Geometric Ergodicity and β-mixing of the Periodic Multivariate BEKK-GARCH Model

Abstract

This paper introduces the new class of periodic multivariate GARCH models in their periodic BEKK specification. Semi-polynomial Markov chains combined with algebraic geometry are used to obtain some properties like irreducibility. We impose weak conditions to obtain the strict periodic stationarity and the geometric ergodicity of the process, via the theory of positive linear operators on a cone: it is supposed that zero belongs to the support of the driving noise density which is absolutely continuous with respect to the Lebesgue measure and the spectral radius of a matrix built from the periodic coefficients of the model is smaller than one.