Multivariate GARCH estimation via a Bregman-proximal trust-region method

TL;DR

This paper introduces a novel Bregman-proximal trust-region method for estimating high-dimensional multivariate GARCH models, effectively handling overparameterization and complex constraints, with promising computational results.

Contribution

It develops a general estimation framework for multivariate GARCH models in any dimension using a Bregman-proximal trust-region approach, addressing overparameterization and nonlinear constraints.

Findings

Method demonstrates strong computational performance.

Handles high-dimensional GARCH estimation efficiently.

Preliminary experiments show promising results.

Abstract

The estimation of multivariate GARCH time series models is a difficult task mainly due to the significant overparameterization exhibited by the problem and usually referred to as the "curse of dimensionality". For example, in the case of the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the dimensionality of the problem. Moreover, these parameters are subjected to convoluted nonlinear constraints necessary to ensure, for instance, the existence of stationary solutions and the positive semidefinite character of the conditional covariance matrices used in the model design. So far, this problem has been addressed in the literature only in low dimensional cases with strong parsimony constraints. In this paper we propose a general formulation of the estimation problem in any dimension and develop a Bregman-proximal trust-region method for its solution. The Bregman-proximal approach allows us 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 performances of the proposed approach.