Efficient evaluation of multidimensional time-varying density forecasts, with applications to risk management

We propose two simple evaluation methods for time-varying density forecasts of continuous higher-dimensional random variables. Both methods are based on the probability integral transformation for unidimensional forecasts. The first method tests multinormal densities and relies on the rotation of the coordinate system. The advantages of the second method are not only its applicability to arbitrary continuous distributions, but also the evaluation of the forecast accuracy in specific regions of its domain, as defined by the user's interest. We show that the latter property is particularly useful for evaluating a multidimensional generalization of the Value at Risk. In both simulations and an empirical study, we examine the performances of the two tests.