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

TL;DR

This paper develops asymptotic theory for the extremogram and cross-extremogram of bivariate GARCH(1,1) processes, analyzing tail behavior and dependence in financial return data.

Contribution

It introduces new asymptotic results for extremograms in bivariate GARCH models and applies them to real financial data for model validation.

Findings

Tail indices of GARCH components may differ depending on parameters

Residual extremograms support iid innovation hypothesis

Significant extremal dependence observed at lag zero

Abstract

In this paper, we derive some 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 5-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 iid 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.