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Extremes

, Volume 21, Issue 2, pp 261–284 | Cite as

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

  • Rasmus Søndergaard Pedersen
  • Olivier Wintenberger
Article

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.

Keywords

Stochastic recurrence equations Markov processes Regular variation Multivariate ARCH Asymptotic properties Geometric ergodicity 

AMS 2000 Subject Classifications

60G70 60G10 60H25 39A50 

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Notes

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.

References

  1. Alsmeyer, G.: On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results. J. Theor. Probab. 16, 217–247 (2003)MathSciNetCrossRefMATHGoogle Scholar
  2. 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)MathSciNetCrossRefMATHGoogle Scholar
  3. Avarucci, M., Beutner, E., Zaffaroni, P.: On moment conditions for quasi-maximum likelihood estimation of multivariate ARCH models. Econ. Theory 29, 545–566 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. Basrak, B., Davis, R.A., Mikosch, T.: A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908–920 (2002a)MathSciNetCrossRefMATHGoogle Scholar
  5. Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Process. Appl. 99, 95–115 (2002b)MathSciNetCrossRefMATHGoogle Scholar
  6. Basrak, B., Segers, J.: Regularly varying multivariate time series. Stoch. Process. Appl. 119, 1055–1080 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. Basrak, B., Tafro, A.: A complete convergence theorem for stationary regularly varying multivariate time series. Extremes 19, 549–560 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. Bauwens, L., Laurent, S., Rombouts, J.V.K.: Multivariate GARCH models: A survey. J. Appl. Econ. 21, 79–109 (2006)MathSciNetCrossRefGoogle Scholar
  9. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.L.: Statistics of extremes: theory and applications. Wiley, NJ (2006)MATHGoogle Scholar
  10. Boussama, F., Fuchs, F., Stelzer, R.: Stationarity and geometric ergodicity of BEKK multivariate GARCH models. Stoch. Process. Appl. 121, 2331–2360 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 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)MathSciNetCrossRefMATHGoogle Scholar
  12. 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
  13. Damek, E., Matsui, M., Świa̧tkowski, W.: Componentwise different tail solutions for bivariate stochastic recurrence equations, arXiv:1706.05800 (2017)
  14. 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)MathSciNetCrossRefMATHGoogle Scholar
  15. Davis, R.A., Mikosch, T.: The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Stat. 26, 2049–2080 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. de Haan, L., Resnick, S.I.: Limit theory for multivariate sample extremes. Z. Wahrscheinlichkeitstheorie Verwandte Geb. 40, 317–337 (1977)MathSciNetCrossRefMATHGoogle Scholar
  17. 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)MathSciNetCrossRefMATHGoogle Scholar
  18. 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)MathSciNetCrossRefMATHGoogle Scholar
  19. Engle, R.F., Kroner, K.F.: Multivariate simultaneous generalized ARCH. Econ. Theory 11, 122–150 (1995)MathSciNetCrossRefGoogle Scholar
  20. 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)MathSciNetCrossRefMATHGoogle Scholar
  21. Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1, 126–166 (1991)MathSciNetCrossRefMATHGoogle Scholar
  22. Janssen, A., Segers, J.: Markov tail chains. J. Appl. Probab. 51, 1133–1153 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)MathSciNetCrossRefMATHGoogle Scholar
  24. Kulik, R., Soulier, P., Wintenberger, O.: The tail empirical process of regularly varying functions of geometrically ergodic Markov chains. arXiv:1511.04903 (2015)
  25. Leadbetter, M., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes, Springer series in statistics. Springer-Verlag, Berlin (1983)CrossRefMATHGoogle Scholar
  26. 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)MathSciNetCrossRefMATHGoogle Scholar
  27. Matsui, M., Mikosch, T.: The extremogram and the cross-extremogram for a bivariate GARCH(1,1) process. Adv. Appl. Probab. 48, 217–233 (2016)MathSciNetCrossRefGoogle Scholar
  28. Mikosch, T., Wintenberger, O.: Precise large deviations for dependent regularly varying sequences. Probab. Theory Relat. Fields 156, 851–887 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. Nelson, D.B.: Stationarity and persistence in the GARCH(1,1) model. Econ. Theory 6, 318–334 (1990)MathSciNetCrossRefGoogle Scholar
  30. Nielsen, H.B., Rahbek, A.: Unit root vector autoregression with volatility induced stationarity. J. Empir. Financ. 29, 144–167 (2014)CrossRefGoogle Scholar
  31. Pedersen, R.S.: Targeting estimation of CCC-GARCH models with infinite fourth moments. Econ. Theory 32, 498–531 (2016)MathSciNetCrossRefMATHGoogle Scholar
  32. Pedersen, R.S., Rahbek, A.: Multivariate variance targeting in the BEKK-GARCH model. Econ. J. 17, 24–55 (2014)MathSciNetGoogle Scholar
  33. Perfekt, R.: Extreme value theory for a class of Markov chains with values in ℝd. Adv. Appl. Probab. 29, 138–164 (1997)MathSciNetCrossRefMATHGoogle Scholar
  34. Resnick, S.I.: Heavy-tail phenomena: probabilistic and statistical modeling, Springer Science & Business Media (2007)Google Scholar
  35. Segers, J.: Generalized Pickands estimators for the extreme value index. J. Stat. Plan. Infer. 128, 381–396 (2005)MathSciNetCrossRefMATHGoogle Scholar
  36. Stărică, C.: Multivariate extremes for models with constant conditional correlations. J. Empir. Financ. 6, 515–553 (1999)CrossRefGoogle Scholar
  37. 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

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CopenhagenCopenhagen KDenmark
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagen ØDenmark
  3. 3.Université Pierre et Marie Curie, LSTAParisFrance

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