Multivariate extremes for models with constant conditional correlations

Abstract

Models with constant conditional correlations are versatile tools for describing the behavior of multivariate time series of financial returns. Mathematically speaking, they are solutions of a special class of stochastic recurrence equations (SRE). The extremal behavior of general solutions of SRE has been studied in detail by Kesten [Kesten, H., 1973. Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131, 207–248] and Perfekt [Perfekt, R., 1997. Extreme value theory for a class of Markov chains with values in R d . Advances in Applied Probability 29, 138–164]. The central concept to understanding the joint extremal behavior of such multivariate time series is the multivariate regular variation spectral measure. In this paper, we propose an estimator for the spectral measure associated with solutions of SRE and prove its consistency. Our estimator is the tail empirical measure of the multivariate time series. Successful use of the estimator depends on a good choice of k, the number of upper order statistics contributing to the empirical measure. We introduce a new criteria for the choice of k based on a scaling property of the spectral measure. We investigate the performance of our estimation technique on exchange rate time series from HFDF96 data set. The estimated spectral measure is used to calculate probabilities of joint extreme returns and probabilities of large movements in an exchange rate conditional on the occurrence of extreme returns in another exchange rate. We find a high level of dependence between the extreme movements of most of the currencies in the EU. We also investigate the changes in the level of dependence between the extreme returns of pairs of currencies as the sampling frequency decreases. When at least one return is extreme, a strong dependence between the components is present already at the 4-hour level for most of the European currencies.