Probability distributions for realized covariance measures

Realized covariance measures (RCs) are an essential input for assessing the risks of different investment allocations. Thus, it is useful to model and forecast them. To this end, a realistic distributional assumption is needed. In this paper, we compare all probability distributions that have so far in the literature been applied to time-series of RCs. We derive them in a unified framework based on their stochastic representations using random lower and upper triangular matrices. These matrices are composed of standard normal distributions in the off-diagonal elements and χ -distributions on the diagonals (Barlett matrices). Furthermore, we derive a novel family of probability distributions (the t -Riesz distribution family), which has a property we call tail-homogeneity . This property means that in crisis periods, i.e. large RCs, our distribution family realistically assumes high dependence between the individual entries of the RCs. We show theoretically how all the distributions differ from each other, and how they are related to each other. In the empirical part, we demonstrate how the theoretical differences translate into differences in fit and forecasting performance. We show that our novel distribution family achieves the best fit. Out-of-sample forecasting comparisons further corroborate the excellent performance of our novel distribution family.