Correct Specification and Identification of Nonparametric Transformation Models

This paper derives necessary and sufficient conditions for nonparametric transformation models to be (i) correctly specified, and (ii) identified. Our correct specification conditions come in a form of partial differential equations; when satisfied by the true distribution, they ensure that the observables are indeed generated from a nonparametric transformation model. Our nonparametric identification result is global; we derive it under conditions that are substantially weaker than full independence. In particular, we show that a completeness assumption combined with independence with respect to one of the regressors suffices for the model to be identified.