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Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. (English) Zbl 1056.62028

From the paper: An autoregressive process with Markov regime, or Markov-switching autoregression, is a bivariate process \(\{(X_k,Y_k)\}\), where \(\{X_k\}\) is a Markov chain on a state space \({\mathcal X}\) and, conditional on \(\{X_k\}\), \(\{Y_k\}\) is an inhomogeneous \(s\)-order Markov chain on a state space \({\mathcal Y}\) such that the conditional distribution of \(Y_n\) only depends on \(X_n\) and lagged \(Y\)’s. The process \(\{X_k\}\), usually referred to as the regime, is not observable and inference has to be carried out in terms of the observable process \(\{Y_k\}\).
We consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.

MSC:

62F12 Asymptotic properties of parametric estimators
62M05 Markov processes: estimation; hidden Markov models
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M09 Non-Markovian processes: estimation
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