Threshold Regression with Endogeneity

This paper studies estimation in threshold regression with endogeneity. Three key results di¤er from those in regular models. First, both the threshold point and the threshold e¤ect parameters are shown to be identi\u85ed without the need for instrumentation. Second, in partially linear threshold models, both parametric and nonparametric components rely on the same data, which prima facie suggests identi cation failure. But, as shown here, the discontinuity structure of the threshold itself supplies identifying information for the parametric coe ¢ cients without the need for extra randomness in the regressors. Third, instrumentation plays di¤erent roles in the estimation of the system parameters, delivering identi\u85cation for the structural coe ¢ cients in the usual way, but raising convergence rates for the threshold e¤ect parameters and improving e ¢ ciency for the threshold point. Simulation studies corroborate the theory and the asymptotics. An empirical application is conducted to explore the e¤ects of 401(k) retirement programs on savings, illustrating the relevance of threshold models in treatmen