  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.
