Full Bayesian Analysis of ARX Models Under Scale-Mixtures of Normal Errors: An Application to Solar Radiation in Najran, Saudi Arabia

Abstract

Autoregressive models with exogenous variables (ARX) constitute a fundamental family of time series tools with broad applicability across engineering, environmental science, and finance. A persistent limitation of standard Bayesian treatments is the Gaussian error assumption, which frequently proves inadequate when dealing with real data displaying heavy tails or occasional extreme observations. To overcome this shortcoming, the present paper develops a complete Bayesian inferential framework for ARX models under the scale-mixtures of normal (SMN) error distribution, integrating model identification, parameter estimation, and multi-step-ahead prediction within a unified scheme. A stochastic search variable selection (SSVS) procedure is adapted to perform simultaneous selection of the autoregressive order and the active exogenous regressors by assigning binary latent indicators to each candidate coefficient. Mixture-of-normals priors are specified for the dynamic and exogenous coefficients and an inverse-gamma prior for the error scale, while Bernoulli priors govern the latent selection indicators. These choices yield tractable full conditional posterior distributions: multivariate normal for the complete coefficient vector, inverse-gamma for the scale, and Bernoulli for the indicators. The conditional predictive distribution of future observations is also multivariate normal. For SMN-specific mixing parameters whose conditionals lack standard forms, Metropolis–Hastings steps are embedded within the Gibbs sampler. An extensive simulation study evaluates recovery accuracy across three SMN distributions and several ARX configurations. The methodology is then applied to the forecasting of daily global horizontal irradiance (GHI) in Najran, southwestern Saudi Arabia, using clear-sky GHI as an exogenous covariate, demonstrating the practical value of the proposed framework in a renewable energy context.