Missing Endogenous Variables in Conditional Moment Restriction Models

We estimate finite-dimensional parameters in conditional moment restriction (CMR) models when at least one of the endogenous variables (outcomes and/or explanatory variables) in the model is missing for some individuals in the sample. We demonstrate that efficiency gains in estimation occur if and only if there is at least one endogenous variable—included in or excluded from the CMR model—that is nonmissing (observed for all individuals in the sample), which we show characterizes informative imputation. We propose a semiparametrically efficient estimator which is also “doubly robust.” To illustrate the insights our estimator can provide in empirical applications with large sample sizes, we artificially induce missingness in the female labor supply model of Angrist and Evans. Despite medium levels of missingness in female labor income (the outcome) and a sample size exceeding 200,000 observations, the inverse propensity score weighted generalized method of moments (GMM) estimator finds only a statistically insignificant negative effect of having a third child (the endogenous regressor) on labor income. In contrast, our efficient estimator yields point estimates of this effect that are not only comparable to the GMM estimates but are also statistically significant.