A matrix-variate log-normal model for covariance matrices

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 Normal 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. 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 bias correction based on a second-order Taylor expansion is proposed. An application to real data and a simulation study assess the validity of the proposed approach.