Asymptotic Properties of the Maximum Likelihood Estimator for Markov-switching Observation-driven Models

Abstract

A Markov-switching observation-driven model is a stochastic process $((S_t,Y_t){)}_{t\in \mathbb{Z}}$ where $(S_t{)}_{t\in \mathbb{Z}}$ is an unobserved Markov chain on a finite set and $(Y_t{)}_{t\in \mathbb{Z}}$ is an observed stochastic process such that the conditional distribution of $Y_t$ given $(Y_{\tau }{)}_{\tau \le t-1}$ and $(S_{\tau }{)}_{\tau \le t}$ depends on $(Y_{\tau }{)}_{\tau \le t-1}$ and $S_t$. In this paper, we prove consistency and asymptotic normality of the maximum likelihood estimator for such model. As a special case, we also give conditions under which the maximum likelihood estimator for the widely applied Markov-switching generalised autoregressive conditional heteroscedasticity model introduced by Haas, Mittnik, and Paolella (2004b) is consistent and asymptotically normal.