Asymptotic Behavior of the Maximum Likelihood Estimator for General Markov Switching Models

Abstract

: Motivated by studying the asymptotic properties of the parameter estimator in switching linear state space models, switching GARCH models, switching stochastic volatility models, and recurrent neural networks, we investigate the maximum likelihood estimator for general Markov switching models. To this end, we first propose an innovative matrix-valued Markovian iterated function system (MIFS) representation for the likelihood function. Then, we express the derivatives of the MIFS as a composition of random matrices. To the best of our knowledge, this is a new method in the literature. Using this useful device, we establish the strong consistency and asymptotic normality of the maximum likelihood estimator under some regularity conditions. Furthermore, we characterize the Fisher information as the inverse of the asymptotic variance.