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On the tail behavior of a class of multivariate conditionally heteroskedastic processes

Author

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  • Rasmus Søndergaard Pedersen

    (Department of Economics [Copenhagen] - Faculty of Social Sciences [Copenhagen] - UCPH - University of Copenhagen = Københavns Universitet)

  • Olivier Wintenberger

    (Department of Mathematical Sciences [Copenhagen] - Faculty of Science [Copenhagen] - UCPH - University of Copenhagen = Københavns Universitet, LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

Abstract

Conditions for geometric ergodicity of multivariate autoregressive conditional heteroskedasticity (ARCH) processes, with the so-called BEKK (Baba, Engle, Kraft, and Kroner) parametrization, are considered. We show for a class of BEKK-ARCH processes that the invariant distribution is regularly varying. In order to account for the possibility of different tail indices of the marginals, we consider the notion of vector scaling regular variation, in the spirit of Perfekt (1997, Advances in Applied Probability, 29, pp. 138-164). The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices.

Suggested Citation

  • Rasmus Søndergaard Pedersen & Olivier Wintenberger, 2017. "On the tail behavior of a class of multivariate conditionally heteroskedastic processes," Post-Print hal-01436267, HAL.
    • Handle: RePEc:hal:journl:hal-01436267
    Note: View the original document on HAL open archive server: https://hal.science/hal-01436267v3
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    References listed on IDEAS

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    1. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(1), pages 122-150, February.
    2. 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.
    3. Boussama, Farid & Fuchs, Florian & Stelzer, Robert, 2011. "Stationarity and geometric ergodicity of BEKK multivariate GARCH models," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2331-2360, October.
    4. Pedersen, Rasmus Søndergaard, 2016. "Targeting Estimation Of Ccc-Garch Models With Infinite Fourth Moments," Econometric Theory, Cambridge University Press, vol. 32(2), pages 498-531, April.
    5. Einmahl, J.H.J. & de Haan, L.F.M. & Piterbarg, V.I., 2001. "Nonparametric estimation of the spectral measure of an extreme value distribution," Other publications TiSEM c3485b9b-a0bd-456f-9baa-0, Tilburg University, School of Economics and Management.
    6. Avarucci, Marco & Beutner, Eric & Zaffaroni, Paolo, 2013. "On Moment Conditions For Quasi-Maximum Likelihood Estimation Of Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 29(3), pages 545-566, June.
    7. Igor Vaynman & Brendan K. Beare, 2014. "Stable Limit Theory for the Variance Targeting Estimator," Advances in Econometrics, in: Essays in Honor of Peter C. B. Phillips, volume 33, pages 639-672, Emerald Group Publishing Limited.
    8. 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.
    9. Kulik, Rafał & Soulier, Philippe & Wintenberger, Olivier, 2019. "The tail empirical process of regularly varying functions of geometrically ergodic Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4209-4238.
    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. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    12. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
    13. Paul D. Feigin & Richard L. Tweedie, 1985. "Random Coefficient Autoregressive Processes:A Markov Chain Analysis Of Stationarity And Finiteness Of Moments," Journal of Time Series Analysis, Wiley Blackwell, vol. 6(1), pages 1-14, January.
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    Cited by:

    1. Andreas Hetland, 2018. "The Stochastic Stationary Root Model," Econometrics, MDPI, vol. 6(3), pages 1-33, August.

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    Keywords

    Stochastic recurrence equations; Markov processes; regular variation; multivariate ARCH; asymptotic properties; geometric ergodicity;
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