Robust estimation for moment condition models with data missing not at random

We consider estimation for parameters defined through moment conditions when data are missing not at random. The missingness mechanism cannot be determined from the data alone, and inference under missingness not at random may be sensitive to unverifiable assumptions about the missingness mechanism. To add protection against model misspecification, we posit multiple models for the response probability and propose a weighting estimator with calibrated weights. Assuming the conditional distribution of the outcome given covariates is correctly modeled, we show that if any one of the multiple models for the response probability is correctly specified, the proposed estimator is consistent for the true value. A simulation study confirms that our estimator has multiple robustness when the outcome data is missing not at random. The method is also applied to an application.