason (Last Week): Omitted Variables
• Another: Simultaneity
  • Causality runs in “wrong” direction
  • \(Y_2\) causes \(Y_1\), but \(Y_1\) also causes \(Y_2\)
  • Graph representing relationship is not acyclic
• Economic equilibria: system of variables mutually determined
  • Ex. 1: Supply and demand
  • Ex. 2: Game theory

Simultaneous equations systems (Code)

library(dagitty) #Library to create causal graphs
library(ggdag) #library to plot causal graphs
simeqgraph<-dagify(Y1~Y2+Z1,Y2~Y1+Z2) #create graph
#Set position of nodes 
  coords<-list(x=c(Y2 = 1, Y1 = 0, Z1=-1, Z2=2),
               y=c(Y2 = 0, Y1 = 0, Z1=0,Z2=0)) 
  coords_df<-coords2df(coords)
  coordinates(simeqgraph)<-coords2list(coords_df)
#Plot causal graph
ggdag(simeqgraph)+theme_dag_blank()+
  labs(title="Simultaneous System with Instruments") 

Simultaneous equations systems

• Represent by system of structural equations \[Y_1=\beta_{11}Y_2+\beta_{12}Z_1+u_{1}\] \[Y_2=\beta_{21}Y_1+\beta_{22}Z_2+u_{2}\]

• E.g. Supply and demand
  • \(Y_1\) is price, \(Y_2\) is quantity
  • Equation 1 is supply curve
    • As quantity sold increases, ask a higher price
  • Equation 2 is demand curve
    • As price goes up, demand less quantity
  • \(Z_1\) are forces shifting supply curve: e.g., price of inputs
  • \(Z_2\) are forces shifting demand curve: e.g. prices of substitutes
  • \(u_1,u_2\) are unobserved supply/demand shifters

Simultaneity Bias

• Substitute equation 1 in equation 2 \[Y_2=\beta_{21}(\beta_{11}Y_2+\beta_{12}Z_1+u_{1})+\beta_{22}Z_2+u_{2}\]
• Solve for reduced form for \(Y_2\) in terms of \(Z\) \[Y_2=\frac{1}{1-\beta_{21}\beta_{11}}(\beta_{21}\beta_{12}Z_1+\beta_{22}Z_2+\beta_{21}u_{1}+u_{2})\]
• Valid if \(\beta_{21}\beta_{11}\neq 1\)
• Now consider estimating equation 1 by OLS

\[Y_1=\beta_{11}Y_2+\beta_{12}Z_1+u_{1}\]

• \(Y_2\) contains \(u_1\) and so is correlated with \(u_1\)

Handling simultaneity bias

• Similar to omitted variables
  • Unobserved demand shifts directly affect quantity and also price
• Difference is that simultaneity means any term which shifts demand will affect quantity sold in addition to price,
  • Control strategy not possible even if all variables observed
  • If there is any variation in \(u_1\) at all, it will be correlated with \(Y_2\)
• But we CAN isolate variation due only to supply shifts \(Z_2\)
  • Estimate by 2SLS with \(Z_2\) as excluded instrument
  • Assumes \(Z_2\) only affects supply and not demand

General simultaneous systems

• Can have a system with \(\ell\) equations and \(\ell\) endogenous variables \[Y_1=\beta_{11}Y_2+\ldots+\beta_{1\ell}Y_{\ell}+Z_1^\prime\beta_{1\ell+1}+u_{1}\] \[Y_2=\beta_{21}Y_1+\beta_{22}Y_3+\ldots+\beta_{2\ell}Y_{\ell}+Z_2^\prime\beta_{2\ell+1}+u_{2}\] \[\ldots\] \[Y_{\ell}=\beta_{\ell 1}Y_1 + \beta_{\ell 2}Y_2 +\ldots+ \beta_{\ell\ell}Y_{\ell-1} +Z_{\ell}^\prime\beta_{\ell \ell+1} +u_{\ell}\]

• Represent full set of causal relationships between \((Y_1,\ldots,Y_\ell)\) and also effect of exogenous variables \(Z\) not determined inside the system

• \(Z_1^\prime\) through \(Z_{\ell}^\prime\) may be vectors containing shared elements if one variable affects multiple variables

