Beyond dimension two: A test for higher-order tail risk

In practice, multivariate dependencies between extreme risks are often only assessed in a pairwise way. We propose a test to detect when tail dependence is truly high{dimensional and bivariate simplications would produce misleading results. This occurs when a signicant portion of the multivariate dependence structure in the tails is of higher dimension than two. Our test statistic is based on a decomposition of the stable tail dependence function, which is standard in extreme value theory for describing multivariate tail dependence. The asymptotic properties of the test are provided and a bootstrap based nite sample version of the test is suggested. A simulation study documents the good performance of the test for standard sample sizes. In an application to international government bonds, we detect a high tail{risk and low return situation during the last decade which can essentially be attributed to increased higher{order tail risk. We also illustrate the empirical consequences from ignoring higher-dimensional tail risk.