1. Instrumental Variables
Methods of Economic Investigation
Lecture 15

2. 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

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

4. 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

5. 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)

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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:

13. 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

14. 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:

15. 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

16. Next time
Issues with IV
Heterogeneity
Weak Instruments