Bayesian Estimation of Autoregressive Models with Exogenous Variables Under Scale-Mixtures of Normal Errors

Autoregressive models with exogenous variables (ARX) constitute a fundamental class of dynamic regression models used extensively for time series analysis across a wide range of applications. A pervasive limitation of the existing Bayesian analyses of ARX models is their near-exclusive reliance on the Gaussian error assumption, which is routinely violated in empirical applications exhibiting heavy-tailed innovations, distributional outliers, or excess kurtosis. To address this deficiency, we develop a rigorous Bayesian estimation framework for these models whose errors are drawn from the scale-mixtures of normal (SMN) family, which is a rich, symmetric, heavy-tailed class of distributions. Exploiting the hierarchical stochastic representation of the SMN family through observation-specific latent scale-mixing variables, the ARX model is embedded in an augmented data structure that restores Gaussian conditional structure. Under three distinct prior formulations—namely, normal-gamma, Zellner’s g-prior, and Jeffreys’ prior—we derive closed-form full conditional posterior distributions for the ARX coefficient vector and the error scale parameter, which follow multivariate normal and inverse-gamma distributions, respectively. In addition, for the SMN-specific shape parameters, we derive the full conditional posteriors for each distribution in the family, and some of them are non-standard distributions handled by embedding Metropolis-Hastings steps within the Gibbs sampler. The resulting hybrid MCMC algorithm is validated through a comprehensive simulation study spanning three ARX model configurations and all three SMN special cases. A real macroeconomic application to US consumer price inflation demonstrates the practical utility of the framework, confirming heavy-tailed residuals and yielding precise, well-calibrated posterior estimates.