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On a Threshold Double Autoregressive Model

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  • Dong Li
  • Shiqing Ling
  • Rongmao Zhang

Abstract

This article first proposes a score-based test for a double autoregressive model against a threshold double autoregressive (AR) model. It is an asymptotically distribution-free test and is easy to implement in practice. The article further studies the quasi-maximum likelihood estimation of a threshold double autoregressive model. It is shown that the estimated threshold is n -consistent and converges weakly to a functional of a two-sided compound Poisson process and the remaining parameters are asymptotically normal. Our results include the asymptotic theory of the estimator for threshold AR models with autoregressive conditional heteroscedastic (ARCH) errors and threshold ARCH models as special cases, each of which is also new in literature. Two portmanteau-type statistics are also derived for checking the adequacy of fitted model when either the error is nonnormal or the threshold is unknown. Simulation studies are conducted to assess the performance of the score-based test and the estimator in finite samples. The results are illustrated with an application to the weekly closing prices of Hang Seng Index. This article also includes the weak convergence of a score-marked empirical process on the space under an α-mixing assumption, which is independent of interest.

Suggested Citation

  • Dong Li & Shiqing Ling & Rongmao Zhang, 2016. "On a Threshold Double Autoregressive Model," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(1), pages 68-80, January.
    • Handle: RePEc:taf:jnlbes:v:34:y:2016:i:1:p:68-80
    DOI: 10.1080/07350015.2014.1001028
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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. Shiqing Ling & Dong Li, 2008. "Asymptotic inference for a nonstationary double AR (1) model," Biometrika, Biometrika Trust, vol. 95(1), pages 257-263.
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    7. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
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    9. Rabemananjara, R & Zakoian, J M, 1993. "Threshold Arch Models and Asymmetries in Volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(1), pages 31-49, Jan.-Marc.
    10. Shiqing Ling, 2004. "Estimation and testing stationarity for double‐autoregressive models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 63-78, February.
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    Cited by:

    1. Zhu, Huafeng & Zhang, Xingfa & Liang, Xin & Li, Yuan, 2017. "On a vector double autoregressive model," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 86-95.
    2. Gourieroux, Christian & Jasiak, Joann, 2019. "Robust analysis of the martingale hypothesis," Econometrics and Statistics, Elsevier, vol. 9(C), pages 17-41.
    3. Li, Dong & Tao, Yuxin & Yang, Yaxing & Zhang, Rongmao, 2023. "Maximum likelihood estimation for α-stable double autoregressive models," Journal of Econometrics, Elsevier, vol. 236(1).
    4. Guo, Shaojun & Li, Dong & Li, Muyi, 2019. "Strict stationarity testing and GLAD estimation of double autoregressive models," Journal of Econometrics, Elsevier, vol. 211(2), pages 319-337.

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