Nonparametric identification and estimation of transformation models

This paper derives sufficient conditions for nonparametric transformation models to be identified and develops estimators of the identified components. Our nonparametric identification result is global, and allows for endogenous regressors. In particular, we show that a completeness assumption combined with conditional independence with respect to one of the regressors suffices for the model to be nonparametrically identified. The identification result is also constructive in the sense that it yields explicit expressions of the functions of interest. We show how natural estimators can be developed from these expressions, and analyze their theoretical properties. Importantly, it is demonstrated that different normalizations of the model lead to different asymptotic properties of the estimators with one normalization in particular resulting in an estimator for the unknown transformation function that converges at a parametric rate. A test for whether a candidate regressor satisfies the conditional independence assumption required for identification is developed. A Monte Carlo experiment illustrates the performance of our method in the context of a duration model with endogenous regressors.