Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime

Randal Douc, Èric Moulines and Tobias Rydén

Abstract:: An autoregressive process with Markov regimes is an autoregressive process for which the regression function at each time point is given by a non-observable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly non-stationary process of this kind for which the hidden state space in compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.

Key words:: asymptotic normality, autoregressive process, consistency, geometric ergodicity, hidden Markov model, identifiability, maximum likelihood, switching autoregression