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QML estimation of a class of multivariate GARCH models without moment conditions on the observed process

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  • Francq, Christian
  • Zakoian, Jean-Michel

Abstract

We establish the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of a class of multivariate GARCH processes. The conditions are mild and coincide with the minimal ones in the univariate case. In particular, contrary to the current literature on the estimation of multivariate GARCH models, no moment assumption is made on the observed process. Instead, we require strict stationarity, for which a necessary and sufficient condition is established.

Suggested Citation

  • Francq, Christian & Zakoian, Jean-Michel, 2010. "QML estimation of a class of multivariate GARCH models without moment conditions on the observed process," MPRA Paper 20779, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:20779
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    References listed on IDEAS

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    1. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, vol. 19(2), pages 280-310, April.
    2. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(1), pages 70-86, February.
    3. He, Changli & Teräsvirta, Timo, 2004. "An Extended Constant Conditional Correlation Garch Model And Its Fourth-Moment Structure," Econometric Theory, Cambridge University Press, vol. 20(5), pages 904-926, October.
    4. Silvennoinen, Annastiina & Teräsvirta, Timo, 2007. "Multivariate GARCH models," SSE/EFI Working Paper Series in Economics and Finance 669, Stockholm School of Economics, revised 18 Jan 2008.
    5. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    6. Robert F. Engle & Kevin Sheppard, 2001. "Theoretical and Empirical properties of Dynamic Conditional Correlation Multivariate GARCH," NBER Working Papers 8554, National Bureau of Economic Research, Inc.
    7. Tomoaki Nakatani & Timo Terasvirta, 2009. "Testing for volatility interactions in the Constant Conditional Correlation GARCH model," Econometrics Journal, Royal Economic Society, vol. 12(1), pages 147-163, March.
    8. 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.
    9. Luc Bauwens & Sébastien Laurent & Jeroen V. K. Rombouts, 2006. "Multivariate GARCH models: a survey," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(1), pages 79-109, January.
    10. Hafner, Christian M. & Preminger, Arie, 2009. "Asymptotic Theory For A Factor Garch Model," Econometric Theory, Cambridge University Press, vol. 25(2), pages 336-363, April.
    11. Engle, Robert, 2002. "Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 339-350, July.
    12. Comte, F. & Lieberman, O., 2003. "Asymptotic theory for multivariate GARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 61-84, January.
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    Cited by:

    1. Rasmus S. Pedersen & Anders Rahbek, 2014. "Multivariate variance targeting in the BEKK–GARCH model," Econometrics Journal, Royal Economic Society, vol. 17(1), pages 24-55, February.
    2. Sucarrat, Genaro & Grønneberg, Steffen & Escribano, Alvaro, 2016. "Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 582-594.
    3. Caporin, Massimiliano & McAleer, Michael, 2014. "Robust ranking of multivariate GARCH models by problem dimension," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 172-185.
    4. Michael McAleer & Massimiliano Caporin, 2011. "Ranking Multivariate GARCH Models by Problem Dimension:An Empirical Evaluation," KIER Working Papers 778, Kyoto University, Institute of Economic Research.
    5. Massimiliano Caporin & Paolo Paruolo, 2015. "Proximity-Structured Multivariate Volatility Models," Econometric Reviews, Taylor & Francis Journals, vol. 34(5), pages 559-593, May.

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    More about this item

    Keywords

    Asymptotic Normality; Conditional Heteroskedasticity; Consistency; Constant Conditional Correlation; Multivariate GARCH; Quasi Maximum Likelihood Estimation; Strict Stationarity Condition;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics

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