Identification, Doubly Robust Estimation, and Semiparametric Efficiency Theory of Nonignorable Missing Data With a Shadow Variable

We consider identification and estimation with an outcome missing not at random (MNAR). We study an identification strategy based on a so-called shadow variable. A shadow variable is assumed to be correlated with the outcome, but independent of the missingness process conditional on the outcome and fully observed covariates. We describe a general condition for nonparametric identification of the full data law under MNAR using a valid shadow variable. Our condition is satisfied by many commonly-used models; moreover, it is imposed on the complete cases, and therefore has testable implications with observed data only. We describe semiparametric estimation methods and evaluate their performance on both simulation data and a real data example. We characterize the semiparametric efficiency bound for the class of regular and asymptotically linear estimators, and derive a closed form for the efficient influence function.