Instrumental Variables: Introduction Methods of Economic Investigation Lecture 14

Last Time  Review of Causal Effects Defining our types of estimates:  ATE (hypothetical)  TOT (can get this if SB = 0)  ITT (use this if we’ve got compliance problems) Methods  Experiment (gold standard but can’t always get it)  Fixed Effects (assumption on within group variation)  Difference-in-Differences (assumption on parallel trends)  Propensity Score Matching (assumption on relationship between observables, unobseravables, and treatment)

Today’s Class  Introduction to Instrumental Variables What are they How do we estimate IV Tests for specification/fit

Recap of the problem  There is some part of the error that we don’t observe (maybe behavioral parameters, maybe simultaneously determined component, etc.)  This component might not be: Fixed within a group Fixed over time/space Related to observables  BUT…this component IS correlated with the treatment/variable of interest

Our Treatment Effects Model  Consider the following model to estimate the effect of treatment S on some outcome Y: Y = αX + ρS + η  Our Treatment here is S Think of the example of schooling How much more will you earn if you go to college? Can’t observe true underlying ability which is correlated with college attendance decision and future earnings

What’s correlated and what’s not  The model we want to estimate: Y i = αX + ρs i +γA i + v i  We have that: E[sv] = 0 (by assumption) E[Av] = 0 (by construction)  The idea: if A could be observed, we’d just include it in the regression and be done

The Instrument…. Assigned to Treatment (S=1) A B Not Assigned Treatment (S=0) A H =1 A L =0 A H =1 ITT compares all of A to all of B: this mixes up the compliers (A H =S=1; A L =S=0) and the non-compliers (A H =1, S=0; A L =0, S=1)

Introducing Instruments  The problem: How to estimate ρ when A is not observed A is related to Y Cov(AS) ≠0  The solution: find something that is Correlated with S [“Monotonicity”] Uncorrelated with any other determinant of the outcome variable Y [“Exclusion Restriction”]

How does IV work  Call our instrument z  Our two instrument characteristics can be re-written as E[z S] ≠0 E[z η] = 0  Then from our equations we can write our population estimate of ρ as:

The Instrument…. Assigned to Treatment (S=1) A B Not Assigned Treatment (S=0) A H =1 A L =0 A H =1 Using the Instrument, we can determine where the partition is: then we can compare the part of A which was “randomly assigned (A H =S=1) to the part of B that is randomly assigned (A L =S=0)

Simplest case for IV  Homogeneous treatment effects (same ρ for all i )  Dummy Variable for instrument z= 1 with probability q Can break-up continuous instruments into sets of dummy variables or use GLS to generalize  For now—don’t worry about covariates Simple extension: just include these in both stages Simplify our notation later…

Return to LATE  Using z as a dummy that’s 1 with probability q Cov(Y, z) = {E[Y | z = 1] – E[Y | z = 0]}q(1 – q) Cov(s, z) = {E[s | z = 1] – E[s | z = 0]}q(1 – q)  Can rewrite ρ as:  Should look familiar: it’s our LATE estimate

Another type of intuition  Remember that E[η | S] ≠ 0 (that’s why we’re in this mess) E[Y | S] ≠ ρE[S] Can condition on Z, rather than S  By the “exclusion restriction” property of our instrument E[η | Z] = 0  So now can estimate ρ because E[Y | z] = ρE[S | z] If Z is binary, then this simplifies to our Wald estimator

IV estimate intuition  The only reason for a relationship between z and Y is the relationship between z and X  In dummy variable specification: this is just rescaling the reduced form difference in means (E[Y | z=1] – E[Y| z=0]) by the first stage difference in means (E[S | z=1] – E[S| z=0])

 To see why this is, Think about our “structural equations” Y i = αX + ρs i + η i  We can estimate ρ by getting the ratio of two different coefficients First stage: s i = π 10 X + π 11 z i + ξ 1i Reduced form: y i = π 20 X + π 21 z i + ξ 2i How does IV work: Regression Intuition Exogenous instrument Endogenous Exogenous Covariates

Rewriting the Structural equation Plug in the values from the first stage: Y i = αX + ρs i + η i = αX + ρ [π 10 X + π 11 z i + ξ 1i ] + η i = [α + ρπ 10 ]X + ρ π 11 z i + ρ ξ 1i + η i = π 20 X + π 21 z i + ξ 2i = αX + ρ[π 10 X + π 11 z i ] + + ξ 2i Fitted value in the population regression of s on z (and X) Coefficient population regression of y on s, and also on the fitted value of S (and the X’s)

Population vs. Estimates  If we had the entire population, we could measure the relationship between z and S and obtain the true π’s  Using these π’s we could then obtain the true ρ  Unfortunately, most of the time, we have finite samples

Estimating 2SLS  In practice, use finite samples to obtain fitted value Consistent estimate of parameters from OLS Use these parameters to construct fitted value  Then use this fitted value to construct second stage estimating equation  Can get consistent estimates because covariates and fitted values are independent of η (by assumption) Independent of (by construction)

Bias in 2SLS  2SLS is biased—we’ll talk about this in detail next time but the general idea is: We must estimate the first stage (e.g. ) In practice, the first-stage estimates reflect some of the randomness in the endogenous variable (e.g. S) This randomness generates finite-sample correlations between first-stage fitted values and second stage errors  Endogeneous variable correlated with the second stage errors  Some of that is left in the first stage fitted value  Asymptotically this bias goes to zero but in finite sample might not

Next time:  Issues with IV estimates Return to Consistency: what about bias? Weak instruments Heterogeneous Treatment Effects