IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v115y2005i9p1557-1582.html
   My bibliography  Save this article

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

Author

Listed:
  • 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.

Suggested Citation

  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(05)00055-4
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Christian Francq & Michel Roussignol & Jean‐Michel Zakoian, 2001. "Conditional Heteroskedasticity Driven by Hidden Markov Chains," Journal of Time Series Analysis, Wiley Blackwell, vol. 22(2), pages 197-220, March.
    2. Ling, Shiqing & McAleer, Michael, 2002. "NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS," Econometric Theory, Cambridge University Press, vol. 18(3), pages 722-729, June.
    3. Jing Zhang & Robert A. Stine, 2001. "Autocovariance Structure of Markov Regime Switching Models and Model Selection," Journal of Time Series Analysis, Wiley Blackwell, vol. 22(1), pages 107-124, January.
    4. Dueker, Michael J, 1997. "Markov Switching in GARCH Processes and Mean-Reverting Stock-Market Volatility," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 26-34, January.
    5. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    6. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    7. Francq, C. & Zakoian, J. -M., 2001. "Stationarity of multivariate Markov-switching ARMA models," Journal of Econometrics, Elsevier, vol. 102(2), pages 339-364, June.
    8. Tim Bollerslev, 1988. "On The Correlation Structure For The Generalized Autoregressive Conditional Heteroskedastic Process," Journal of Time Series Analysis, Wiley Blackwell, vol. 9(2), pages 121-131, March.
    9. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    10. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus, 2000. "Stationary Arch Models: Dependence Structure And Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 16(1), pages 3-22, February.
    11. Bai, Xuezheng & Russell, Jeffrey R. & Tiao, George C., 2003. "Kurtosis of GARCH and stochastic volatility models with non-normal innovations," Journal of Econometrics, Elsevier, vol. 114(2), pages 349-360, June.
    12. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
    13. Hillebrand, Eric, 2005. "Neglecting parameter changes in GARCH models," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 121-138.
    14. Cai, Jun, 1994. "A Markov Model of Switching-Regime ARCH," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(3), pages 309-316, July.
    15. Hamilton, James D. & Susmel, Raul, 1994. "Autoregressive conditional heteroskedasticity and changes in regime," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 307-333.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Augustyniak, Maciej, 2014. "Maximum likelihood estimation of the Markov-switching GARCH model," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 61-75.
    2. Dimitris N. Politis & Dimitrios D. Thomakos, 2007. "NoVaS Transformations: Flexible Inference for Volatility Forecasting," Working Paper series 44_07, Rimini Centre for Economic Analysis.
    3. 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).
    4. 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.
    5. 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.
    6. 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.
    7. Pelletier, Denis, 2006. "Regime switching for dynamic correlations," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 445-473.
    8. 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.
    9. 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.
    10. 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.
    11. Krämer, Walter, 2008. "Long memory with Markov-Switching GARCH," Economics Letters, Elsevier, vol. 99(2), pages 390-392, May.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. Fayçal Hamdi & Saïd Souam, 2018. "Mixture periodic GARCH models: theory and applications," Empirical Economics, Springer, vol. 55(4), pages 1925-1956, December.
    17. 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.
    18. 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.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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).
    2. Gao, Guangyuan & Ho, Kin-Yip & Shi, Yanlin, 2020. "Long memory or regime switching in volatility? Evidence from high-frequency returns on the U.S. stock indices," Pacific-Basin Finance Journal, Elsevier, vol. 61(C).
    3. Halkos, George & Tzirivis, Apostolos, 2018. "Effective energy commodities’ risk management: Econometric modeling of price volatility," MPRA Paper 90781, University Library of Munich, Germany.
    4. Franses,Philip Hans & Dijk,Dick van, 2000. "Non-Linear Time Series Models in Empirical Finance," Cambridge Books, Cambridge University Press, number 9780521779654.
    5. Degiannakis, Stavros & Xekalaki, Evdokia, 2004. "Autoregressive Conditional Heteroskedasticity (ARCH) Models: A Review," MPRA Paper 80487, University Library of Munich, Germany.
    6. 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.
    7. Eduardo Rossi, 2010. "Univariate GARCH models: a survey (in Russian)," Quantile, Quantile, issue 8, pages 1-67, July.
    8. 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.
    9. Shi, Yanlin & Feng, Lingbing, 2016. "A discussion on the innovation distribution of the Markov regime-switching GARCH model," Economic Modelling, Elsevier, vol. 53(C), pages 278-288.
    10. Ho, Kin-Yip & Shi, Yanlin & Zhang, Zhaoyong, 2013. "How does news sentiment impact asset volatility? Evidence from long memory and regime-switching approaches," The North American Journal of Economics and Finance, Elsevier, vol. 26(C), pages 436-456.
    11. Haas, Markus & Mittnik, Stefan, 2008. "Multivariate regimeswitching GARCH with an application to international stock markets," CFS Working Paper Series 2008/08, Center for Financial Studies (CFS).
    12. BAUWENS, Luc & HAFNER, Christian & LAURENT, Sébastien, 2011. "Volatility models," LIDAM Discussion Papers CORE 2011058, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
      • Bauwens, L. & Hafner, C. & Laurent, S., 2012. "Volatility Models," LIDAM Reprints ISBA 2012028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
      • Bauwens, L. & Hafner C. & Laurent, S., 2011. "Volatility Models," LIDAM Discussion Papers ISBA 2011044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. repec:dau:papers:123456789/2285 is not listed on IDEAS
    14. McAleer, Michael & Medeiros, Marcelo C., 2008. "A multiple regime smooth transition Heterogeneous Autoregressive model for long memory and asymmetries," Journal of Econometrics, Elsevier, vol. 147(1), pages 104-119, November.
    15. Pelletier, Denis, 2006. "Regime switching for dynamic correlations," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 445-473.
    16. Mihaela Craioveanu & Eric Hillebrand, 2012. "Level changes in volatility models," Annals of Finance, Springer, vol. 8(2), pages 277-308, May.
    17. Zhang, Yue-Jun & Yao, Ting & He, Ling-Yun & Ripple, Ronald, 2019. "Volatility forecasting of crude oil market: Can the regime switching GARCH model beat the single-regime GARCH models?," International Review of Economics & Finance, Elsevier, vol. 59(C), pages 302-317.
    18. Szabolcs Blazsek & Anna Downarowicz, 2013. "Forecasting hedge fund volatility: a Markov regime-switching approach," The European Journal of Finance, Taylor & Francis Journals, vol. 19(4), pages 243-275, April.
    19. Ardia, David & Bluteau, Keven & Boudt, Kris & Catania, Leopoldo, 2018. "Forecasting risk with Markov-switching GARCH models:A large-scale performance study," International Journal of Forecasting, Elsevier, vol. 34(4), pages 733-747.
    20. Alizadeh, Amir H. & Nomikos, Nikos K. & Pouliasis, Panos K., 2008. "A Markov regime switching approach for hedging energy commodities," Journal of Banking & Finance, Elsevier, vol. 32(9), pages 1970-1983, September.
    21. Hotta, Luiz Koodi & Trucíos Maza, Carlos César & Pereira, Pedro L. Valls & Zevallos Herencia, Mauricio Henrique, 2024. "Forecasting VaR and ES through Markov-switching GARCH models: does the specication matter?," Textos para discussão 567, FGV EESP - Escola de Economia de São Paulo, Fundação Getulio Vargas (Brazil).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:115:y:2005:i:9:p:1557-1582. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.