Inference in Instrumental Variables Analysis with Heterogeneous Treatment Effects

Abstract

We study inference in an instrumental variables model with heterogeneous treatment effects and possibly many instruments and/or covariates. In this case two-step estimators such as the two-stage least squares (TSLS) or versions of the jackknife instrumental variables (JIV) estimator estimate a particular weighted average of the local average treatment effects. The weights in these estimands depend on the first-stage coefficients, and either the sample or population variability of the covariates and instruments, depending on whether they are treated as fixed (conditioned upon) or random. We give new asymptotic variance formulas for the TSLS and JIV estimators, and propose consistent estimators of these variances. The heterogeneity of the treatment effects generally increases the asymptotic variance. Moreover, when the treatment effects are heterogeneous, the conditional asymptotic variance is smaller than the unconditional one. Our results are also useful when the treatment effects are constant, because they provide the asymptotic distribution and valid standard errors for the estimators that are robust to the presence of many covariates.