Bayesian modeling and forecasting of seasonal autoregressive models with scale-mixtures of normal errors

Most of existing Bayesian analysis methods of time series with seasonal pattern are based on the normality assumption; however, most of the real time series violate this assumption. With assuming the scale-mixtures of normal (SMN) distribution for the model errors, we introduce the Bayesian estimation and prediction of seasonal autoregressive (SAR) models, using the Gibbs sampler and Metropolis-Hastings algorithms. The SMN distribution is a general class that includes different symmetric heavy-tailed distributions as special cases, such as the Student’s t, slash and contaminated normal distributions. With employing different priors for the SAR parameters, we derive the full conditional posterior distributions of the SAR coefficients and scale parameter to be the multivariate normal and inverse gamma, respectively, and the conditional predictive distribution of the future observations to be the multivariate normal. For the other parameters related to the SMN distribution, we derive their conditional posteriors to be in a closed form but some of them are not standard distributions. Using the derived closed-form conditional posterior and predictive distributions, we propose the Gibbs sampler with the Metropolis-Hastings algorithm to approximate empirically the marginal posterior and predictive distributions. We introduce an extensive simulation study and a real application in order to evaluate the accuracy of the proposed MCMC algorithm.