Detecting Tail Risk Differences in Multivariate Time Series

Abstract

We derive functional central limit theory for tail index estimates in multivariate time series under mild conditions on the extremal dependence between the components. We use this result to also derive convergence results for extreme value-at-risk and extreme expected shortfall estimates. This allows us to construct tests for equality of ‘tail risk’ in multivariate data, which can be useful in a number of empirical contexts. In constructing test statistics, we avoid estimating long-run variances by using self-normalization. Size and power of the tests for equal ‘tail risk’ are assessed in simulations. An empirical application to exchange returns illustrates the practical usefulness of the tests.