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Inference in Threshold Models

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Abstract

This paper develops new statistical inference methods for the parameters in threshold regression models. In particular, we develop a test for homogeneity of the threshold parameter and a test for linear restrictions on the regression coefficients. The tests are built upon a transformed partial-sum process after re-ordering the observations based on the rank of the threshold variable, which recasts the cross-sectional threshold problem into the time-series structural break analogue. The asymptotic distributions of the test statistics are derived using this novel approach, and the finite sample properties are studied in Monte Carlo simulations. We apply the new tests to the tipping point problem studied by Card, Mas, and Rothstein (2008), and statistically justify that the location of the tipping point varies across tracts.

Suggested Citation

  • Yoonseok Lee & Yulong Wang, 2020. "Inference in Threshold Models," Center for Policy Research Working Papers 223, Center for Policy Research, Maxwell School, Syracuse University.
    • Handle: RePEc:max:cprwps:223
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    File URL: https://surface.syr.edu/cpr/255/
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    References listed on IDEAS

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    1. Seo, Myung Hwan & Linton, Oliver, 2007. "A smoothed least squares estimator for threshold regression models," Journal of Econometrics, Elsevier, vol. 141(2), pages 704-735, December.
    2. Graham Elliott & Ulrich K. Müller & Mark W. Watson, 2015. "Nearly Optimal Tests When a Nuisance Parameter Is Present Under the Null Hypothesis," Econometrica, Econometric Society, vol. 83, pages 771-811, March.
    3. Caner, Mehmet & Hansen, Bruce E., 2004. "Instrumental Variable Estimation Of A Threshold Model," Econometric Theory, Cambridge University Press, vol. 20(5), pages 813-843, October.
    4. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    5. Elliott, Graham & Muller, Ulrich K., 2007. "Confidence sets for the date of a single break in linear time series regressions," Journal of Econometrics, Elsevier, vol. 141(2), pages 1196-1218, December.
    6. David Card & Alexandre Mas & Jesse Rothstein, 2008. "Tipping and the Dynamics of Segregation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 123(1), pages 177-218.
    7. Sokbae Lee & Yuan Liao & Myung Hwan Seo & Youngki Shin, 2018. "Factor-Driven Two-Regime Regression," Papers 1810.11109, arXiv.org, revised Sep 2020.
    8. Jushan Bai & Pierre Perron, 1998. "Estimating and Testing Linear Models with Multiple Structural Changes," Econometrica, Econometric Society, vol. 66(1), pages 47-78, January.
    9. Lee, Yoonseok & Wang, Yulong, 2023. "Threshold regression with nonparametric sample splitting," Journal of Econometrics, Elsevier, vol. 235(2), pages 816-842.
    10. Andrews, Donald W K, 1993. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Econometrica, Econometric Society, vol. 61(4), pages 821-856, July.
    11. Li, Dong & Ling, Shiqing, 2012. "On the least squares estimation of multiple-regime threshold autoregressive models," Journal of Econometrics, Elsevier, vol. 167(1), pages 240-253.
    12. Gonzalo, Jesus & Pitarakis, Jean-Yves, 2002. "Estimation and model selection based inference in single and multiple threshold models," Journal of Econometrics, Elsevier, vol. 110(2), pages 319-352, October.
    13. Yu, Ping, 2012. "Likelihood estimation and inference in threshold regression," Journal of Econometrics, Elsevier, vol. 167(1), pages 274-294.
    14. Hidalgo, Javier & Lee, Jungyoon & Seo, Myung Hwan, 2019. "Robust inference for threshold regression models," Journal of Econometrics, Elsevier, vol. 210(2), pages 291-309.
    15. Yoonseok Lee & Yulong Wang, 2020. "Nonparametric Sample Splitting," Center for Policy Research Working Papers 222, Center for Policy Research, Maxwell School, Syracuse University.
    16. Park, Joon Y. & Phillips, Peter C.B., 1999. "Asymptotics For Nonlinear Transformations Of Integrated Time Series," Econometric Theory, Cambridge University Press, vol. 15(3), pages 269-298, June.
    17. Andrews, Donald W K & Ploberger, Werner, 1994. "Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative," Econometrica, Econometric Society, vol. 62(6), pages 1383-1414, November.
    18. Hansen Bruce E., 1997. "Inference in TAR Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 2(1), pages 1-16, April.
    19. Lee, Sokbae & Seo, Myung Hwan & Shin, Youngki, 2011. "Testing for Threshold Effects in Regression Models," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 220-231.
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    Cited by:

    1. Isaiah Andrews & Toru Kitagawa & Adam McCloskey, 2024. "Inference on Winners," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 139(1), pages 305-358.
    2. Andrews, Isaiah & Kitagawa, Toru & McCloskey, Adam, 2021. "Inference after estimation of breaks," Journal of Econometrics, Elsevier, vol. 224(1), pages 39-59.

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    More about this item

    Keywords

    Threshold Regression; Test; Homogeneous Threshold; Linear Restriction; Tipping Point;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C24 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Truncated and Censored Models; Switching Regression Models; Threshold Regression Models

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