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On stationarity and ergodicity of the bilinear model with applications to GARCH models

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  • Dennis Kristensen

Abstract

We establish sufficient conditions for the bilinear time‐series model to be strictly stationary and ergodic in terms of its associated Lyapunov exponent. In two special cases, we verify that the conditions are also necessary. We then use these results to give necessary and sufficient conditions for stationarity of specific generalized autoregressive conditionally heteroskedastic (GARCH) models which can be written as a bilinear model, including linear GARCH, Power GARCH, EGARCH among others. These results generalize the ones found in the studies of, among others, Bougerol and Picard [Journal of Econometrics 52 (1992) 115], Duan [Journal of Econometrics 79 (1997) 97] and Nelson [Econometric Theory 6 (1990) 318]. In many cases, the conditions are weaker than the ones found elsewhere in the literature.

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  • Dennis Kristensen, 2009. "On stationarity and ergodicity of the bilinear model with applications to GARCH models," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(1), pages 125-144, January.
    • Handle: RePEc:bla:jtsera:v:30:y:2009:i:1:p:125-144
    DOI: 10.1111/j.1467-9892.2008.00603.x
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    Cited by:

    1. Martinez Oscar & Olmo Jose, 2012. "A Nonlinear Threshold Model for the Dependence of Extremes of Stationary Sequences," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 16(3), pages 1-39, September.
    2. Kristensen Dennis & Rahbek Anders, 2009. "Asymptotics of the QMLE for Non-Linear ARCH Models," Journal of Time Series Econometrics, De Gruyter, vol. 1(1), pages 1-38, April.
    3. Bibi, Abdelouahab & Lescheb, Ines, 2012. "On general periodic time-varying bilinear processes," Economics Letters, Elsevier, vol. 114(3), pages 353-357.
    4. Shiqing Ling & Liang Peng & Fukang Zhu, 2015. "Inference For A Special Bilinear Time-Series Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 36(1), pages 61-66, January.
    5. Iqbal Owadally, 2014. "Tail risk in pension funds: an analysis using ARCH models and bilinear processes," Review of Quantitative Finance and Accounting, Springer, vol. 43(2), pages 301-331, August.
    6. Abdelhakim Aknouche & Nadia Rabehi, 2010. "On an independent and identically distributed mixture bilinear time‐series model," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(2), pages 113-131, March.
    7. Andreou, Elena, 2016. "On the use of high frequency measures of volatility in MIDAS regressions," Journal of Econometrics, Elsevier, vol. 193(2), pages 367-389.
    8. Predrag M. Popović & Hassan S. Bakouch, 2020. "A bivariate integer-valued bilinear autoregressive model with random coefficients," Statistical Papers, Springer, vol. 61(5), pages 1819-1840, October.
    9. Andreou, Elena, 2016. "On the use of high frequency measures of volatility in MIDAS regressions," CEPR Discussion Papers 11307, C.E.P.R. Discussion Papers.
    10. Bibi, Abdelouahab & Ghezal, Ahmed, 2015. "Consistency of quasi-maximum likelihood estimator for Markov-switching bilinear time series models," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 192-202.
    11. Marcella Niglio & Cosimo Damiano Vitale, 2012. "Local Unit Roots and Global Stationarity of TARMA Models," Methodology and Computing in Applied Probability, Springer, vol. 14(1), pages 17-34, March.
    12. Abdelouahab Bibi & Ahmed Ghezal, 2016. "On periodic time-varying bilinear processes: structure and asymptotic inference," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 25(3), pages 395-420, August.
    13. Elena Andreou, 2016. "On the use of high frequency measures of volatility in MIDAS regressions," University of Cyprus Working Papers in Economics 03-2016, University of Cyprus Department of Economics.
    14. Zongwu Cai & Xiyuan Liu, 2020. "A Functional-Coefficient VAR Model for Dynamic Quantiles with Constructing Financial Network," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 202017, University of Kansas, Department of Economics, revised Oct 2020.

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