Discrete endogenous variables in weakly separable models

This paper contains an extension of the identification method proposed in Jun et al. (2011), hereafter JPX, which is based on a generated collection of sets, that is a 'Dynkin system'. We demonstrate the usefulness of this extension in the context of the model proposed by Vytlacil and Yildiz (2007), hereafter VY. VY formulate a fully non-parametric 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 non-binary-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.