Characterization of shallow volcanoclastic deposits by turning ray seismic tomography: an application to the Naples urban area

Dynamic models with parameters that are allowed to depend on the state of a hidden Markov chain have become a popular tool for modelling time series subject to changes in regime. An important question that arises in applications involving such modelsis how to determine the number of states required for the model to be an adequate characterization of the observed data. In this paper, we investigate the properties of alternative procedures that can be used to determine the state dimension of a Markovswitching autoregressive model. These include procedures that exploit the ARMA representation which Markov-switching processes admit, as well as procedures that are based on optimization of complexity-penalized likelihood measures. Our Monte Carlo analysis reveals that such procedures estimate the state dimension correctly, provided that the parameter changes are not too small and the hidden Markov chain is fairly persistent. The use of the various methods is also illustrated by means of empirical examples.