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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.

References

REFERENCES

Acemoglu, D., Johnson, S., & Robinson, J.A. (2001) The colonial origins of comparative development: An empirical investigation. American Economic Review 91, 1369–1401.CrossRefGoogle Scholar
Bai, J. (1997) Estimation of a change point in multiple regression models. Review of Economics and Statistics 79, 551–563.CrossRefGoogle Scholar
Bhattacharya, P.K. & Brockwell, P.J. (1976) The minimum of an additive process with applications to signal estimation and storage theory. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 37, 51–75.CrossRefGoogle Scholar
Billingsley, P. (1961) The Lindeberg–Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788–792.Google Scholar
Boldea, O., Hall, A.R., & Han, S. (2012) Asymptotic distribution theory for break point estimators in models estimated via 2SLS. Econometric Reviews 31, 1–33.CrossRefGoogle Scholar
Caner, M. & Hansen, B.E. (2004) Instrumental variable estimation of a threshold model. Econometric Theory 20, 813–843.CrossRefGoogle Scholar
Chan, K.S. (1993) Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Annals of Statistics 21, 520–533.CrossRefGoogle Scholar
Chan, K.S. & Tsay, R.S. (1998) Limiting properties of the least squares estimator of a continuous threshold autoregressive model. Biometrika 85, 413–426.CrossRefGoogle Scholar
Christopoulos, D., McAdam, P., & Tzavalis, E. (2021) Dealing with endogeneity in threshold models using copulas. Journal of Business & Economic Statistics 39, 166–178.CrossRefGoogle Scholar
Egert, B. & Morales-Zumaquero, A. (2008) Exchange rate regimes, foreign exchange volatility and export performance in central and eastern Europe: Just another blur project? Review of Development Economics 12, 577–593.CrossRefGoogle Scholar
Frankel, J. & Romer, D. (1999) Does trade cause growth? American Economic Review 89, 379–399.CrossRefGoogle Scholar
Gouriéroux, C., Monfort, A., Renault, E., & Trognon, A. (1987) Generalized residuals. Journal of Econometrics 34, 5–32.CrossRefGoogle Scholar
Grier, K.B. & Smallwood, A.D. (2013) Exchange rate shocks and trade: A multivariate GARCH-M approach. Journal of International Money and Finance 37, 282–305.CrossRefGoogle Scholar
Hall, A.R., Han, S., & Boldea, O. (2012) Inference regarding multiple structural changes in linear models with endogenous regressors. Journal of Econometrics 170, 281–302.CrossRefGoogle ScholarPubMed
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and its Application. Academic Press.Google Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.CrossRefGoogle Scholar
Hansen, B.E. (2000) Sample splitting and threshold estimation. Econometrica 68, 575–603.CrossRefGoogle Scholar
Hansen, B.E. (2011) Threshold autoregression in economics. Statistics and Its Interface 4, 123–127.CrossRefGoogle Scholar
Hansen, B.E. (2017) Regression kink with an unknown threshold . Journal of Business & Economics Statistics 35, 228–240.CrossRefGoogle Scholar
Heckman, J.J. (1979) Sample selection bias as a specification error. Econometrica 47, 153–161.CrossRefGoogle Scholar
Kim, J. & Pollard, D. (1990) Cube root asymptotics. Annals of Statistics 18, 191–219.CrossRefGoogle Scholar
Kourtellos, A., Stengos, T., & Sun, Y. (2022) Endogeneity in semiparametric threshold regression. Econometric Theory 38, 562–595.CrossRefGoogle Scholar
Kourtellos, A., Stengos, T., & Tan, C.-M. (2016) Structural threshold regression. Econometric Theory 32, 827–860.CrossRefGoogle Scholar
Murphy, K.M. & Topel, R.H. (1985) Estimation and inference in two-step econometric models. Journal of Business & Economic Statistics 3, 370–379.CrossRefGoogle Scholar
Navarro, S. (2008) Control functions. In Durlauf, S.N. & Blume, L.E. (eds.), The New Palgrave Dictionary of Economics, 2nd ed., pp. 20–28. Palgrave Macmillan.Google Scholar
Newey, W.K. (1984) A method of moments interpretation of sequential estimators. Economics Letters 14, 201–206.CrossRefGoogle Scholar
Newey, W.K. & McFadden, D.L. (1994) Large sample estimation and hypothesis testing. In Eagle, R.F. & McFadden, D.L. (eds.), Handbook of Econometrics, vol. 4, pp. 2111–2245. Elsevier Science B.V. CrossRefGoogle Scholar
Papageorgiou, C. (2002) Trade as a threshold variable for multiple regimes. Economics Letters 77, 85–91.CrossRefGoogle Scholar
Perron, P. & Qu, Z. (2006) Estimating restricted structural change models. Journal of Econometrics 134, 373–399.CrossRefGoogle Scholar
Perron, P. & Yamamoto, Y. (2014) A note on estimating and testing for multiple structural changes in models with endogenous regressors via 2SLS. Econometric Theory 30, 491–507.CrossRefGoogle Scholar
Petruccelli, J.D. (1992) On the approximation of time series by threshold autoregressive models. Sankhya Series B 54, 54–61.Google Scholar
Seo, M.H. & Shin, Y. (2016) Dynamic panels with threshold effect and endogeneity. Journal of Econometrics 195, 169–186.CrossRefGoogle Scholar
Tan, C.M. (2010) No one true path: Uncovering the interplay between geography, institutions, and fractionalization in economic development. Journal of Applied Econometrics 25(7), 1100–1127.CrossRefGoogle Scholar
Tenreyro, S. (2007) On the trade impact of nominal exchange rate volatility. Journal of Development Economics 82, 485–508.CrossRefGoogle Scholar
Wooldridge, J.M. (2015) Control function methods in applied econometrics. Journal of Human Resources 50, 420–445.CrossRefGoogle Scholar
Yu, P. (2012) Likelihood estimation and inference in threshold regression. Journal of Econometrics 167, 274–294.CrossRefGoogle Scholar
Yu, P. (2013) Inconsistency of 2SLS estimators in threshold regression with endogeneity. Economics Letters 120, 532–536.CrossRefGoogle Scholar
Yu, P. (2014) The bootstrap in threshold regression. Econometric Theory 30, 676–714.CrossRefGoogle Scholar
Yu, P. (2015) Adaptive estimation of the threshold point in threshold regression. Journal of Econometrics 189, 83–100.CrossRefGoogle Scholar
Yu, P., Liao, Q., & Phillips, P.C.B. (2018) Inference and Specification Testing in Threshold Regression with Endogeneity. Cowles Foundation Discussion Paper No. 2209, Yale University.CrossRefGoogle Scholar
Yu, P. & Phillips, P.C.B. (2018) Threshold regression with endogeneity. Journal of Econometrics 203, 50–68.CrossRefGoogle Scholar
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