Autocovariance structure of powers of switching-regime ARMA Processes

Christian Francq, Jean-Michel Zakoïan
2002 E S A I M: Probability & Statistics  
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. Mathematics Subject
more » ... thematics Subject is an independent and identically distributed sequence of K-dimensional centered variables, with E(η t η T t ) = Ω, the covariance matrix Ω being nonsingular. In addition, assume that (η t ) is independent of (∆ t ). Hence ( t ) is a white noise. The model (1) is called Markov-switching ARMA(p, q) model. Markov switching models (MSM) can be viewed as extensions of the hidden Markov models (HMM) introduced by Baum and Petrie [1]. By contrast with the HMM's, the observations of a MSM are not independent random variables conditional on the Markov chain. These models have found a variety of applications in econometrics since the paper by Hamilton [5]. In a recent paper, Francq and Zakoïan [4] have established necessary and sufficient conditions for the existence
doi:10.1051/ps:2002014 fatcat:ovwghfvyb5atjazlujfnqufw5y