Autocovariance structure of Markov regime switching models and model selection (Q2722255)

The autocovariance structure of a model belonging to a general class of second-order stationary Markov regime switching processes being that of a vector autoregressive moving-average (VARMA) model whose orders \(p\) and \(q\) are bounded above by elementary functions of the number of Markov regimes \(k\) is investigated. The results apply to models with both mean-variance switching and switching among autoregressive regimes, unifying and extending previous work. In the case of a mean-variance switching process the orders \(p,q<k.\) For models switching among autoregressions, the bounds are elementary functions of the dimension of the process, the number of regimes and the maximum order of autoregression. The sample autocovariances are more easily calculated than maximum likelihood estimates of the model parameters. These bounds can be very valuable in model selection.NEWLINENEWLINENEWLINEIn particular, the paper's results yield an estimate of the lower bound on the number of regimes. Such a lower bound is particularly relevant in the light of a result by \textit{D.L. Donoho} [Ann. Stat. 16, No. 4, 1390-1420 (1988; Zbl 0665.62040)], who discusses the inability to build two-sided confidence intervals for the complexity of certain models.