Propensity model selection with nonignorable nonresponse and instrument variable

Handling data with nonignorable missing responses is difficult because of the identifiability issue caused by a nonignorable nonresponse. An effective approach described in the literature is to impose a parametric model on the nonresponse propensity (while the conditional distribution of the response, given covariates, is totally unspecified). Then, use a nonresponse instrument, which is a useful covariate vector that can be excluded from the propensity, given the response and other covariates. However, how to find a nonresponse instrument from a given set of covariates is not well addressed. In addition, we may want to select a parametric propensity model from a set of candidate models. Therefore, we propose a simultaneous propensity model and instrument selection criterion. In the presence of a nonignorable nonresponse, the proposed method consistently selects the most compact correct parametric propensity model and instrument from a group of candidate models, assuming one of these candidate models is correct and an instrument exists. Simulation results show that our proposed method works quite well. A real-data example is presented for illustration.