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

TL;DR

This paper establishes the consistency and asymptotic normality of the maximum likelihood estimator for Markov-switching observation-driven models, including specific conditions for the widely used MS-GARCH model.

Contribution

It provides the first rigorous proof of the asymptotic properties of the MLE in these models, extending to popular MS-GARCH applications.

Findings

MLE is consistent under certain conditions

MLE is asymptotically normal for these models

Applicable to MS-GARCH models used in finance

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 $(Y_\tau)_{\tau \leq t-1}$ and $(S_\tau)_{\tau \leq t}$ depends on $(Y_\tau)_{\tau \leq t-1}$ 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.