Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T20:38:11.308Z Has data issue: false hasContentIssue false
  • English
  • Français

TESTING FOR HOMOGENEOUS THRESHOLDS IN THRESHOLD REGRESSION MODELS

Published online by Cambridge University Press:  28 October 2022

Yoonseok Lee Open the ORCID record for Yoonseok Lee [Opens in a new window]  and
Yulong Wang Open the ORCID record for Yulong Wang [Opens in a new window]
Yoonseok Lee
Affiliation:
Syracuse University
Yulong Wang*
Affiliation:
Syracuse University
*
Address correspondence to Yulong Wang, Department of Economics and Center for Policy Research, Syracuse University, 127 Eggers Hall, Syracuse, NY 13244, USA; e-mail: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper develops a test for homogeneity of the threshold parameter in threshold regression models. The test has a natural interpretation from time series perspectives and can also be applied to test for additional change points in the structural break models. The limiting distribution of the test statistic is derived, and the finite sample properties are studied in Monte Carlo simulations. We apply the new test to the tipping point problem studied by Card, Mas, and Rothstein (2008, Quarterly Journal of Economics 123, 177–218) and statistically justify that the location of the tipping point varies across tracts.

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 3 , June 2024 , pp. 608 - 651
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), 2022. Published by Cambridge University Press

Footnotes

We thank the editor (Peter Phillips), the Co-Editor (Simon Lee), and three anonymous referees for very constructive comments and suggestions. We also thank Ulrich Müller, Bo Honoré, Mark Watson, Kirill Evidokimov, Myung Hwan Seo, Zhijie Xiao, and participants at numerous seminar/conference presentations for very helpful discussions. Lee acknowledges financial support from the CUSE grant. Wang acknowledges financial support from the Appleby-Mosher grant.

