Time series models for realized covariance matrices based on the matrix-F distribution

We propose a new Conditional BEKK matrix-F (CBF) model for the time-varying realized covariance (RCOV) matrices. This CBF model is capable of capturing heavy-tailed RCOV, which is an important stylized fact but could not be handled adequately by the Wishart-based models. To further mimic the long memory feature of the RCOV, a special CBF model with the conditional heterogeneous autoregressive (HAR) structure is introduced. Moreover, we give a systematical study on the probabilistic properties and statistical inferences of the CBF model, including exploring its stationarity, establishing the asymptotics of its maximum likelihood estimator, and giving some new inner-product-based tests for its model checking. In order to handle a large dimensional RCOV matrix, we construct two reduced CBF models -- the variance-target CBF model (for moderate but fixed dimensional RCOV matrix) and the factor CBF model (for high dimensional RCOV matrix). For both reduced models, the asymptotic theory of the estimated parameters is derived. The importance of our entire methodology is illustrated by simulation results and two real examples.