Consistent Ranking of Multivariate Volatility Models

A large number of parameterizations have been proposed to model conditional variance dynamics in a multivariate framework. This paper examines the ranking of multivariate volatility models in terms of their ability to forecast out-of-sample conditional variance matrices. We investigate how sensitive the ranking is to alternative statistical loss functions which evaluate the distance between the true covariance matrix and its forecast. The evaluation of multivariate volatility models requires the use of a proxy for the unobservable volatility matrix which may shift the ranking of the models. Therefore, to preserve this ranking conditions with respect to the choice of the loss function have to be discussed. To do this, we extend the conditions defined in Hansen and Lunde (2006) to the multivariate framework. By invoking norm equivalence we are able to extend the class of loss functions that preserve the true ranking. In a simulation study, we sample data from a continuous time multivariate diffusion process to illustrate the sensitivity of the ranking to different choices of the loss functions and to the quality of the proxy. An application to three foreign exchange rates, where we compare the forecasting performance of 16 multivariate GARCH specifications, is provided.