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.

中文翻译：

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基于矩阵 F 分布的已实现协方差矩阵的时间序列模型

我们为时变已实现协方差 (RCOV) 矩阵提出了一种新的条件 BEKK 矩阵-F (CBF) 模型。该 CBF 模型能够捕获重尾 RCOV，这是一个重要的程式化事实，但基于 Wishart 的模型无法充分处理。为了进一步模仿 RCOV 的长记忆特征，引入了具有条件异构自回归 (HAR) 结构的特殊 CBF 模型。此外，我们系统地研究了 CBF 模型的概率特性和统计推断，包括探索其平稳性，建立其最大似然估计量的渐近性，并为其模型检查提供一些新的基于内积的测试。为了处理大维的 RCOV 矩阵，我们构建了两个简化的 CBF 模型——方差-目标 CBF 模型（用于中等但固定维的 RCOV 矩阵）和因子 CBF 模型（用于高维 RCOV 矩阵）。对于这两种简化模型，推导出了估计参数的渐近理论。仿真结果和两个真实示例说明了我们整个方法的重要性。