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 (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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alsmeyer, G.: On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results. J. Theor. Probab. 16, 217–247 (2003)
Alsmeyer, G., Mentemeier, S.: Tail behaviour of stationary solutions of random difference equations: The case of regular matrices. J. Differ. Equ. Appl. 18, 1305–1332 (2012)
Avarucci, M., Beutner, E., Zaffaroni, P.: On moment conditions for quasi-maximum likelihood estimation of multivariate ARCH models. Econ. Theory 29, 545–566 (2013)
Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908–920 (2002a)
Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115 (2002b)
Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080 (2009)
Basrak, B., Tafro, A.: A complete convergence theorem for stationary regularly varying multivariate time series. Extremes 19, 549–560 (2016)
Bauwens, L., Laurent, S., Rombouts, J.V.K.: Multivariate GARCH models: A survey. J. Appl. Econ. 21, 79–109 (2006)
Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.L.: Statistics of extremes: theory and applications. Wiley, NJ (2006)
Boussama, F., Fuchs, F., Stelzer, R.: Stationarity and geometric ergodicity of BEKK multivariate GARCH models. Stoch. Process. Appl. 121, 2331–2360 (2011)
Buraczewski, D., Damek, E., Guivarc’h, Y., Hulanicki, A., Urban, R.: Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Relat. Fields 145, 385–420 (2009)
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)
Damek, E., Matsui, M., Świa̧tkowski, W.: Componentwise different tail solutions for bivariate stochastic recurrence equations, arXiv:1706.05800 (2017)
Davis, R.A., Hsing, T.: Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23, 879–917 (1995)
Davis, R.A., Mikosch, T.: The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Stat. 26, 2049–2080 (1998)
de Haan, L., Resnick, S.I.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie Verwandte Geb. 40, 317–337 (1977)
de Haan, L., Resnick, S.I., Rootzén, H., de Vries, C.G.: Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stoch. Process. Appl. 32, 213–224 (1989)
Einmahl, J.H., de Haan, L., Piterbarg, V.I.: Nonparametric estimation of the spectral measure of an extreme value distribution. Ann. Stat. 29, 1401–1423 (2001)
Engle, R.F., Kroner, K.F.: Multivariate simultaneous generalized ARCH. Econ. Theory 11, 122–150 (1995)
Feigin, P., Tweedie, R.: Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal. 6, 1–14 (1985)
Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166 (1991)
Janssen, A., Segers, J.: Markov tail chains. J. Appl. Probab. 51, 1133–1153 (2014)
Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)
Kulik, R., Soulier, P., Wintenberger, O.: The tail empirical process of regularly varying functions of geometrically ergodic Markov chains. arXiv:1511.04903 (2015)
Leadbetter, M., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes, Springer series in statistics. Springer-Verlag, Berlin (1983)
Lindskog, F., Resnick, S.I., Roy, J.: Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps. Probab. Surv. 11, 270–314 (2014)
Matsui, M., Mikosch, T.: The extremogram and the cross-extremogram for a bivariate GARCH(1,1) process. Adv. Appl. Probab. 48, 217–233 (2016)
Mikosch, T., Wintenberger, O.: Precise large deviations for dependent regularly varying sequences. Probab. Theory Relat. Fields 156, 851–887 (2013)
Nelson, D.B.: Stationarity and persistence in the GARCH(1,1) model. Econ. Theory 6, 318–334 (1990)
Nielsen, H.B., Rahbek, A.: Unit root vector autoregression with volatility induced stationarity. J. Empir. Financ. 29, 144–167 (2014)
Pedersen, R.S.: Targeting estimation of CCC-GARCH models with infinite fourth moments. Econ. Theory 32, 498–531 (2016)
Pedersen, R.S., Rahbek, A.: Multivariate variance targeting in the BEKK-GARCH model. Econ. J. 17, 24–55 (2014)
Perfekt, R.: Extreme value theory for a class of Markov chains with values in ℝd. Adv. Appl. Probab. 29, 138–164 (1997)
Resnick, S.I.: Heavy-tail phenomena: probabilistic and statistical modeling, Springer Science & Business Media (2007)
Segers, J.: Generalized Pickands estimators for the extreme value index. J. Stat. Plan. Infer. 128, 381–396 (2005)
Stărică, C.: Multivariate extremes for models with constant conditional correlations. J. Empir. Financ. 6, 515–553 (1999)
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)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pedersen, R.S., Wintenberger, O. On the tail behavior of a class of multivariate conditionally heteroskedastic processes. Extremes 21, 261–284 (2018). https://doi.org/10.1007/s10687-017-0307-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-017-0307-3
Keywords
- Stochastic recurrence equations
- Markov processes
- Regular variation
- Multivariate ARCH
- Asymptotic properties
- Geometric ergodicity