Instrumental Variables Methods of Economic Investigation Lecture 15

Last Time Introduction to Instrumental Variables Correlation with variable of interest Exclusion restriction Interpretation of IV with homogeneous treatment effects Gives us a Wald estimate Nice/well-defined properties of OLS

Today’s Class Uses for 2SLS Heterogeneous Treatment Effects Experiments with compliance issues Omitted Variable Bias Heterogeneous Treatment Effects LATE framework Interpretation

Review of Instrumental Variables Two characteristics Instrument (Z) is correlated with (S) Must be that S is always increasing (or always decreasing) If it changed signs, then the first stage prediction wouldn’t work Instrument (Z) is uncorrelated with other determinants of the outcome (Y) This means Z is uncorrelated with unobservables that affect Y The only way Z affects Y is through S

Steps to Estimate IV-1 Step 1: The Structural Equation Y =ρS + η Problems: S correlated with η OLS estimates won’t recover causal effect of S on Y Step 2: Find an Instrument Correlated with S Uncorrelated with η (and so uncorrelated with the unobservables)

Steps to Estimate IV-2 Step 3: Estimate the First Stage S = πZ + ν Can estimate this with OLS Want to test to see if π is significant—will return to this in the case of weak instruments where α is close to zero Step 4: Obtain the fitted values This is the component of S that is unrelated to the error term in the structural equation

Steps to Estimate IV -3 Step 5: Estimate the Second Stage This is using the fitted value, i.e. the predicted value of S given the instrument Z The fitted value captures the component of S that is uncorrelated with the error If we want to recover β take the OLS estimate from the second stage b and divide it by the coefficient from the first stage α

Goal: Average Effect of S on Y (ATE) Various uses for IV Goal: Average Effect of S on Y (ATE) Omitted Variable Non-Experimental Experimental Perfect Compliance Imperfect Compliance Matching Diff-in-diff IV Fixed Effects IV Perfect Compliance Imperfect Compliance Measurement Error IV IV

Things to worry about Is my instrument really uncorrelated with other determinants of the outcome? How do I interpret my IV estimate? What if I think there are heterogeneous treatment effects? How strong does my first stage have to be for this to all work? We’ll deal with each of these issues

Can we test the exclusion restriction This is the assumption of the model and often cannot be formally tested The reduced form gives some information on the reasonableness of the assumption Knowing π and the OLS biased estimated of ρ we might gut check how reasonable it is for the only effect of Z to be through S If we have multiple instruments, we can test using the overidentification test

Overidentification Model is overidentified if we have: # instruments variables > # endogenous variables Models with exactly same number of instruments as endogenous variables are just identified If the model is overidentified we can test the quality of the fit

Testing Model Fit Suppose we have Q instruments and define (this is our first stage RHS variable) Define and as before The residuals from the second stage can be defined as:

2SLS Residual Terms We assume that η is orthogonal to Z so that E[Zi η(Γ)]=0 The sample analog of this is In finite samples, this won’t be exactly zero 2SLS fits the value of Γ making this closest to zero This has an asymptotic distribution of where

The Minimand There is an underlying Method of Moments way to illustrate this but we’ll ignore that for now. Basic idea is to minimize the quadratic form of the vector mN(Γ) The optimal weighting matrix to estimate this is Λ-1 and then the equation to be minimized is:

The Overidentification Test Intuition: is mn(g) close enough to zero for us to believe that Z uncorrelated with the error (other unobservable stuff) Null hypothesis: E[ηZ]=0 distributed Χ2(Q-1) Can also test this directly Estimate the just-identified version for the Q instruments Test that the estimate coefficients are statistically indistinguishable

Next time Issues with IV Heterogeneity Weak Instruments