Estimating the Endogenous Threshold Regression Based on Instruments

This paper proposes two groups of estimators in endogenous threshold regression based on instruments. The first estimator is a two-stage least squares estimator where the threshold point can be interpreted as a quantile of the threshold variable. This method is invariant to the endogeneity of threshold variable; also, the correlation between the threshold variable and other variables such as other covariates and the error term is helpful to identification. The estimator of the threshold point is asymptotically normal and correlated with slope estimators. The second group of estimators are two control function estimators. We first use a simple example to show that the control function estimator of threshold point in Kourtellos, Stengos and Tan (2016, Econometric Theory 32, 827-860) is not consistent unless the endogeneity level of the threshold variable is small compared to the threshold effect. We correct the control function in their estimator to generate a consistent estimator of threshold point; this method can be treated as an extension of the two-stage least squares in Caner and Hansen (2004, Econometric Theory 20, 813-843). We further put forward a new control function approach which can be treated as an extension of the classical control function approach in endogenous linear regression. Both approaches explore some threshold effect information in conditional variance beyond that in conditional mean. These two control function estimators are not invariant to the endogeneity of threshold variable, but have faster convergence rate than the first estimator and are asymptotically independent of the slope estimators. Simulation studies show that the second control function estimator and the corresponding confidence interval for the threshold point dominate the other methods.