We propose a modeling framework for time-varying covariance matrices based on the assumption that the logarithm of a realized covariance matrix follows a matrix-variate oNrmal distribution. By operating in the space of symmetric matrices, the approach guarantees positive definiteness without imposing parameter constraints beyond stationarity. The conditional mean of the logarithmic covariance matrix is specified through a BEKK-type structure that can be rewritten as a diagonal vector representation, yielding a parsimonious specification that mitigates the curse of dimensionality. Estimation is performed by maximum likelihood exploiting properties of matrix-variate Normal distributions and expressing the scale parameter matrix as a function of the location matrix. The covariance matrix is recovered via the matrix exponential. Since this transformation induces an upward bias, an approximate, time-specific bias correction based on a second-order Taylor expansion is proposed. The framework is flexible and applicable to a wide class of problems involving symmetric positive definite matrices.

翻译：我们提出了一种基于已实现协方差矩阵对数服从矩阵变量正态分布假设的时变协方差矩阵建模框架。通过在对称矩阵空间中进行操作，该方法在仅施加平稳性约束而不引入额外参数限制的条件下保证了正定性。对数协方差矩阵的条件均值通过BEKK型结构进行设定，该结构可重写为对角向量表示形式，从而产生能够缓解维度灾难的简约参数设定。估计过程利用矩阵变量正态分布的性质，并将尺度参数矩阵表达为位置矩阵的函数，通过极大似然方法实现。协方差矩阵通过矩阵指数运算恢复。由于该变换会引入向上偏差，本文提出了一种基于二阶泰勒展开的近似时变偏差校正方法。该框架具有灵活性，适用于涉及对称正定矩阵的广泛问题类别。