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I have the following model:

$y_i = \alpha + \beta d_i + g(x_i)+e_i$

Since I have reason to believe that $d_i$ is endogenous, I am using 3 plausibly exogenous variables ($z_1,z_2,z_3$) to instrument it. Both $d_i$ and my instruments are continuous variables. My first stage regression takes the form

$d_i = \gamma+\gamma_1 z_1+\gamma_2 z_2+ \gamma_3 z_3+f(x_i) + u_i$

Question1: What is the interpretation of Local Average Treatment effect when there is one continuous instrument (say just $z_1$)? Is it the treatment effect on all $i$'s whose $d_i$ changes due to a change in $z_1$? Can I interpret that as the average treatment effect of $d_i$ over the support of $z_1$?

Question 2: When I have multiple instuments, like in my case, does the LATE become a weighted average of LATEs from each instrument?

FightMilk
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1 Answers1

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To answer your question 1: when there's only 1 continuous instrument $z_1$ and no $x$, the 2SLS estimator yields a weighted average of individual treatment effects $\beta_i$ (assuming treatment effect heterogeneity), where the weights are the individual first-stage coefficient $\gamma_{1,i}$.

To see why this is the case, see Nick Huntington-Klein (2020) section 2.1 and appendix 1 for a simple illustration.

To answer your question 2: yes but conditionally. Imbens and Angrist (1994) (Theorem 2) show that when treatment is binary and instrument $Z$ is discrete, then the IV estimator yields a convex combination of LATEs at each discrete level of $Z$, under regularity conditions. Furthermore, Mogstad, Torgovitsky, and Walters (2021) show that in presence of multiple instruments, 2SLS estimator also yields a weighted average of LATEs, under a mild, testable "partial monotonicity" assumption.

peacefulzephyr
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