Consistency of the maximum likelihood estimate for non-homogeneous Markov–switching models

Abstract

Many nonlinear time series models have been proposed in the last decades. Among them, the models with regime switchings provide a class of versatile and interpretable models which have received a particular attention in the literature. In this paper, we consider a large family of such models which generalize the well known Markov-switching AutoRegressive (MS-AR) by allowing non-homogeneous switching and encompass Threshold AutoRegressive (TAR) models and prove the consistency of the maximum likelihood estimator under general conditions. We show that these conditions apply to specific but representative models with non-homogeneous Markov switchings. The famous MacKenzie River lynx dataset is used to illustrate one of these models.

Maximum likelihood estimation in hidden Markov models with inhomogeneous noise

“…The investigation of strong consistency of maximum likelihood estimation in inhomogeneous HMMs is less developed. In [18] and [19] the MLE in inhomogeneous Markov switching models is studied. There, the transition probabilities are also influenced by the observations, but the inhomogeneity there is different from the time-dependent inhomogeneity considered in our work, since the conditional law is not changing over time.…”

Sparse vector Markov switching autoregressive models. Application to multivariate time series of temperature

“…As far as we know, no analogous CLT is available for non homogeneous MSVAR models yet. Indeed, the most recent theoritical results for non homogeneous MSVAR are due to Ailliot and Pène (Ailliot and Pène, 2015) who have demonstrated that non homogeneous MSVAR models verify a property of ergodicity and that the estimator of maximum likelihood is consistent. So, one can not directly generalize the asymptotic consistency result to non homogeneous MSVAR models.…”

Consistency of the maximum likelihood estimator in hidden Markov models with trends

“…Note that some models that are presented as time-inhomogeneous in the literature are actually homogeneous according to our definition. This is the case of the Markovswitching models considered since the work of Hamilton (1989) and more recently by Pouzo et al (2016) and Ailliot and Pene (2015) for instance. These models are a generalization of HMM where the hidden state X t depends both of the previous hidden state X t−1 and on previous observations, let's say Y t−1 for an order one model, and where the observation Y t depends both on the corresponding hidden state X t and on previous observations.…”