Reduced form in simultaneous equation models

• Reduced form
  • All \(Y\) variables as functions of only \(Z\) variables
  • Exists under some restrictions on coefficients
• Need a unique solution to this system of equations
  • E.g., supply and demand curves must touch exactly once
  • Otherwise model of data not well defined
• If supply and demand curves exactly overlap, price and quantity could take multiple values
  • Need more info to describe system
• If model complete and well defined, can calculate probability law of variables inside the system in terms of external, or exogenous, disturbances to system

Estimation of simultaneous equations models

• Can estimate one equation in system by 2SLS estimation
  • If reduced form exists and
  • There are terms \(Z\) that enter into other equations but not directly into this one
• Need at least as many excluded \(Z\) terms as included endogenous variables \(Y\) to estimate any one equation consistently
• May only be able to estimate some equations in system
• Execute in R equation by equation using ivreg
• Command to do all at once is systemfit in systemfit library

Measurement Error

• One more source of endogeneity
• \(X^{*}\) is assigned randomly, but we don’t observe it directly
• Instead observe noisy proxy \(X=X^{*}+\epsilon\)
  • \(\epsilon\perp X^{*}\) is nonsystematic error in measurement process
  • Maybe survey taker sometimes writes numbers slightly off
  • Maybe tool used to measure sometimes picks up unrelated noise
• We care about coefficient of \(Y=\beta_0+\beta_1X^{*} + e\)
• How bad is it to estimate instead regression of \(Y\) on \(X\)?

Measurement-error bias

• OLS Estimate is \[\beta_{OLS}=\frac{Cov(X,Y)}{Var(X)}\]
• True object of interest is \[\beta=\frac{Cov(X^*,Y)}{Var(X^*)}\] \[\beta_{OLS}=\frac{Cov(X^{*}+\epsilon,Y)}{Var(X^{*}+\epsilon)}\] \[=\frac{Cov(X^{*},Y)+Cov(\epsilon,Y)}{Var(X^{*})+Var(\epsilon)}\]
• By randomness of error, \(Cov(\epsilon,Y)=0\) \[=\frac{Cov(X^{*},Y)}{Var(X^{*})+Var(\epsilon)}\]
• \(\left|\beta_{OLS}\right|<\left|\beta\right|\)
• Magnitude smaller since divided by larger number

Intuition

• Attenuation bias
• Measurement error makes OLS estimate smaller than true effect
• Limiting case: \(X\) is pure noise: no information about \(X^{*}\) at all
  • Then \(Y\) is unrelated to \(X\)
  • Coefficient becomes 0
• Equivalent to a form of endogeneity
• \(Y= \beta_0 +\beta_1 X^{*} + e = \beta_0 + \beta_1 (X-\epsilon) + e = \beta_0 + \beta_1 X-\beta_1\epsilon + e = \beta_0 + \beta_1 X+u\)
• Here \(u=e-\beta_1\epsilon\), \(E[Xu]=E[X^{*} +\epsilon,e-\beta_1\epsilon]=-\beta_1 E[\epsilon^2]\neq 0\)

Solving Measurement error bias

• If we have an IV correlated with \(X^*\) but uncorrelated with the measurement error in it, we can estimate effect by IV
• Special case: 2 noisy measurements: \(X_1=X^{*}+ \epsilon_1\), \(X_2=X^{*}+\epsilon_2\) \(\epsilon_1\perp\epsilon_2\)
• \(Cov(X_1,X_2)=Var(X^{*})\neq 0\) and \(Cov(X_2,Y-X_1\beta_1)=0\), since measurement errors are independent
• Example: if survey-takers sometimes misrecord a response, survey same households twice, use one survey as IV for other
• Compare alternate strategy
  • Take 2 noisy measurements and average
  • Variance of measurement error decreases, but doesn’t disappear
    • OLS still inconsistent
  • Unlike IV which gives estimate which gives consistent estimator even if error variance large

Instrumental Variables: Overview

• Endogeneity causes bias in OLS estimates of linear model
  • Omitted variables
  • Simultaneity
  • Measurement Error
• IV estimator handles all issues
• Instrument \(Z\) must be exogenous and relevant
  • Exclusion: no direct effect of (some) \(Z\) on \(Y_1\)
  • \(Z\) correlated with endogenous regressor \(Y_{-1}\)