  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.