References

REFERENCES

Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61(4), 821–856.CrossRefGoogle Scholar
Andrews, D.W.K. & Ploberger, W. (1994) Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62(6), 1383–1414.CrossRefGoogle Scholar
Bae, J., Jun, D., & Levental, S. (2010) The uniform CLT for martingale difference arrays under the uniformly Integrable entropy. Bulletin of the Korean Mathematical Society 47(1), 39–51.CrossRefGoogle Scholar
Bai, J., Lumsdaine, R.L., & Stock, J.H. (1998) Testing for and dating common breaks in multivariate time series. Review of Economic Studies 65(3), 395–432.CrossRefGoogle Scholar
Bai, J. & Perron, P. (1998) Estimating and testing linear models with multiple structural changes. Econometrica 66(1), 47–78.CrossRefGoogle Scholar
Bhattacharya, P.K. (1974) Convergence of sample paths of normalized sums of induced order statistics. Annals of Statistics 2(5), 1034–1039.CrossRefGoogle Scholar
Bhattacharya, P.K. (1984) 18 induced order statistics: Theory and applications. Handbook of Statistics 4, 383–403.CrossRefGoogle Scholar
Caner, M. & Hansen, B.E. (2001) Threshold autoregressions with a unit root. Econometrica 69(6), 1555–1596.CrossRefGoogle Scholar
Caner, M. & Hansen, B.E. (2004) Instrumental variable estimation of a threshold model. Econometric Theory 20(5), 813–843.CrossRefGoogle Scholar
Card, D., Mas, A., & Rothstein, J. (2008) Tipping and the dynamics of segregation. Quarterly Journal of Economics 123(1), 177–218.CrossRefGoogle Scholar
Casini, A. & Perron, P. (2021a) Continuous record Laplace-based inference about the break date in structural change models. Journal of Econometrics 224(1), 3–21.CrossRefGoogle Scholar
Casini, A. & Perron, P. (2021b) Continuous Record Asymptotics for Change-Point Models. Working paper. Preprint, arXiv:1803.10881 Google Scholar
Casini, A. & Perron, P. (2022) Generalized Laplace inference in multiple change-points models. Econometric Theory 38(1), 35–65.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory . Oxford University Press.CrossRefGoogle Scholar
Elliott, G. & Müller, U.K. (2007) Confidence sets for the date of a single break in linear time series regressions. Journal of Econometrics 141(2), 1196–1218.CrossRefGoogle Scholar
Elliott, G. & Müller, U.K. (2014) Pre and post break parameter inference. Journal of Econometrics 180(2), 141–157.CrossRefGoogle Scholar
Elliott, G., Müller, U.K., & Watson, M.W. (2015) Nearly optimal tests when a nuisance parameter is present under the null hypothesis. Econometrica 83(2), 771–811.CrossRefGoogle Scholar
Gonzalo, J. & Pitarakis, J. (2002) Estimation and model selection based inference in single and multiple threshold models. Journal of Econometrics 110(2), 319–352.CrossRefGoogle Scholar
Hansen, B.E. (1996) Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64(2), 413–430.CrossRefGoogle Scholar
Hansen, B.E. (2000) Sample splitting and threshold estimation. Econometrica 68(3), 575–603.CrossRefGoogle Scholar
Hidalgo, J., Lee, J., & Seo, M.H. (2019) Robust inference for threshold regression models. Journal of Econometrics 210(2), 291–309.CrossRefGoogle Scholar
Jiang, L., Wang, X., & Yu, J. (2018) New distribution theory for the estimation of structural break point in mean. Journal of Econometrics 205(1), 156–176.CrossRefGoogle Scholar
Jiang, L., Wang, X., & Yu, S. (2020) In-fill asymptotic theory for structural break point in autoregression: A unified theory. Econometric Reviews 40(4), 359–386.CrossRefGoogle Scholar
Kosorok, M.R. (2008) Introduction to Empirical Processes and Semiparametric Inference . Springer.CrossRefGoogle Scholar
Kourtellos, A., Stengos, T., & Tan, C.M. (2016) Structural threshold regression. Econometric Theory 32(4), 827–860.CrossRefGoogle Scholar
Lee, S., Liao, Y., Seo, M.H., & Shin, Y. (2021) Factor-driven two-regime regression. Annals of Statistics 49(3), 1656–1678.CrossRefGoogle Scholar
Lee, S., Seo, M.H., & Shin, Y. (2011) Testing for threshold effects in regression models. Journal of the American Statistical Association 106(493), 220–231.CrossRefGoogle Scholar
Li, D. & Ling, S. (2012) On the least squares estimation of multiple-regime threshold autoregressive models. Journal of Econometrics 167(1), 240–253.CrossRefGoogle Scholar
Li, H. & Müller, U.K. (2009) Valid inference in partially unstable general method of moment models. Review of Economic Studies 76(1), 343–365.CrossRefGoogle Scholar
Li, Q. & Racine, J.S. (2007) Nonparametric Econometrics: Theory and Practice . Princeton University Press.Google Scholar
Miao, K., Su, L., & Wang, W. (2020) Panel threshold regressions with latent group structures. Journal of Econometrics 214(2), 451–481.CrossRefGoogle Scholar
Nyblom, J. (1989) Testing for the Constancy of parameters over time. Journal of the American Statistical Association 84(405), 223–230.CrossRefGoogle Scholar
Schelling, T.C. (1971) Dynamic models of segregation. Journal of Mathematical Sociology 1(2), 143–186.CrossRefGoogle Scholar
Sen, P.K. (1976) A note on invariance principles for induced order statistics. Annals of Probability 4(3), 474–479.CrossRefGoogle Scholar
Seo, M.H. & Linton, O. (2007) A smooth least squares estimator for threshold regression models. Journal of Econometrics 141(2), 704–735.CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes with Applications to Statistics . Springer.CrossRefGoogle Scholar
Yang, S.S. (1981) Linear functions of concomitants of order statistics with application to nonparametric estimation of a regression function. Journal of the American Statistical Association 76(375), 658–662.CrossRefGoogle Scholar
Yang, S.S. (1985) A smooth nonparametric estimator of a quantile function. Journal of the American Statistical Association 80(392), 1004–1011.CrossRefGoogle Scholar
Yu, P. (2012) Likelihood estimation and inference in threshold regression. Journal of Econometrics 167(1), 274–294.CrossRefGoogle Scholar
Yu, P. & Fan, X. (2021) Threshold regression with a threshold boundary. Journal of Business and Economic Statistics 39(4), 953–971.CrossRefGoogle Scholar
Yu, P. & Phillips, P. (2018) Threshold regression with Endogeneity. Journal of Econometrics 203(1), 50–68.CrossRefGoogle Scholar