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On the least squares estimation of multiple-regime threshold autoregressive models

Author

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  • Li, Dong
  • Ling, Shiqing

Abstract

This paper studies the least squares estimator (LSE) of the multiple-regime threshold autoregressive (TAR) model and establishes its asymptotic theory. It is shown that the LSE is strongly consistent. When the autoregressive function is discontinuous over each threshold, the estimated thresholds are n-consistent and asymptotically independent, each of which converges weakly to the smallest minimizer of a one-dimensional two-sided compound Poisson process. The remaining parameters are n-consistent and asymptotically normal. The theory of Chan (1993) is revisited and a numerical approach is proposed to simulate the limiting distribution of the estimated threshold via simulating a related compound Poisson process. Based on the numerical result, one can construct a confidence interval for the unknown threshold. This issue is not straightforward and has remained as an open problem since the publication of Chan (1993). This paper provides not only a solution to this long-standing open problem, but also provides methodological contributions to threshold models. Simulation studies are conducted to assess the performance of the LSE in finite samples. The results are illustrated with an application to the quarterly U.S. real GNP data over the period 1947–2009.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:econom:v:167:y:2012:i:1:p:240-253
    DOI: 10.1016/j.jeconom.2011.11.006
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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. Coakley, Jerry & Fuertes, Ana-Maria & Perez, Maria-Teresa, 2003. "Numerical issues in threshold autoregressive modeling of time series," Journal of Economic Dynamics and Control, Elsevier, vol. 27(11-12), pages 2219-2242, September.
    3. Bruce E. Hansen, 2000. "Sample Splitting and Threshold Estimation," Econometrica, Econometric Society, vol. 68(3), pages 575-604, May.
    4. Li, C W & Li, W K, 1996. "On a Double-Threshold Autoregressive Heteroscedastic Time Series Model," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(3), pages 253-274, May-June.
    5. Gonzalo, Jesus & Wolf, Michael, 2005. "Subsampling inference in threshold autoregressive models," Journal of Econometrics, Elsevier, vol. 127(2), pages 201-224, August.
    6. Koop, Gary & Potter, Simon M, 1999. "Dynamic Asymmetries in U.S. Unemployment," Journal of Business & Economic Statistics, American Statistical Association, vol. 17(3), pages 298-312, July.
    7. Jushan Bai & Pierre Perron, 2003. "Computation and analysis of multiple structural change models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 18(1), pages 1-22.
    8. Mehmet Caner & Bruce E. Hansen, 2001. "Threshold Autoregression with a Unit Root," Econometrica, Econometric Society, vol. 69(6), pages 1555-1596, November.
    9. Christian Genest & Bruno Rémillard, 2004. "Test of independence and randomness based on the empirical copula process," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(2), pages 335-369, December.
    10. 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.
    11. Hansen Bruce E., 1997. "Inference in TAR Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 2(1), pages 1-16, April.
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    Citations

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    Cited by:

    1. Li, Dong & Ling, Shiqing & Zakoïan, Jean-Michel, 2015. "Asymptotic inference in multiple-threshold double autoregressive models," Journal of Econometrics, Elsevier, vol. 189(2), pages 415-427.
    2. Chen, Haiqiang, 2015. "Robust Estimation And Inference For Threshold Models With Integrated Regressors," Econometric Theory, Cambridge University Press, vol. 31(04), pages 778-810, August.
    3. repec:bla:jtsera:v:38:y:2017:i:1:p:99-119 is not listed on IDEAS
    4. Jesús Gonzalo & Jean-Yves Pitarakis, 2013. "Estimation and inference in threshold type regime switching models," Chapters,in: Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 8, pages 189-205 Edward Elgar Publishing.
    5. Li, Dong & Tong, Howell, 2016. "Nested sub-sample search algorithm for estimation of threshold models," LSE Research Online Documents on Economics 68880, London School of Economics and Political Science, LSE Library.
    6. Andros Kourtellos & Thanasis Stengos & Yiguo Sun, 2017. "Endogeneity in Semiparametric Threshold Regression," University of Cyprus Working Papers in Economics 10-2017, University of Cyprus Department of Economics.
    7. repec:wyi:journl:002203 is not listed on IDEAS
    8. Yu, Ping, 2015. "Adaptive estimation of the threshold point in threshold regression," Journal of Econometrics, Elsevier, vol. 189(1), pages 83-100.
    9. Ke Zhu & Wai Keung Li & Philip L. H. Yu, 2017. "Buffered Autoregressive Models With Conditional Heteroscedasticity: An Application to Exchange Rates," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(4), pages 528-542, October.
    10. Dong Li & Shiqing Ling & Jean-Michel Zakoian, 2013. "Asymptotic Inference in Multiple-Threshold Nonlinear Time Series Models," Working Papers 2013-51, Center for Research in Economics and Statistics.
    11. Yang, Yaxing & Ling, Shiqing, 2017. "Self-weighted LAD-based inference for heavy-tailed threshold autoregressive models," Journal of Econometrics, Elsevier, vol. 197(2), pages 368-381.
    12. Sokbae Lee & Yuan Liao & Myung Hwan Seo & Youngki Shin, 2016. "Oracle Estimation of a Change Point in High Dimensional Quantile Regression," Papers 1603.00235, arXiv.org, revised Dec 2016.
    13. Chan, Ngai Hang & Yau, Chun Yip & Zhang, Rong-Mao, 2015. "LASSO estimation of threshold autoregressive models," Journal of Econometrics, Elsevier, vol. 189(2), pages 285-296.

    More about this item

    Keywords

    Asymptotic distribution; Compound Poisson process; Least squares estimation; Multiple-regime TAR model;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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