Explicit minimal representation of variance matrices, and its implication for dynamic volatility models

Abstract

We propose a minimal representation of variance matrices of dimension k, where parameterization and positive-definiteness conditions are both explicit. Then, we apply it to the specification of dynamic multivariate volatility processes. Compared to the most parsimonious unrestricted formulation currently available, the required number of covariance parameters (hence processes) is reduced by about a half, which makes them estimable in full parametric generality if needed. Our conditions are easy to implement: there are only k of them, and they are explicit and univariate. To illustrate, we forecast minimum-variance portfolios and show that risk is always reduced (by a factor of 2 to 3 in spite of us using the simplest dynamics) compared to the standard benchmark used in finance, while also improving returns on the investment. Because of our representation, we do not get the usual dimensionality problems of existing unrestricted models, and the performance relative to the benchmark is actually improved substantially as k increases.