Semiparametric efficiency bounds for seemingly unrelated conditional moment restrictions

This paper addresses the problem of semiparametric efficiency bounds for conditional moment restriction models with different conditioning variables. We characterize such an efficiency bound, that in general is not explicit, as a limit of explicit efficiency bounds for a decreasing sequence of unconditional (marginal) moment restriction models. An iterative procedure for approximating the efficient score when this is not explicit is provided. Our theoretical results complete and extend existing results in the literature, provide new insight for the theory of semiparametric efficiency bounds literature and open the door to new applications. In particular, we investigate a class of regression-like (mean regression, quantile regression,...) models with missing data.