Essays on threshold regression with endogeneity

Abstract

The first chapter proposes three inference methods for the threshold point in endogenous threshold regression and two specification tests for the presence of endogeneity and threshold effects without relying on the instrumentation of covariates. The first inference 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 do not require instrumentation. These three inference methods are applicable regardless of the endogeneity of the threshold variable. Both specification tests are score-type tests; especially, 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. The second chapter studies control function 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 magnitude of the threshold effect. We then correct the control function in the STR estimation to generate a consistent estimator using a method that extends the two-stage least squares procedure in Caner and Hansen (2004, Econometric Theory 20, 813-843). After that, we further develop the second approach which can be treated as an extension of the classical control function approach in endogenous linear regression. Both approaches explore the threshold effects in conditional variance beyond that in the conditional mean. Given the threshold point estimates, we propose two types of estimates for the slope parameters. The first type is a by-product of our control function approaches, and the second type employs the generalized method of moment (GMM) procedures based on two new sets of moment conditions. In terms of the performance of estimator and its confidence interval, our simulation studies, in conjunction with the asymptotic analyses, show that the second control function approach for the threshold point and the associated second GMM approach for slope parameters dominate the other methods. The third chapter investigates the relationship between firms' productivity and their probability of exporting with a model which combines the threshold regression proposed in Chapter 1 and probit regression. With cross-sectional data spanning from 2005 to 2007 in Chinese manufacturing, we start with composing a measure of total factor productivity for each firm within the sample. Then, we search for any possible thresholds in the relationship between our constructed measure and firms' foreign market participation behavior. Last, based on the detection outcome, we study any changes in the relationship of interest. Empirical results examine, once again, the validity of our methods proposed in Chapter 1 in estimating threshold points. Moreover, compelling evidence is implied on favoring the existence of threshold effects among the correlation between export possibility and productivity among the manufacturing industries. Specifically, the co-movement of the two variables directs oppositely in the two sides of the threshold point. Therefore, failing to recognize the threshold effect would inevitably lead to a mismeasure of the relationship.