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NEW CONTROL FUNCTION APPROACHES IN THRESHOLD REGRESSION WITH ENDOGENEITY

Published online by Cambridge University Press:  16 March 2023

Ping Yu ,
Qin Liao  and
Peter C. B. Phillips
Ping Yu*
Affiliation:
University of Hong Kong
Qin Liao
Affiliation:
University of Hong Kong
Peter C. B. Phillips
Affiliation:
Yale University, University of Auckland, University of Southampton, and Singapore Management University
*
Address correspondence to Ping Yu, Faculty of Business and Economics, The University of Hong Kong, Pokfulam, Hong Kong; e-mail: [email protected].
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Abstract

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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.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 5 , October 2024 , pp. 1065 - 1119
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

Footnotes

We thank the Co-Editor, Simon Lee, and three referees for helpful comments on earlier versions of this paper. We also thank Chirok Han, Bruce Hansen, Shengjie Hong, Zhongxiao Jia, Hiroyuki Kasahara, Seojeong Lee, Hongyi Li, Jaimie Lien, Sangsoo Park, Wenwu Wang, Jason Wu, Zhiguo Xiao, Chuancun Yin, and seminar participants at CUHK, ESWC2020, QNU, UWAC2018, Korea University, and Tsinghua University for their helpful suggestions. Phillips and Yu acknowledge support from the GRF of Hong Kong Government under Grant No. 17520716, and Phillips acknowledges research support under NSF Grant No. SES 18-50860 and the Kelly Fund at the University of Auckland.

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