Bayesian estimation of switching ARMA models

Switching ARMA processes have recently appeared as an efficient modelling to nonlinear time-series models, because they can represent multiple or heterogeneous dynamics through simple components. The levels of dependence between the observations are double: at a first level, the parameters of the model are selected by a Markovian procedure. At a second level, the next observation is generated according to a standard time-series model. When the model involves a moving average structure, the complexity of the resulting likelihood function is such that simulation techniques, like those proposed by Shephard (1994, Biometrika 81, 115–131) and Billio and Monfort (1998, Journal of Statistical Planning and Inference 68, 65–103), are necessary to derive an inference on the parameters of the model. We propose in this paper a Bayesian approach with a non-informative prior distribution developed in Mengersen and Robert (1996, Bayesian Statistics 5. Oxford University Press, Oxford, pp. 255–276) and Robert and Titterington (1998, Statistics and Computing 8(2), 145–158) in the setup of mixtures of distributions and hidden Markov models, respectively. The computation of the Bayes estimates relies on MCMC techniques which iteratively simulate missing states, innovations and parameters until convergence. The performances of the method are illustrated on several simulated examples. This work also extends the papers by Chib and Greenberg (1994, Journal of Econometrics 64, 183–206) and Chib (1996, Journal of Econometrics 75(1), 79–97) which deal with ARMA and hidden Markov models, respectively.