On the tail behavior of a class of multivariate conditionally heteroskedastic processes
- 113 Downloads
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 (VSRV), closely related to non-standard regular variation. The characterization of the tail behavior of the processes is used for deriving the asymptotic properties of the sample covariance matrices.
KeywordsStochastic recurrence equations Markov processes Regular variation Multivariate ARCH Asymptotic properties Geometric ergodicity
AMS 2000 Subject Classifications60G70 60G10 60H25 39A50
Unable to display preview. Download preview PDF.
We are grateful for comments and suggestions from the editor-in-chief (Thomas Mikosch), an associate editor, and two referees, which have led to a much improved manuscript. Moreover, we thank Sebastian Mentemeier for valuable comments. Pedersen greatly acknowledges funding from the Carlsberg Foundation. Financial support by the ANR network AMERISKA ANR 14 CE20 0006 01 is gratefully acknowledged by Wintenberger.
- Buraczewski, D., Damek, E., Mikosch, T.: Stochastic Models with Power-Law Tails: The Equation X = AX + B, Springer Series in Operations Research and Financial Engineering, Springer International Publishing (2016)Google Scholar
- Damek, E., Matsui, M., Świa̧tkowski, W.: Componentwise different tail solutions for bivariate stochastic recurrence equations, arXiv:1706.05800 (2017)
- Kulik, R., Soulier, P., Wintenberger, O.: The tail empirical process of regularly varying functions of geometrically ergodic Markov chains. arXiv:1511.04903 (2015)
- Resnick, S.I.: Heavy-tail phenomena: probabilistic and statistical modeling, Springer Science & Business Media (2007)Google Scholar
- Vaynman, I., Beare, B.K.: Stable limit theory for the variance targeting estimator. In: Essays in Honor of Peter C. B. Phillips, ed. by Y. Chang, T. B. Fomby, and J. Y. Park, Emerald Group Publishing Limited, vol. 33 of Advances in Econometrics, chap. 24, pp. 639–672. (2014)Google Scholar