Estimating a Nonparametric Triangular Model with Binary Endogenous Regressors

We consider identification and estimation in a nonparametric triangular system with a binary endogenous regressor and nonseparable errors. For identification we take a control function approach utilizing the Dynkin system idea developed in Jun, Pinkse, and Xu (2011, JPX11) and extended in Kédagni and Mourifie (2014, KM14), by which we articulate various tradeoff relations among continuity, monotonicity, and differentiability. For estimation, we use the idea of local instruments under smoothness assumptions, as in e.g. Carneiro and Lee (2009, CL09), Heckman and Vytlacil (1999) but we do not assume additive separability in latent variables. Our estimator uses nonparametric kernel regression techniques and its statistical properties are derived using the functional delta method. We establish that it is n–consistent and has a limiting normal distribution. We apply the method to estimate the returns on a college education. Unlike existing work, notably CL09 and Carneiro, Heckman, and Vytlacil (2011, CHV11), we find that returns on a college education are consistently positive. The returns curves we estimate are moreover inconsistent with the shape restrictions imposed in those papers.