Missing Endogenous Variables in Conditional Moment Restriction Models

We consider the estimation of finite dimensional parameters identified via a system of conditional moment equalities when at least one of the endogenous variables (outcomes and/or explanatory variables) is missing at random for some individuals in the sample. We derive the semiparametric efficiency bound for estimating the parameters and use it to demonstrate that efficiency gains occur only if there exists at least one endogenous variable that is nonmissing, i.e., observed for all individuals in the sample. We show how to construct “doubly robust” estimators and propose an estimator that achieves the efficiency bound. A simulation study reveals that our estimator works well in medium-sized samples for point estimation as well as for inference. To see what insights our estimator can deliver in empirical applications with very large sample sizes, we revisit the female labor supply model of Angrist and Evans (1998) and show that if there is even medium missingness in female labor income (the outcome variable), then having more than 200,000 observations is not enough for a researcher using inverse propensity score weighted GMM to find a statistically significant negative effect of having a 3rd child (the endogenous explanatory variable) on labor income. In contrast, our semiparametrically efficient estimator can deliver point estimates of this effect that are comparable to the GMM estimates as well as being statistically significant.