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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 highdimensional and bivariate simpli cations would produce misleading results. This occurs when a signi cant 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 tailrisk 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., address = Berlin, author = Carsten Bormann and Melanie Schienle and Julia Schaumburg, copyright = http://www.econstor.eu/dspace/Nutzungsbedingungen, keywords = C01; C46; C58; 330; decomposition of tail dependence; multivariate extreme values; stable tail dependence function; subsample bootstrap; tail correlation, language = eng, number = 2014-042, publisher = Humboldt University of Berlin, Collaborative Research Center 649 - Economic Risk, title = Beyond dimension two: A test for higher-order tail risk, type = SFB 649 Discussion Paper, url = https://hdl.handle.net/10419/103804, year = 2014