Full Bayesian analysis of seasonal autoregressive models under scale-mixtures of normal errors

Most of existing Bayesian analysis approaches of seasonal autoregressive (SAR) models are based on the normality assumption; however, the real time series might violate this assumption. In order to fill this research gap, in this paper, we first assume the scale-mixtures of normal (SMN) distribution for the SAR errors, and then we introduce a full Bayesian analysis of these models, including identification, estimation, and prediction. Our approach combines Gibbs sampler and Metropolis-Hastings algorithms along with a stochastic search variable selection (SSVS) method. We introduce latent variables for the SAR lags, and we also employ a mixture-normal prior for the SAR coefficients, an inverse-gamma prior for the scale parameter, and a Bernoulli prior for the latent variables. These priors yield full conditional posterior and predictive densities that are either standard distributions or have closed-form expressions. Specifically, the conditional posteriors of the SAR coefficients, scale parameter, and latent variables are multivariate normal, inverse-gamma, and Bernoulli, respectively, while the conditional predictive distribution for future observations is multivariate normal. For SMN-related parameters, the conditional posteriors are also derived in closed form, though some are non-standard. Leveraging these results, we design a Gibbs sampler with embedded Metropolis–Hastings steps and SSVS to empirically approximate the joint posterior and predictive distributions, enabling accurate multi-step-ahead forecasts. Finally, we evaluate the accuracy of the proposed MCMC algorithm through both simulation studies and a real-data application.