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The L2-structures of standard and switching-regime GARCH models

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  • Francq, Christian
  • ZakoI¨an, Jean-Michel

Abstract

This paper analyzes the probabilistic structure of Markov-switching GARCH(p,q) models, in which the volatility process is driven by a finite state-space Markov chain. We give necessary and sufficient conditions for the existence of moments of any order. We find that the squares and higher order powers of the process have the L2 structures of ARMA processes, and hence admit ARMA representations. These results are applicable to standard GARCH models and have statistical implications in terms of order identification and parameter estimation.

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  • Francq, Christian & ZakoI¨an, Jean-Michel, 2005. "The L2-structures of standard and switching-regime GARCH models," Stochastic Processes and their Applications, Elsevier, vol. 115(9), pages 1557-1582, September.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:9:p:1557-1582
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    Cited by:

    1. Haas, Markus, 2010. "Covariance forecasts and long-run correlations in a Markov-switching model for dynamic correlations," Finance Research Letters, Elsevier, vol. 7(2), pages 86-97, June.
    2. Abdelhakim Aknouche & Christian Francq, 2022. "Stationarity and ergodicity of Markov switching positive conditional mean models," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 436-459, May.
    3. Nazim Regnard & Jean‐Michel Zakoïan, 2010. "Structure and estimation of a class of nonstationary yet nonexplosive GARCH models," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(5), pages 348-364, September.
    4. Haas, Markus, 2008. "The autocorrelation structure of the Markov-switching asymmetric power GARCH process," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1480-1489, September.
    5. Krämer, Walter, 2008. "Long memory with Markov-Switching GARCH," Economics Letters, Elsevier, vol. 99(2), pages 390-392, May.
    6. Maddalena Cavicchioli, 2016. "Statistical Analysis Of Mixture Vector Autoregressive Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1192-1213, December.
    7. Lee, Oesook & Lee, Jungwha, 2014. "The functional central limit theorem for the multivariate MS–ARMA–GARCH model," Economics Letters, Elsevier, vol. 125(3), pages 331-335.
    8. Augustyniak, Maciej, 2014. "Maximum likelihood estimation of the Markov-switching GARCH model," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 61-75.
    9. Fayçal Hamdi & Saïd Souam, 2018. "Mixture periodic GARCH models: theory and applications," Empirical Economics, Springer, vol. 55(4), pages 1925-1956, December.
    10. Dimitris Politis & Dimitrios Thomakos, 2007. "NoVaS Transformations: Flexible Inference for Volatility Forecasting," Working Papers 0005, University of Peloponnese, Department of Economics.
    11. Ataurima Arellano, Miguel & Rodríguez, Gabriel, 2020. "Empirical modeling of high-income and emerging stock and Forex market return volatility using Markov-switching GARCH models," The North American Journal of Economics and Finance, Elsevier, vol. 52(C).
    12. 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.
    13. Chunliang Deng & Xingfa Zhang & Yuan Li & Qiang Xiong, 2020. "Garch Model Test Using High-Frequency Data," Mathematics, MDPI, vol. 8(11), pages 1-17, November.
    14. Melike E. Bildirici & Memet Salman & Özgür Ömer Ersin, 2022. "Nonlinear Contagion and Causality Nexus between Oil, Gold, VIX Investor Sentiment, Exchange Rate and Stock Market Returns: The MS-GARCH Copula Causality Method," Mathematics, MDPI, vol. 10(21), pages 1-16, October.
    15. Pelletier, Denis, 2006. "Regime switching for dynamic correlations," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 445-473.
    16. Bildirici, Melike & Ersin, Özgür, 2012. "Nonlinear volatility models in economics: smooth transition and neural network augmented GARCH, APGARCH, FIGARCH and FIAPGARCH models," MPRA Paper 40330, University Library of Munich, Germany, revised May 2012.
    17. 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.
    18. Carol Alexander & Emese Lazar, 2009. "Modelling Regime‐Specific Stock Price Volatility," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 71(6), pages 761-797, December.
    19. Francq, Christian & ZakoI¨an, Jean-Michel, 2008. "Deriving the autocovariances of powers of Markov-switching GARCH models, with applications to statistical inference," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 3027-3046, February.
    20. Aknouche, Abdelhakim & Almohaimeed, Bader & Dimitrakopoulos, Stefanos, 2024. "Noising the GARCH volatility: A random coefficient GARCH model," MPRA Paper 120456, University Library of Munich, Germany, revised 15 Mar 2024.

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