The extremogram: a correlogram for extreme events

We consider a strictly stationary sequence of random vectors whose finite-dimensional distributions are jointly regularly varying with some positive index. This class of processes includes among others ARMA processes with regularly varying noise, GARCH processes with normally or student distributed noise, and stochastic volatility models with regularly varying multiplicative noise. We define an analog of the autocorrelation function, the extremogram, which only depends on the extreme values in the sequence. We also propose a natural estimator for the extremogram and study its asymptotic properties under �-mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram. 1. Measures of extremal dependence in a strictly stationary sequence The motivation for this research comes from the problem of choosing between two popular and commonly used families of models, the generalized autoregressive conditional heteroscedastic (GARCH) process and the heavy-tailed stochastic volatility (SV) process, for modeling a particular financial time series. Both GARCH and SV models possess the stylized features exhibited by log- returns of financial assets. Specifically, these time series have heavy-tailed marginal distributions, are dependent but uncorrelated, and display stochastic volatility. The latter property is manifested via the often slow decay of the sample autocorrelation function (ACF) of the absolute values and squares of the time series. Since both GARCH and SV models can be chosen to have virtually identical behavior in the tails of the marginal distribution and in the ACF of the squares of the process, it is difficult for a given time series of returns to decide between the two models on the basis of routine time series diagnostic tools. The problem of finding probabilistically reasonable and statistically estimable measures of ex- tremal dependence in a strictly stationary sequence is to some extent an open one. In classical time series analysis, which mostly deals with second order structure of stationary sequences, the ACF is a well accepted object for describing meaningful information about serial dependence. The ACF is sometimes over-valued as a tool for measuring dependence especially if one is only interested in extremes. It does of course determine the distribution of a stationary Gaussian sequence, but for non-Gaussian and non-linear time series the ACF often provides little insight into the dependence structure of the process. This is particularly the case when one considers heavy-tailed non-linear time series such as the GARCH model. In this case, the estimation of the ACF via the sample ACF is also rather imprecise and even misleading since the asymptotic confidence bands are typ- ically larger than the estimated autocorrelations, see for example the results in Basrak et al. (1) for bilinear processes; Davis and Mikosch (12), Mikosch and Stuaricua (26) and Basrak et al. (2) for ARCH and GARCH processes, Resnick (32) for teletraffic models. 1.1. The extremal index. The asymptotic behavior of the extremes leads to one clear difference between GARCH and SV processes. It was shown in Davis and Mikosch (12), Basrak et al. (2), Davis and Mikosch (13) (see also Breidt and Davis (7) for the light-tailed SV case) that GARCH processes exhibit extremal clustering (i.e., clustering of extremes), while SV processes lack this