Robust inference and testing of continuity in threshold regression models

This paper is concerned with inference in regression models with either a kink or a jump at an unknown threshold, particularly when we do not know whether the kink or jump is the true specification. One of our main results shows that the statistical properties of the estimator of the threshold parameter are substantially different under the two settings, with a slower rate of convergence under the kink design, and more surprisingly slower than if the correct kink specification were employed in the estimation. We thus propose two testing procedures to distinguish between them. Next, we develop a robust inferential procedure that does not require prior knowledge on whether the regression model is kinky or jumpy. Furthermore, we propose to construct confidence intervals for the unknown threshold by the bootstrap test inversion, also known as grid bootstrap. Finite sample performances of the bootstrap tests and the grid bootstrap confidence intervals are examined and com- pared against tests and confidence intervals based on the asymptotic distribution through Monte Carlo simulations. Finally, we implement our procedure to an economic empirical application.