Stationarity and Geometric Ergodicity of BEKK Multivariate GARCH Models

TL;DR

This paper establishes conditions for the existence of stationary and geometrically ergodic BEKK multivariate GARCH models, using algebraic geometry and spectral radius criteria.

Contribution

It provides the first comprehensive conditions for stationarity and geometric ergodicity of BEKK GARCH models, including algebraic geometric analysis.

Findings

Conditions on the noise distribution for stationarity

Spectral radius criterion for GARCH coefficients

Use of algebraic geometry in Markov chain analysis

Abstract

Conditions for the existence of strictly stationary multivariate GARCH processes in the so-called BEKK parametrisation, which is the most general form of multivariate GARCH processes typically used in applications, and for their geometric ergodicity are obtained. The conditions are that the driving noise is absolutely continuous with respect to the Lebesgue measure and zero is in the interior of its support and that a certain matrix built from the GARCH coefficients has spectral radius smaller than one. To establish the results semi-polynomial Markov chains are defined and analysed using algebraic geometry.