Abstract
Modelling multiple seasonal time series is a pervasive challenge across diverse domains, including energy, transportation, and financial markets. Existing Bayesian methods for seasonal vector autoregressive (SeVAR) models treat lag-order identification, coefficient estimation, and multi-step forecasting as separate tasks, lacking a unified framework. To address this limitation, we develop a comprehensive Gibbs sampling scheme capable of executing all three tasks simultaneously for SeVAR processes. We embed the SeVAR model in a hierarchical Bayes mixture framework in the spirit of group (block) stochastic search variable selection (Block-SSVS). Rather than assigning inclusion indicators to individual coefficients, a single Bernoulli-distributed binary indicator is attached to each lag matrix, ensuring that all entries within a block are either jointly retained or jointly shrunk toward zero. Following the free-parameter approximation standard in the literature, the cross (regular×seasonal) lag blocks are treated as unrestricted free parameters rather than as the exact products implied by the multiplicative operator, so the fitted model is a seasonally structured subset VAR over the candidate lag set generated by that operator. A block-structured spike-and-slab matrix normal prior governs the stacked coefficient matrix conditional on these block indicators and the innovation precision matrix, and a Wishart prior completes the precision-matrix specification. Within this hierarchical structure, we establish the closed-form full conditional posteriors for all model parameters. Specifically, we derive that the coefficient matrix follows a matrix normal distribution, the precision matrix yields a Wishart posterior, and each block indicator follows a Bernoulli posterior governed by the joint, matrix-variate evidence of its block. Furthermore, the one-step and multi-step conditional predictive distributions are established as a multivariate normal structure. Leveraging these results, we construct a stochastic simulation-based recursion to produce multi-step forecasts. The resulting posterior block-inclusion probabilities furnish an automatic best-subset SeVAR model at no additional computational cost beyond the Gibbs sampler itself. A controlled Monte Carlo study confirms the accuracy of the procedure for identification, estimation, and prediction. In addition, an application to hourly solar radiation in Najran, Saudi Arabia, demonstrates the framework’s identification capacity on genuinely bivariate stochastic data, with forecast accuracy reported at the data’s own predictability floor.