Estimation and testing of kink regression model with endogenous regressors

Kink regression model which assumes continuity at the threshold point has wide applications in statistics and economics. Existing estimation methods are obtained under a rather important assumption that the errors are mean independent of the threshold variable, namely, it is exogenous. However, endogeneity can arise as a result of omitted variables, lagged dependent variable, measurement error and many other sources. In this paper, we consider the estimation and testing for the kink regression model with endogenous threshold variable and possible other endogenous regressors. We find that the conventional form of 2SLS estimator is inconsistent as the expectation of a nonlinear function is not generally the function of expectations. The continuity feature of the regression prompts us to try method based on the GMM principle using nonlinear instruments. We derive the asymptotic properties using a different way from the usual GMM estimators as the objective function is not smooth with respect to the threshold parameter. A sup-Wald test for the presence of kink effect is established and a bootstrap procedure to gain the p value is introduced. Monte Carlo simulations show that the proposed estimator and testing procedure perform well. The proposed procedures is also illustrated using an empirical application.