The extremogram: A correlogram for extreme events

TL;DR

This paper introduces the extremogram, a new correlogram for analyzing extreme events in stationary sequences, with theoretical properties and applications to models like ARMA, GARCH, and stochastic volatility.

Contribution

It defines the extremogram as an analog of autocorrelation for extremes, proposes an estimator, and studies its asymptotic behavior under mixing conditions.

Findings

Extremogram effectively captures dependence in extreme events.

The estimator is asymptotically normal under certain conditions.

Applications to various models demonstrate the extremogram's utility.

Abstract

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 depends only on the extreme values in the sequence. We also propose a natural estimator for the extremogram and study its asymptotic properties under $\alpha$-mixing. We show asymptotic normality, calculate the extremogram for various examples and consider spectral analysis related to the extremogram.