This paper studies control function (CF) approaches in endogenous threshold regression where the threshold variable is allowed to be endogenous. We first use a simple example to show that the structural threshold regression (STR) estimator of the threshold point in Kourtellos, Stengos and Tan (2016, Econometric Theory 32, 827–860) is inconsistent unless the endogeneity level of the threshold variable is low compared to the threshold effect. We correct the CF in the STR estimator to generate our first CF estimator using a method that extends the two-stage least squares procedure in Caner and Hansen (2004, Econometric Theory 20, 813–843). We develop our second CF estimator which can be treated as an extension of the classical CF approach in endogenous linear regression. Both these approaches embody threshold effect information in the conditional variance beyond that in the conditional mean. Given the threshold point estimates, we propose new estimates for the slope parameters. The first is a by-product of the CF approach, and the second type employs generalized method of moment (GMM) procedures based on two new sets of moment conditions. Simulation studies, in conjunction with the limit theory, show that our second CF estimator and confidence interval for the threshold point together with the associated second GMM estimator and confidence interval for the slope parameter dominate the other methods. We further apply the new estimation methodology to an empirical application from international trade to illustrate its usefulness in practice.
This paper studies the estimation and inferences in panel threshold regression with unobserved individual-specific threshold effects which is important from the practical perspective and is a distinguishing feature from traditional linear panel data models. It is shown that the within-regime differencing in the static model or the within-regime first-differencing in the dynamic model cannot generate consistent estimators of the threshold, so the correlated random effects models are suggested to handle the endogeneity in such general panel threshold models. We provide a unified estimation and inference framework that is valid for both the static and dynamic models and regardless of whether the unobserved individual-specific threshold effects exist or not. Especially, we propose alternative inference methods for the model parameters, which have better theoretical properties than the existing methods. Simulation studies and an empirical application illustrate the usefulness of our new estimation and inference methodology in practice.
We propose three new methods of inference for the threshold point in endogenous threshold regression and two speciﬁcation tests designed to assess the presence of endogeneity and threshold eﬀects without necessarily relying on instrumentation of the covariates. The ﬁrst inferential method is a parametric two-stage least squares method and is suitable when instruments are available. The second and third methods are based on smoothing the objective function of the integrated diﬀerence kernel estimator in diﬀerent ways and these methods do not require instrumentation. All three methods are applicable irrespective of endogeneity of the threshold variable. The two speciﬁcation tests are constructed using a score-type principle. 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. Simulations show good ﬁnite sample performance of these procedures and the methods provide flexibility in testing and inference for practitioners working with threshold models.
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.