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

  • Muneya Matsui (a1) and Thomas Mikosch (a2)


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.


Corresponding author

Department of Business Administration, Nanzan University, 18 Yamazato-cho Showa-ku Nagoya, 466‒8673, Japan. Email address:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK--2100 Copenhagen, Denmark. Email address:


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[1] Basrak, B. and Segers, J. (2009).Regularly varying multivariate time series.Stoch. Process. Appl. 119,10551080.
[2] Basrak, B.,Davis, R. A. and Mikosch, T. (2002).A characterization of multivariate regular variation.Ann. Appl. Prob. 12,908920.
[3] Basrak, B.,Davis, R. A. and Mikosch, T. (2002).Regular variation of GARCH processes.Stoch. Process. Appl. 99,95115.
[4] Bingham, N. H.,Goldie, C. M. and Teugels, J. L. (1987).Regular Variation. Cambridge University Press.
[5] Bollerslev, T. (1986).Generalized autoregressive conditional heteroskedasticity.J. Econometrics 31,307327.
[6] Bollerslev, T. (1990).Modelling the coherence in short-run nominal exchange rates: a multivariate generalized ARCH model.Rev. Econom. Statist. 72,498505.
[7] Boman, J. and Lindskog, F. (2009).Support theorems for the Radon transform and Cramér-Wold theorems.J. Theoret. Prob. 22,683710.
[8] Bougerol, P. and Picard, N. (1992).Stationarity of GARCH processes and of some non-negative time series.J. Econometrics 52,115127.
[9] Brandt, A. (1986).The stochastic equation Y n+1=A n Y n +B n with stationary coefficients.Adv. Appl. Prob. 18,211220.
[10] Breiman, L. (1965). On some limit theorems similar to the arc-sin law.Theory Prob. Appl. 10,323331.
[11] Brockwell, P. J. and Davis, R. A. (1991).Time Series: Theory and Methods,2nd edn.Springer,New York.
[12] Davis, R. A. and Hsing, T. (1995).Point process and partial sum convergence for weakly dependent random variables with infinite variance.Ann. Prob. 23,879917.
[13] Davis, R. A. and Mikosch, T. (2009).The extremogram: a correlogram for extreme events.Bernoulli 15,9771009.
[14] Davis, R. A.,Mikosch, T. and Cribben, I. (2012).Towards estimating extremal serial dependence via the boostrapped extremogram.J. Econometrics 170,142152.
[15] Davis, R. A.,Mikosch, T. and Zhao, Y. (2013).Measures of serial extremal dependence and their estimation.Stoch. Process. Appl. 123,25752602.
[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. 100,15381550.
[17] Francq, C. and Zakoïan, J. M. (2012).QML estimation of a class of multivariate asymmetric GARCH models.Econometric Theory 28,179206.
[18] Goldie, C. M. (1991).Implicit renewal theory and tails of solutions of random equations.Ann. Appl. Prob. 1,126166.
[19] Jeantheau, T. (1998).Strong consistency of estimators for multivariate ARCH models.Econometric Theory 14,7086.
[20] Jessen, A. H. and Mikosch, T. (2006).Regularly varying functions.Publ. Inst. Math. (Beograd) (N.S.) 80(94),171192.
[21] Kesten, H. (1973).Random difference equations and renewal theory for products of random matrices.Acta Math. 131,207248.
[22] Lancaster, P. (1969).Theory of Matrices.Academic Press,New York.
[23] Ling, S. and McAleer, M. (2003).Asymptotic theory for a vector ARMA-GARCH model.Econometric Theory 19,280310.
[24] Matsui, M. and Mikosch, T. (2015).The extremogram and the cross-extremogram for a bivariate GARCH(1,1) process. Preprint. Avaialable at
[25] McNeil, A. J.,Frey, R. and Embrechts, P. (2015).Quantitative Risk Management: Concepts, Techniques and Tools. Revised edn.,Princeton University Press.
[26] Mikosch, T. and Stărică, C. (2000).Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process.Ann. Statist. 28,14271451.
[27] Nelson, D. B. (1990).Stationarity and persistence in the GARCH(1,1) model.Econometric Theory 6,318334.
[28] Resnick, S. I. (1987).Extreme Values, Regular Variation, and Point Processes.Springer,New York.
[29] Resnick, S. I. (2007).Heavy-Tail Phenomena: Probabilistic and Statistical Modeling.Springer,New York.
[30] Stărică, C. (1999).Multivariate extremes for models with constant conditional correlations.J. Empirical Finance 6,515553.


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

  • Muneya Matsui (a1) and Thomas Mikosch (a2)


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