Measuring and testing tail equivalence

We call two copulas (lower) tail equivalent if their first-order approximations in the lower 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 orders are the same but tail order parameters are non-identical. Therefore, the measure admits attribution of the difference in two tails to that in tail orders or in tail order parameters. We derive asymptotic properties of the proposed measure, and then propose a novel statistical test for tail equivalence. The 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 global 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.