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...

Violation of the efficient market hypothesis (EMH) in a specific market may lead to construction of bubbles which is a signal of inefficiencies. Although speculative bubbles soon decay, if they exist for a long time, they will lead to financial crises. Early warning systems (EWSs) are designed...