0, α > 0, the η t are independent standard normal random variables and Elog |φ + η t √α| ⩾ 0. We show that the maximum likelihood estimator of (φ, α) is consistent and asymptotically normal. Combination of this result with that in Ling ([11]) for the stationary case gives the asymptotic normality of the maximum likelihood estimator of φ for any φ in the real line, with a root-n rate of convergence. This is in contrast to the results for the classical ar(1) model, corresponding to α = 0. Copyright 2008, Oxford University Press."> 0, α > 0, the η t are independent standard normal random variables and Elog |φ + η t √α| ⩾ 0. We show that the maximum likelihood estimator of (φ, α) is consistent and asymptotically normal. Combination of this result with that in Ling ([11]) for the stationary case gives the asymptotic normality of the maximum likelihood estimator of φ for any φ in the real line, with a root-n rate of convergence. This is in contrast to the results for the classical ar(1) model, corresponding to α = 0. Copyright 2008, Oxford University Press."> 0, α > 0, the η t are independent standard normal random variables and Elog |φ + η t √α| ⩾ 0. We show that the maximum likeliho">
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Asymptotic inference for a nonstationary double AR (1) model

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  • Shiqing Ling
  • Dong Li
Abstract
We investigate the nonstationary double ar(1) model, where ω > 0, α > 0, the η t are independent standard normal random variables and Elog |φ + η t √α| ⩾ 0. We show that the maximum likelihood estimator of (φ, α) is consistent and asymptotically normal. Combination of this result with that in Ling ([11]) for the stationary case gives the asymptotic normality of the maximum likelihood estimator of φ for any φ in the real line, with a root-n rate of convergence. This is in contrast to the results for the classical ar(1) model, corresponding to α = 0. Copyright 2008, Oxford University Press.

Suggested Citation

  • Shiqing Ling & Dong Li, 2008. "Asymptotic inference for a nonstationary double AR (1) model," Biometrika, Biometrika Trust, vol. 95(1), pages 257-263.
    • Handle: RePEc:oup:biomet:v:95:y:2008:i:1:p:257-263
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    1. Djogbenou, Antoine & Inan, Emre & Jasiak, Joann, 2023. "Time-varying coefficient DAR model and stability measures for stablecoin prices: An application to Tether," Journal of International Money and Finance, Elsevier, vol. 139(C).
    2. Aknouche, Abdelhakim & Al-Eid, Eid M. & Hmeid, Aboubakry M., 2011. "Offline and online weighted least squares estimation of nonstationary power ARCH processes," Statistics & Probability Letters, Elsevier, vol. 81(10), pages 1535-1540, October.
    3. 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.
    4. Abdelhakim Aknouche, 2015. "Quadratic random coefficient autoregression with linear-in-parameters volatility," Statistical Inference for Stochastic Processes, Springer, vol. 18(2), pages 99-125, July.
    5. Karapanagiotidis, Paul, 2013. "Empirical evidence for nonlinearity and irreversibility of commodity futures prices," MPRA Paper 56801, University Library of Munich, Germany.
    6. Zhu Huafeng & Zhang Xingfa & Liang Xin & Li Yuan, 2018. "Moving Average Model with an Alternative GARCH-Type Error," Journal of Systems Science and Information, De Gruyter, vol. 6(2), pages 165-177, April.
    7. Min Chen & Dong Li & Shiqing Ling, 2014. "Non-Stationarity And Quasi-Maximum Likelihood Estimation On A Double Autoregressive Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 35(3), pages 189-202, May.
    8. Tao, Yubo & Phillips, Peter C.B. & Yu, Jun, 2019. "Random coefficient continuous systems: Testing for extreme sample path behavior," Journal of Econometrics, Elsevier, vol. 209(2), pages 208-237.
    9. 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.
    10. Nielsen, Heino Bohn & Rahbek, Anders, 2014. "Unit root vector autoregression with volatility induced stationarity," Journal of Empirical Finance, Elsevier, vol. 29(C), pages 144-167.
    11. Guodong Li & Qianqian Zhu & Zhao Liu & Wai Keung Li, 2017. "On Mixture Double Autoregressive Time Series Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(2), pages 306-317, April.
    12. Christian Francq & Jean-Michel Zakoian, 2013. "Inference in Non Stationary Asymmetric Garch Models," Working Papers 2013-11, Center for Research in Economics and Statistics.
    13. Abdelhakim Aknouche & Eid Al-Eid, 2012. "Asymptotic inference of unstable periodic ARCH processes," Statistical Inference for Stochastic Processes, Springer, vol. 15(1), pages 61-79, April.
    14. Christian Gouriéroux & Jean-Michel Zakoïan, 2017. "Local explosion modelling by non-causal process," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 737-756, June.
    15. Tao, Yubo & Phillips, Peter C.B. & Yu, Jun, 2017. "Random Coefficient Continuous Systems: Testing for Extreme Sample Path Behaviour," Economics and Statistics Working Papers 18-2017, Singapore Management University, School of Economics.
    16. 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.
    17. Huan Gong & Dong Li, 2020. "On the three‐step non‐Gaussian quasi‐maximum likelihood estimation of heavy‐tailed double autoregressive models," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(6), pages 883-891, November.
    18. Abdelhakim Aknouche, 2012. "Multistage weighted least squares estimation of ARCH processes in the stable and unstable cases," Statistical Inference for Stochastic Processes, Springer, vol. 15(3), pages 241-256, October.
    19. Aknouche Abdelhakim, 2013. "Two-Stage Weighted Least Squares Estimation of Nonstationary Random Coefficient Autoregressions," Journal of Time Series Econometrics, De Gruyter, vol. 5(1), pages 25-46, January.
    20. Liu, Ji-Chun, 2012. "Structure of a double autoregressive process driven by a hidden Markov chain," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1468-1473.
    21. 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).
    22. Aknouche, Abdelhakim, 2015. "Unified quasi-maximum likelihood estimation theory for stable and unstable Markov bilinear processes," MPRA Paper 69572, University Library of Munich, Germany.
    23. Zhu, Qianqian & Zheng, Yao & Li, Guodong, 2018. "Linear double autoregression," Journal of Econometrics, Elsevier, vol. 207(1), pages 162-174.
    24. Yoon, Gawon, 2016. "Stochastic unit root processes: Maximum likelihood estimation, and new Lagrange multiplier and likelihood ratio tests," Economic Modelling, Elsevier, vol. 52(PB), pages 725-732.

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