New directions in Time Series Analysis (Guest Editor: Philippe Soulier)

Autocovariance structure of powers of switching-regime ARMA Processes

Francq, Christian^a1 and Zakoïan, Jean-Michel^a2

^a1 Université du Littoral-Côte d'Opale, LMPA J. Liouville, Centre Universitaire de la Mi-Voix, 50 rue F. Buisson, BP. 699, 62228 Calais Cedex, France; Christian.Francq@lmpa.univ-littoral.fr.

^a2 Université de Lille 3 and CREST, 15 boulevard Gabriel Péri, 92245 Malakoff Cedex, France; zakoian@ensae.fr.

Abstract

In Francq and Zakoïan [4], we derived stationarity conditions for ARMA(p,q) models subject to Markov switching. In this paper, we show that, under appropriate moment conditions, the powers of the stationary solutions admit weak ARMA representations, which we are able to characterize in terms of p,q, the coefficients of the model in each regime, and the transition probabilities of the Markov chain. These representations are potentially useful for statistical applications.

(Online publication November 15 2002)

Key Words:

• ARMA representation;
• hidden Markov models;
• Markov-switching models;
• identification.

Mathematics Subject Classification:

• 62M10