Extremes of scale mixtures of multivariate time series

TL;DR

This paper studies the extremal behavior of scale mixture models for multivariate time series, deriving new extremal distributions and analyzing tail dependence, with applications to financial data.

Contribution

It extends scale mixture models for multivariate time series by analyzing their extremal properties and tail dependence, providing new methods for constructing multivariate extreme value distributions.

Findings

Extremal index is unit under tail independence.

Derived new multivariate extreme value distributions.

Applied models to financial data.

Abstract

Factor models have large potencial in the modeling of several natural and human phenomena. In this paper we consider a multivariate time series $\mb{Y}_n$, ${n\geq 1}$, rescaled through random factors $\mb{T}_n$, ${n\geq 1}$, extending some scale mixture models in the literature. We analyze its extremal behavior by deriving the maximum domain of attraction and the multivariate extremal index, which leads to new ways to construct multivariate extreme value distributions. The computation of the multivariate extremal index and the characterization of the tail dependence show the interesting property of these models that however much it is the dependence within and between factors $\mb{T}_n$, ${n\geq 1}$, the extremal index of the model is unit whenever $\mb{Y}_n$, ${n\geq 1}$, presents cross-sectional and sequencial tail independence. We illustrate with examples of thinned multivariate time series and multivariate autoregressive processes with random coefficients. An application of these latter to financial data is presented at the end.