Model selection and some extensions of Markov switching models

We study model selection issues and some extensions of Markov switching models. We establish both theoretical and empirical results. We show that the covariance functions of second-order stationary vector Markov switching time series models have vector ARMA(p,q) representations, where the upper bounds for p and q are elementary functions of the number of regimes. This result yields an easily computed method for setting a lower bound on the number of underlying Markov regimes from an estimated autocovariance function. We also propose estimating the number of states of the Markov chain via the number of mixture components of the marginal distribution. Specifically, we use penalized quasi-likelihood estimators with the likelihood calculated from the finite mixtures. We give conditions needed to ensure consistency of the penalized quasi-likelihood estimators. A second step modification of the penalized quasi-likelihood estimator is also proposed to decrease the chance of overestimation and is shown to be strongly consistent. We raise the issue of determining the validity of Markov switching regime in real data sets. We show that it is very hard, in general, to distinguish between Markov switching models and conventional models such as Gaussian ARMA from data without prior subjective knowledge. We also propose a class of generalized Markov regime switching models. The key feature of the models is that the parameters of each regime are subject to stochastic selection even when the regime is known. Under fairly general conditions, it is shown that the ML estimates of distributions which govern the stochastic selection are step functions with a finite number of steps