Structure of a double autoregressive process driven by a hidden Markov chain

This paper considers a new so-called autoregressive process with ARCH(1) errors driven by a hidden Markov chain, Xt+1=α(Δt+1)Xt+ηt+1β(Δt+1)+λ(Δt+1)Xt2,t∈N, where (ηt)(ηt) is a sequence of independent and identically distributed standard normal random variables, and (Δt)(Δt) is a Markov chain with finite state space. Some structural properties of this new autoregressive process are considered. A sufficient condition for the existence of the strictly stationary and geometrically ergodic solution of the process is presented. The condition for this is only E[ln|α(Δt)+ηtλ(Δt)|]<0. Moreover, some simple conditions for the existence of the moments of the process are also derived.