Multivariate GARCH Estimation via a Bregman-Proximal Trust-Region Method

Abstract

The estimation of multivariate GARCH time series models is a difficult task mainly due to the excessive parameterization exhibited by the problem, usually referred to as the "curse of dimensionality." For the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the dimensionality of the problem and, additionally, these parameters are subjected to complex nonlinear constraints. So far, this problem has been addressed only in low dimensional cases with strong parsimony constraints for the diagonal three-dimensional VEC handled with ad-hoc techniques. A general formulation of the estimation problem in any dimension and a Bregman-proximal trust-region method for its solution is proposed. The Bregman-proximal approach allows to handle the constraints in a very efficient and natural way by staying in the primal space and the Trust-Region mechanism stabilizes and speeds up the scheme. Preliminary computational experiments are presented and confirm the very good performance of the proposed approach.

Keywords: multivariate GARCH, VEC model, volatility modeling, multivariate financial time series, Bregman divergences, Burg's divergence, LogDet divergence, constrained optimization

JEL Classification: C5