Calibration tests based on the probability integral transform (PIT) are
routinely used to assess the quality of univariate distributional forecasts.
However, PIT-based calibration tests for multivariate distributional forecasts
face various challenges. We propose two new types of tests based on proper
scoring rules, which overcome these challenges. They arise from a general
framework for calibration testing in the multivariate case, introduced in this
work. The new tests have good size and power properties in simulations and
solve various problems of existing tests. We apply the tests to forecast
distributions for macroeconomic and financial time series data