  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.
