A Geometric GARCH Framework for Covariance Dynamics

Abstract

This paper develops new multivariate GARCH models that respect intrinsic geometric properties of covariance matrix, and are physically meaningful. These models can be specified using either asset returns or realized covariances. New parameter estimation method and performance evaluation methods are also developed, and limitations of existing evaluation methods are addressed. Empirical results suggest that our models outperform existing models such as BEKK and DCC, and realized covariance based models outperform return based models. It turns out that the variation of covariance matrix can be identified by a few principal directions, implying potential for a parsimonious specification of covariance dynamics.

Keywords: Geometric GARCH; Multivariate GARCH; Covariance; Realized Covariance; Principal Geodesic Analysis