Global and Tail Dependence: A Differential Geometry Approach

TL;DR

This paper introduces a novel geometric framework for measuring tail dependence between variables, providing more nuanced and direction-sensitive dependence metrics than traditional coefficients, with applications demonstrated on stock index data.

Contribution

It proposes a new surface-area-based tail dependence measure using differential geometry, enabling refined and asymmetric dependence analysis beyond existing asymptotic coefficients.

Findings

New tail dependence measures differentiate asymmetric dependence.

Measures provide smoother, more detailed dependence taxonomy.

Empirical validation on simulated and stock index data.

Abstract

Measures of tail dependence between random variables aim to numerically quantify the degree of association between their extreme realizations. Existing tail dependence coefficients (TDCs) are based on an asymptotic analysis of relevant conditional probabilities, and do not provide a complete framework in which to compare extreme dependence between two random variables. In fact, for many important classes of bivariate distributions, these coefficients take on non-informative boundary values. We propose a new approach by first considering global measures based on the surface area of the conditional cumulative probability in copula space, normalized with respect to departures from independence and scaled by the difference between the two boundary copulas of co-monotonicity and counter-monotonicity. The measures could be approached by cumulating probability on either the lower left or upper right domain of the copula space, and offer the novel perspective of being able to differentiate asymmetric dependence with respect to direction of conditioning. The resulting TDCs produce a smoother and more refined taxonomy of tail dependence. The empirical performance of the measures is examined in a simulated data context, and illustrated through a case study examining tail dependence between stock indices.