Discrete Endogenous Variables in Weakly Separable Models

This paper contains an extension of the identification method proposed in Jun, Pinkse, and Xu (2009, JPX), which is based on a generated collection of sets, i.e. a Dynkin system. We demonstrate the usefulness of this extension in the context of the model proposed by Vytlacil and Yıldız (2007, VY). VY formulate a fully nonparametric model featuring a nested weakly separable structure in which an endogenous regressor is binary–valued. The extension of the JPX approach considered here allows for nonbinary–valued discrete endogenous regressors and requires weaker support conditions than VY in the binary case, which substantially broadens the range of potential applications of the VY model. In this paper we focus on the binary case for which we provide several alternative simpler sufficient conditions and outline an estimation strategy. Our results allow for the possibility of a nonbinary endogenous regressor, but we provide examples for the nonbinary case in a separate paper.