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We derive order\u2010invariant tests. The new tests are applicable to densities of arbitrary dimensions and can deal with parameter estimation uncertainty and dynamic misspecification. Monte Carlo simulations show that they often have superior power relative to established approaches. We use the tests to evaluate generalized autoregressive conditional heteroskedasticity\u2010based multivariate density forecasts for a vector of stock market returns and macroeconomic forecasts from a Bayesian vector autoregression with time\u2010varying parameters.<\/jats:p>","DOI":"10.1002\/jae.2755","type":"journal-article","created":{"date-parts":[[2020,2,11]],"date-time":"2020-02-11T07:12:36Z","timestamp":1581405156000},"page":"440-456","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Order\u2010invariant tests for proper calibration of multivariate density forecasts"],"prefix":"10.1002","volume":"35","author":[{"given":"Jonas","family":"Dovern","sequence":"first","affiliation":[{"name":"Institute for Economics Friedrich\u2010Alexander University Erlangen\u2010N\u00fcrnberg  Nuremberg Germany"},{"name":"CESifo  Munich Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hans","family":"Manner","sequence":"additional","affiliation":[{"name":"Institute of Economics University of Graz  Graz Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2020,3,26]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.2307\/2938229"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.2307\/2171880"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1162\/003465303322369704"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2007.08.012"},{"key":"e_1_2_8_6_1","first-page":"177","volume-title":"Handbook of applied econometrics and statistical inference","author":"Bera A. 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