Established tests for proper calibration of multivariate density forecasts based on Rosenblatt probability integral transforms can be manipulated by changing the order of variables in the forecasting model. We derive order invariant tests. The new tests are applicable to densities of arbitrary...

We review tests of null hypotheses that consist of many subsidiary null hypotheses, including tests that have not received much attention in the econometrics literature. We study test performance in the context of specification testing for linear regressions based on a Monte Carlo study....

We propose to treat survey-based density expectations as compositional data when testing either for heterogeneity in density forecasts across different groups of agents or for changes over time. Monte Carlo simulations show that the proposed test has more power relative to both a bootstrap...