[Paper Review] Extremal dependence: some contributions

Helena Ferreira, Марта Феррейра|arXiv (Cornell University)|Aug 9, 2011

Financial Risk and Volatility Modeling18 references4 citations

TL;DR

This paper introduces a novel upper-tail dependence function and extremal coefficient of dependence between two sub-vectors of a multivariate random vector, weakening the standard requirement of simultaneous extremal events across all components. By leveraging bivariate techniques and rank-based estimators, the method enables robust assessment of tail dependence in high-dimensional financial data, with proven asymptotic normality, strong consistency, and empirical validation on global market indices showing significant dependence between Europe, USA, and Far East markets.

ABSTRACT

Due to globalization and relaxed market regulation, we have assisted to an increasing of extremal dependence in international markets. As a consequence, several measures of tail dependence have been stated in literature in recent years, based on multivariate extreme-value theory. In this paper we present a tail dependence function and an extremal coefficient of dependence between two random vectors that extend existing ones. We shall see that in weakening the usual required dependence allows to assess the amount of dependence in $d$-variate random vectors based on bidimensional techniques. Very simple estimators will be stated and can be applied to the well-known \emph{stable tail dependence function}. Asymptotic normality and strong consistency will be derived too. An application to financial markets will be presented at the end.

Motivation & Objective

• To address the limitation of existing multivariate tail dependence measures that require all components to be extreme simultaneously, which is overly restrictive and computationally complex.
• To develop a new upper-tail dependence function that assesses dependence between sub-vectors based on the maximum of each sub-vector reaching extreme levels.
• To extend the concept of extremal coefficients to allow for more flexible modeling of tail dependence in high-dimensional settings.
• To derive simple, consistent, and asymptotically normal estimators applicable to the stable tail dependence function and real-world financial data.
• To demonstrate the practical relevance of the proposed measure through an empirical application to international financial market indices.

Proposed method

• Define a new upper-tail dependence function $\Lambda_U^{(I_1|I_2)}(x,y)$ as the limit of conditional probabilities that the maximum of sub-vector $I_1$ exceeds $1 - x/t$ given that the maximum of sub-vector $I_2$ exceeds $1 - y/t$, as $t \to \infty$.
• Introduce the extremal coefficient of dependence $\epsilon_{(I_1,I_2)}$ as $\Lambda_U^{(I_1|I_2)}(1,1)$, extending classical extremal coefficients from bivariate to multivariate sub-vector dependence.
• Use empirical copula-based estimators based on ranks to estimate the extremal coefficient, ensuring robustness to marginal distributional assumptions.
• Establish theoretical properties: strong consistency and asymptotic normality of the estimators via law of large numbers and convergence arguments involving empirical distribution functions.
• Apply the method to monthly maximum log-returns of 9 major stock market indices (1993–2004), grouping markets into Europe, USA, and Far East for analysis.
• Use scatter plots and rank-based estimators to visualize and quantify tail dependence between regional market groups.

Experimental results

Research questions

• RQ1 How can tail dependence be meaningfully measured when only a subset of components in a multivariate vector reaches extreme values?
• RQ2 Can a dependence measure be constructed that weakens the strict requirement of joint extremal behavior across all components, while remaining statistically tractable?
• RQ3 What are the asymptotic properties of estimators for the proposed extremal coefficient of dependence?
• RQ4 How does the new measure compare to existing tail dependence coefficients in capturing cross-market dependence in financial data?
• RQ5 To what extent does the proposed method reveal tail dependence patterns among major global financial markets?

Key findings

• The proposed extremal coefficient of dependence $\epsilon_{(I_1,I_2)}$ is consistently estimated using rank-based estimators, with proven strong consistency and asymptotic normality.
• Empirical application to financial data shows that the USA and Europe exhibit strong tail dependence, with $\widehat{\epsilon}_{(\text{Europe}, \text{USA})} = 1.0083$, indicating high extremal dependence.
• The Far East market shows weaker dependence with other regions, with $\widehat{\epsilon}_{(\text{Europe}, \text{Far East})} = 0.5688$ and $\widehat{\epsilon}_{(\text{USA}, \text{Far East})} = 0.3644$, suggesting lower tail dependence.
• The coefficient $\epsilon_{(\text{Europe}, \text{USA} \cup \text{Far East})} = 1.1259$ indicates that the combined effect of USA and Far East markets on Europe is stronger than the USA alone, reflecting complex interdependence.
• The method successfully captures asymmetric dependence patterns, with the USA showing stronger influence on Europe than vice versa, as reflected in $\epsilon_{(\text{USA}, \text{Europe} \cup \text{Far East})} = 0.9215$.
• The proposed estimators are applicable to the stable tail dependence function and maintain good finite-sample performance, as demonstrated in the financial application.