Deriving the autocovariances of powers of Markov-switching GARCH models, with applications to statistical inference

A procedure is proposed for computing the autocovariances and the ARMA representations of the squares, and higher-order powers, of Markov-switching GARCH models. It is shown that many interesting subclasses of the general model can be discriminated in view of their autocovariance structures. Explicit derivation of the autocovariances allows for parameter estimation in the general model, via a GMM procedure. It can also be used to determine how many ARMA representations are needed to identify the Markov-switching GARCH parameters. A Monte Carlo study and an application to the Standard & Poor index are presented.