A Matrix-Variate Log-Normal Model for Covariance Matrices

TL;DR

The paper tackles the problem of modeling time-varying covariance matrices that must remain positive definite and are high-dimensional in finance. It introduces a matrix-variate log-normal framework in which the log-covariance $\bm{\Gamma}_t$ follows a matrix-variate Normal distribution with location $\bm{M}_t$ and common scale $\bm{U}_t$, and where $\bm{M}_t$ follows a BEKK-type update recast as a diagonal VEC model to reduce dimensionality. Covariances are recovered by $\hat{\bm{C}}_t = \exp(\hat{\bm{M}}_t)$, guaranteeing positive definiteness, with a delta-method-based, time-specific bias correction to counter the upward bias from exponentiation. The framework offers a parsimonious and flexible approach for high-dimensional SPD matrices and is extensible to non-financial SPD problems, especially when dimension reduction or clustering is employed.

Abstract

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.