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The extremogram and the cross-extremogram for a bivariate GARCH(1, 1) process

Part of: Inference from stochastic processes Mathematical economics Stochastic processes

Published online by Cambridge University Press:  25 July 2016

Muneya Matsui  and
Thomas Mikosch
Muneya Matsui*
Affiliation:
Nanzan University
Thomas Mikosch*
Affiliation:
University of Copenhagen
*
Department of Business Administration, Nanzan University, 18 Yamazato-cho Showa-ku Nagoya, 466‒8673, Japan. Email address: [email protected]
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK--2100 Copenhagen, Denmark. Email address: [email protected]
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Abstract

We derive asymptotic theory for the extremogram and cross-extremogram of a bivariate GARCH(1,1) process. We show that the tails of the components of a bivariate GARCH(1,1) process may exhibit power-law behavior but, depending on the choice of the parameters, the tail indices of the components may differ. We apply the theory to five-minute return data of stock prices and foreign-exchange rates. We judge the fit of a bivariate GARCH(1,1) model by considering the sample extremogram and cross-extremogram of the residuals. The results are in agreement with the independent and identically distributed hypothesis of the two-dimensional innovations sequence. The cross-extremograms at lag zero have a value significantly distinct from zero. This fact points at some strong extremal dependence of the components of the innovations.

Keywords

Regular variationbivariate GARCH(1,1)Kesten's theoremextremogramextremal dependence

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62M10: Time series, auto-correlation, regression, etc. 60G10: Stationary processes 91B84: Economic time series analysis
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue A: Probability, Analysis and Number Theory , July 2016 , pp. 217 - 233
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Basrak, B. and Segers, J. (2009).Regularly varying multivariate time series.Stoch. Process. Appl. 119,1055–1080.CrossRefGoogle Scholar
[2] Basrak, B.,Davis, R. A. and Mikosch, T. (2002).A characterization of multivariate regular variation.Ann. Appl. Prob. 12,908–920.CrossRefGoogle Scholar
[3] Basrak, B.,Davis, R. A. and Mikosch, T. (2002).Regular variation of GARCH processes.Stoch. Process. Appl. 99,95–115.CrossRefGoogle Scholar
[4] Bingham, N. H.,Goldie, C. M. and Teugels, J. L. (1987).Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
[5] Bollerslev, T. (1986).Generalized autoregressive conditional heteroskedasticity.J. Econometrics 31,307–327.CrossRefGoogle Scholar
[6] Bollerslev, T. (1990).Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model.Rev. Econom. Statist. 72,498–505.CrossRefGoogle Scholar
[7] Boman, J. and Lindskog, F. (2009).Support theorems for the Radon transform and Cramér-Wold theorems.J. Theoret. Prob. 22,683–710.CrossRefGoogle Scholar
[8] Bougerol, P. and Picard, N. (1992).Stationarity of GARCH processes and of some non-negative time series.J. Econometrics 52,115–127.CrossRefGoogle Scholar
[9] Brandt, A. (1986).The stochastic equation Y n+1=A n Y n +B n with stationary coefficients.Adv. Appl. Prob. 18,211–220.Google Scholar
[10] Breiman, L. (1965). On some limit theorems similar to the arc-sin law.Theory Prob. Appl. 10,323–331.CrossRefGoogle Scholar
[11] Brockwell, P. J. and Davis, R. A. (1991).Time Series: Theory and Methods,2nd edn.Springer,New York.CrossRefGoogle Scholar
[12] Davis, R. A. and Hsing, T. (1995).Point process and partial sum convergence for weakly dependent random variables with infinite variance.Ann. Prob. 23,879–917.CrossRefGoogle Scholar
[13] Davis, R. A. and Mikosch, T. (2009).The extremogram: a correlogram for extreme events.Bernoulli 15,977–1009.CrossRefGoogle Scholar
[14] Davis, R. A.,Mikosch, T. and Cribben, I. (2012).Towards estimating extremal serial dependence via the boostrapped extremogram.J. Econometrics 170,142–152.CrossRefGoogle Scholar
[15] Davis, R. A.,Mikosch, T. and Zhao, Y. (2013).Measures of serial extremal dependence and their estimation.Stoch. Process. Appl. 123,2575–2602.CrossRefGoogle Scholar
[16] Fernández, B. and Muriel, N. (2009).Regular variation and related results for the multivariate GARCH(p,q) model with constant conditional correlations.J. Multivariate Anal. 100,1538–1550.CrossRefGoogle Scholar
[17] Francq, C. and Zakoïan, J. M. (2012).QML estimation of a class of multivariate asymmetric GARCH models.Econometric Theory 28,179–206.CrossRefGoogle Scholar
[18] Goldie, C. M. (1991).Implicit renewal theory and tails of solutions of random equations.Ann. Appl. Prob. 1,126–166.CrossRefGoogle Scholar
[19] Jeantheau, T. (1998).Strong consistency of estimators for multivariate ARCH models.Econometric Theory 14,70–86.CrossRefGoogle Scholar
[20] Jessen, A. H. and Mikosch, T. (2006).Regularly varying functions.Publ. Inst. Math. (Beograd) (N.S.) 80(94),171–192.CrossRefGoogle Scholar
[21] Kesten, H. (1973).Random difference equations and renewal theory for products of random matrices.Acta Math. 131,207–248.CrossRefGoogle Scholar
[22] Lancaster, P. (1969).Theory of Matrices.Academic Press,New York.Google Scholar
[23] Ling, S. and McAleer, M. (2003).Asymptotic theory for a vector ARMA-GARCH model.Econometric Theory 19,280–310.CrossRefGoogle Scholar
[24] Matsui, M. and Mikosch, T. (2015).The extremogram and the cross-extremogram for a bivariate GARCH(1,1) process. Preprint. Avaialable athttp://arxiv.org/abs/1505.05385v1.Google Scholar
[25] McNeil, A. J.,Frey, R. and Embrechts, P. (2015).Quantitative Risk Management: Concepts, Techniques and Tools. Revised edn.,Princeton University Press.Google Scholar
[26] Mikosch, T. and Stărică, C. (2000).Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process.Ann. Statist. 28,1427–1451.CrossRefGoogle Scholar
[27] Nelson, D. B. (1990).Stationarity and persistence in the GARCH(1,1) model.Econometric Theory 6,318–334.CrossRefGoogle Scholar
[28] Resnick, S. I. (1987).Extreme Values, Regular Variation, and Point Processes.Springer,New York.CrossRefGoogle Scholar
[29] Resnick, S. I. (2007).Heavy-Tail Phenomena: Probabilistic and Statistical Modeling.Springer,New York.Google Scholar
[30] Stărică, C. (1999).Multivariate extremes for models with constant conditional correlations.J. Empirical Finance 6,515–553.CrossRefGoogle Scholar