Penalized estimate of the number of states in Gaussian linear AR with Markov regime

Abstract

We deal with the estimation of the regime number in a linear Gaussian autoregressive process with a Markov regime (AR-MR). The problem of estimating the number of regimes in this type of series is that of determining the number of states in the hidden Markov chain controlling the process. We propose a method based on penalized maximum likelihood estimation and establish its strong consistency (almost sure) without assuming previous bounds on the number of states.

Asymptotic properties of autoregressive regime-switching models

“…Hansen derived an asymptotic bound for the distribution of the LRTS based on empirical processes techniques, while Garcia obtained the asymptotic distribution of the LRTS, but under some very restrictive hypothesis. Let us also mention that the consistency of the estimate of the number of regimes was proven recently in a Bayesian framework [28].…”

Jointly determining the state dimension and lag order for Markov‐switching vector autoregressive models

“…The overestimation problem is shared by more conventional criteria such as AIC , BIC , and HQC ; see Remark 6 of Fu and Wu (2020). Consistency of can be achieved when the penalty term admits certain functional forms (Rıos and Rodrıguez, 2008; Fu and Wu, 2020) or under additional restrictions on the model (Olteanu and Rynkiewicz, 2007).…”

A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency

“…In this case, the main problem to be solved is the choice of a good penalty term. For the case of an MS-AR with Gaussian innovation, Ríos and Rodríguez (2008a) considered a penalized likelihood criteria. For finite mixture model, a penalized contrast defined from the Hankel matrices of the first algebraic moments has been taken into account by Dacunha-Castelle and Gassiat (1997).…”