Measuring and testing tail equivalence

TL;DR

This paper introduces a new measure and statistical test for assessing tail equivalence between copulas, with applications to financial data during crises, revealing diverse tail behaviors.

Contribution

It proposes a novel measure and test for tail equivalence of copulas, including asymptotic properties and empirical validation in financial crises.

Findings

Detected non-identical tail behaviors among stock pairs during crises

Validated the test's effectiveness through simulations

Revealed differences in tail behaviors across periods and stocks

Abstract

We call two copulas tail equivalent if their first-order approximations in the tail coincide. As a special case, a copula is called tail symmetric if it is tail equivalent to the associated survival copula. We propose a novel measure and statistical test for tail equivalence. The proposed measure takes the value of zero if and only if the two copulas share a pair of tail order and tail order parameter in common. Moreover, taking the nature of these tail quantities into account, we design the proposed measure so that it takes a large value when tail orders are different, and a small value when tail order parameters are non-identical. We derive asymptotic properties of the proposed measure, and then propose a novel statistical test for tail equivalence. Performance of the proposed test is demonstrated in a series of simulation studies and empirical analyses of financial stock returns in the periods of the world financial crisis and the COVID-19 recession. Our empirical analysis reveals non-identical tail behaviors in different pairs of stocks, different parts of tails, and the two periods of recessions.