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

Part of: Mathematical economics Stochastic processes Inference from 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: mmuneya@nanzan-u.ac.jp
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK--2100 Copenhagen, Denmark. Email address: mikosch@math.ku.dk
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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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