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On the Least Squares Estimation of Multiple-Threshold-Variable Autoregressive Models

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  • Xinyu Zhang
  • Dong Li
  • Howell Tong

Abstract

Most threshold models to-date contain a single threshold variable. However, in many empirical applications, models with multiple threshold variables may be needed and are the focus of this article. For the sake of readability, we start with the Two-Threshold-Variable Autoregressive (2-TAR) model and study its Least Squares Estimation (LSE). Among others, we show that the respective estimated thresholds are asymptotically independent. We propose a new method, namely the weighted Nadaraya-Watson method, to construct confidence intervals for the threshold parameters, that turns out to be, as far as we know, the only method to-date that enjoys good probability coverage, regardless of whether the threshold variables are endogenous or exogenous. Finally, we describe in some detail how our results can be extended to the K-Threshold-Variable Autoregressive (K-TAR) model, K > 2. We assess the finite-sample performance of the LSE by simulation and present two real examples to illustrate the efficacy of our modeling.

Suggested Citation

  • Xinyu Zhang & Dong Li & Howell Tong, 2024. "On the Least Squares Estimation of Multiple-Threshold-Variable Autoregressive Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 42(1), pages 215-228, January.
    • Handle: RePEc:taf:jnlbes:v:42:y:2024:i:1:p:215-228
    DOI: 10.1080/07350015.2023.2174124
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    References listed on IDEAS

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    1. Bruce E. Hansen, 2000. "Sample Splitting and Threshold Estimation," Econometrica, Econometric Society, vol. 68(3), pages 575-604, May.
    2. Haiqiang Chen & Terence Chong & Jushan Bai, 2012. "Theory and Applications of TAR Model with Two Threshold Variables," Econometric Reviews, Taylor & Francis Journals, vol. 31(2), pages 142-170.
    3. 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.
    4. C. S. Wong & W. K. Li, 1998. "A note on the corrected Akaike information criterion for threshold autoregressive models," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(1), pages 113-124, January.
    5. 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.
    6. Jan G. De Gooijer, 2001. "Cross‐validation Criteria for Setar Model Selection," Journal of Time Series Analysis, Wiley Blackwell, vol. 22(3), pages 267-281, May.
    7. Yu, Ping & Phillips, Peter C.B., 2018. "Threshold regression asymptotics: From the compound Poisson process to two-sided Brownian motion," Economics Letters, Elsevier, vol. 172(C), pages 123-126.
    8. Li, Dong & Ling, Shiqing & Li, Wai Keung, 2013. "Asymptotic Theory On The Least Squares Estimation Of Threshold Moving-Average Models," Econometric Theory, Cambridge University Press, vol. 29(3), pages 482-516, June.
    9. Cai, Zongwu, 2002. "Regression Quantiles For Time Series," Econometric Theory, Cambridge University Press, vol. 18(1), pages 169-192, February.
    10. 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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    Cited by:

    1. Ma, Zongguo & Ding, Chenhui & Wang, Xu & Huang, Qiaozhi, 2025. "Carbon emission reduction development, digital economy, and green transformation of China's manufacturing industry," International Review of Financial Analysis, Elsevier, vol. 102(C).

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    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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