Autocontour-based evaluation of multivariate predictive densities

We contribute to the rather sparse literature on multivariate density forecasting by introducing a new framework for the out-of-sample evaluation of multivariate density forecast models which builds on the concept of “autocontours” proposed by González-Rivera, Senyuz, and Yoldas (2011). This approach uniquely combines formal testing with graphical devices. We work with the one-step-ahead quantile residuals, which must be i.i.d. (univariate and multivariate) normal under the null hypothesis of a correct density model. 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 that, in general, a parametric bootstrap provides outstanding finite sample properties. We provide simulation evidence on the finite sample performances of the tests and compare their performances with that of an alternative testing procedure. We also illustrate this methodology by evaluating bivariate density forecasts of the returns on US value and growth portfolios.