Doubly weighted M-estimation for nonrandom assignment and missing outcomes

Abstract

Abstract This article proposes a class of M-estimators that double weight for the joint problems of nonrandom treatment assignment and missing outcomes. Identification of the main parameter of interest is achieved under unconfoundedness and missing at random assumptions with respect to the treatment and sample selection problems, respectively. Given the parametric framework, the asymptotic theory of the proposed estimator is outlined in two parts: first, when the parameter solves an unconditional problem, and second, when it solves a stronger conditional problem. The two parts help to summarize the misspecification scenarios permissible under the given framework and the role played by double weighting in each. As illustrative examples, the article also discusses the estimation of causal parameters like average and quantile treatment effects. With respect to the average treatment effect, this article shows that the proposed estimator is doubly robust. Finally, a detailed application to Calónico and Smith’s (The women of the national supported work demonstration. J Labor Econom. 2017;35(S1):S65–S97.) reconstructed sample from the National Supported Work training program is used to demonstrate the estimator’s performance in empirical settings.