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 crossextremogram 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.
The extremogram and the cross-extremogramIn this paper we conduct an empirical study of extremal serial dependence in a bivariate return series. Our main tools for describing extremal dependence will be the extremogram and the cross-extremogram. For the sake of argument and for simplicity, we restrict ourselves to the bivariate series X t = (X 1,t , X 2,t ) , t ∈ Z. We assume that (X t ) has the structurewhere (Z t ) constitutes an independent and identically distributed (i.i.d.) bivariate noise sequence andwith σ i,t the (nonnegative) volatility of X i,t . We will assume that (X t ) and ( t ) constitute strictly stationary sequences, and that t is predictable with respect to the filtration generated by (Z s ) s≤t . We also assume that Z t = (Z 1,t , Z 2,t ) has mean 0 and covariance matrix (standardized to correlations)where ρ = corr(Z 1,t , Z 2,t ). Later, we will choose parametric models for (X t ), such as univariate GARCH(1, 1) models for both X i,t , i = 1, 2, or a vector GARCH(1,1) model; see Section 2.2 for model descriptions. In the context of these parametric models, the choice of