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Realized covariance matrices (RCs) are an important input to asses the
risks involved in different investment allocations and it is thus useful to model
and forecast them. To this end generalized autoregressive score (GAS) mod-
els are employed in this paper. These models are ideal for comparing different
probability distributions in terms of their ability to model and forecast RCs,
since the dynamic parameters of the conditional observation density are up-
dated by incorporating the shape of the distribution itself (via the scaled
score of the log-likelihood). All probability distributions so far applied to
time series of RCs in the literature are compared and it is shown how they
are related to each other. Furthermore a novel family of probability distribu-
tion, which has a property called “tail homogeneity”, is derived and added
to the comparison. The necessary inputs for the GAS models (Fisher infor-
mation matrix and score) are derived for all distributions. An in-sample fit
comparison confirms previous results that “fat-tailed” distributions outper-
form others and shows that the novel distribution family achieves very good
fit. Out-of-sample forecasting comparisons further corroborate the excellent
performance of the novel distribution family.
