Autocontour-Based Evaluation of Multivariate Predictive Densities

We contribute to the rather thin literature on multivariate density forecasts by introducing a new framework for out-of-sample evaluation of multivariate density forecast models that builds upon the concept of autocontour proposed by Gonzalez-Rivera et al. (2011). This approach uniquely combines formal testing with graphical devices. We work with the one-step-ahead quantile residuals, which under the null hypothesis of a correct density model must be i.i.d. (univariate and multivariate) normal. Their corresponding autocontours are mathematically very tractable and the tests based on them enjoy standard asymptotic properties. We show that parameter uncertainty is asymptotically irrelevant under certain conditions and, in general, a parametric bootstrap provides outstanding finite sample properties. We provide simulation evidence on finite sample performance of the tests and compare their performance with an alternative testing procedure. We also illustrate this methodology by evaluating bivariate density forecasts of the returns on U.S. value and growth portfolios