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.
Ayman A. Amin, Shuhrah A. Alghamdi· Mathematics· 0 citations
Identifying the autoregressive (AR) orders of a multivariate time series is a foundational step whose accuracy governs every downstream modelling and forecasting task. When the series exhibits two simultaneously operating seasonal periodicities, as is routinely observed in hourly electricity loads and intraday financial prices, no principled Bayesian identification framework currently exists. This paper addresses this gap by introducing a complete Bayesian order identification procedure for double-seasonal vector autoregressive (DSVAR) models. These DSVAR models are multivariate processes governed by a multiplicative triple autoregressive operator that jointly captures regular, first-seasonal, and second-seasonal dynamics. We treat the three order indices as unknown discrete parameters and derive closed-form expressions for the joint posterior probability mass function of the order triple. Two complementary prior specifications are considered: a conjugate matrix normal-Wishart prior and Jeffreys’ non-informative prior. The analysis is carried out under the assumption of symmetric, normally distributed errors, which ensures analytical tractability and allows for the posterior probabilities to be evaluated exactly for every admissible combination of orders. Specifically, under each prior, the posterior mass function reduces to explicit determinantal expressions that can be evaluated by a straightforward three-dimensional grid search. Monte Carlo experiments on various DSVAR processes confirm that the proposed technique achieves high identification accuracy even at moderate sample sizes across a range of parameter configurations and prior choices. The proposed Bayesian procedure is benchmarked against the standard Bayesian information criterion (BIC), consistently achieving higher correct identification rates across all cases. Empirical applications to hourly electricity load data from the Czech Republic and Germany, as well as to hourly solar radiation in Najran, Saudi Arabia, demonstrate the practical applicability of the Bayesian identification method.
Ayman A. Amin, F. Almuhayfith· Mathematics· 0 citations
A wide range of real-world multivariate time series encountered in practice exhibit two simultaneous and interacting seasonal cycles, for example hourly electricity demand, intraday financial prices, and sub-daily traffic volumes. Existing Bayesian frameworks for vector autoregressive (VAR) processes accommodate at most a single seasonal periodicity, leaving no established methodology for the double seasonal case commonly observed in high-frequency multivariate data. This paper bridges that gap by introducing the double seasonal VAR (DSVAR) models, which extend the univariate double seasonal literature to a coherent multivariate setting. These models are defined through a multiplicative triple autoregressive operator that naturally accommodates the second seasonal cycle. Under a Gaussian error assumption, we derive a comprehensive and analytically convenient Bayesian framework for both modeling and forecasting of DSVAR processes. We consider two prior families: a conjugate matrix normal-Wishart prior which yields exact closed-form inference, and a Jeffreys’ non-informative prior. Under each prior, we derive the marginal posterior distribution of the coefficient matrix as a matrix-t distribution and the marginal posterior of the precision matrix as a Wishart distribution. Moreover, we derive the predictive distribution of future observations as a multivariate-t with an exact analytic form, together with its highest predictive density regions. The methodology is validated through a Monte Carlo simulation experiment and applied to hourly electricity loads in Czech Republic and Germany, two physically interconnected markets with pronounced intraday and intraweek seasonal cycles. Benchmark comparisons against standard VAR, single-seasonal VAR, and univariate seasonal AR models confirm the substantial forecasting gains delivered by the proposed DSVAR framework at both short and long horizons.
Ayman A. Amin, F. Almuhayfith· Mathematics· 0 citations
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