Identification and Doubly Robust Estimation of Data Missing Not at Random With an Ancillary Variable

We consider identification and estimation with an outcome missing not at random (MNAR). We study an identification strategy based on a so-called {\it ancillary variable}. An ancillary variable is assumed to be correlated with the outcome, but independent of the missingness mechanism conditional on the outcome. We give a necessary and sufficient condition for identification of the full data law given a valid ancillary variable under MNAR, and also sufficient conditions which are convenient to verify in practice. The conditions are satisfied by many commonly-used models, and thus essentially state that lack of identification is not an issue in many situations. Focusing on estimation of an outcome mean, we describe three semiparametric estimation methods: inverse probability weighting, outcome regression and doubly robust estimation. We evaluate the finite sample performance of these estimators via simulations, and apply them to a China Home Pricing dataset extracted from the China Family Panel Survey (CFPS).