This paper studies estimation and speciﬁcation testing in threshold regression with endogeneity. Three key results diﬀer from those in regular models. First, both the threshold point and the threshold eﬀect parameters are shown to be identiﬁed 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 identiﬁcation failure. But, as shown here, the discontinuity structure of the threshold itself supplies identifying information for the parametric coeﬀicients without the need for extra randomness in the regressors. Third, instrumentation plays diﬀerent roles in the estimation of the system parameters, delivering identiﬁcation for the structural coeﬀicients in the usual way, but raising convergence rates for the threshold eﬀect parameters and improving eﬀiciency for the threshold point. Speciﬁcation tests are developed to test for the presence of endogeneity and threshold eﬀects without relying on instrumentation of the covariates. The threshold eﬀect test extends conventional parametric structural change tests to the nonparametric case. A wild bootstrap procedure is suggested to deliver ﬁnite sample critical values for both tests. Simulation studies corroborate the theory and the asymptotics. An empirical application is conducted to explore the eﬀects of 401(k) retirement programs on savings, illustrating the relevance of threshold models in treatment eﬀects evaluation in the presence of endogeneity.
Yu, Ping and Phillips, Peter C.B., "Threshold Regression with Endogeneity" (2014). Cowles Foundation Discussion Papers. 2380.