Threshold Regression Without Distribution Assumption when the Threshold Variable Is Endogenous

In this paper, we propose a semi-parametric approach to estimate threshold models without the reliance of distribution assumption when the threshold variable is endogenous. Early threshold models based on the exogeneity assumption on the threshold variables are widely applied in economics studies. The invalidation of this assumption, however, may bias the estimation and inference. To control for the effects caused by an endogenous threshold variable, we propose a concentrated two-stage least squares method using instrumental variables. This method is based on the local linear smoothers by Fan (1992) and can be simplified as a concentrated kernel-weighted least squares estimation. Particularly, our approach can consistently estimate the threshold model without relying on any distribution assumption and allows us to handle nonlinear effects caused by the endogenous threshold variable in a more flexible setting. We establish consistency of the threshold parameter and the asymptotic properties of the slope coefficients in the proposed method. Monte Carlo simulations are performed to evaluate its finite sample properties. The new method is applied to re-examine the effects of political institutions on economic growth.