Markov-switching Poisson generalized autoregressive conditional heteroscedastic models

We consider a kind of regime-switching autoregressive models for nonnegative integer-valued time series when the conditional distribution given historical information is Poisson distribution. In this type of models the link between the conditional variance (i.e. the conditional mean for Poisson distribution) and its past values as well as the observed values of the Poisson process may be different when an unobservable (hidden) variable, which is a Markovian Chain, takes different states. We study the stationarity and ergodicity of Markov-switching Poisson generalized autoregressive heteroscedastic (MS-PGARCH) models, and give a condition on parameters under which a MS-PGARCH process can be approximated by a geometrically ergodic process. Under this condition we discuss maximum likelihood estimation for MS-PGARCH models. Simulation studies and application to modelling financial count time series are presented to support our methodology.