Bayesian Identification Procedure for Triple Seasonal Autoregressive Models

Triple seasonal autoregressive (TSAR) models have been introduced to model time series date with three layers of seasonality; however, the Bayesian identification problem of these models has not been tackled in the literature. Therefore, in this paper, we have the objective of filling this gap by presenting a Bayesian procedure to identify the best order of TSAR models. Assuming that the TSAR model errors are normally distributed along with employing three priors, i.e., normal-gamma, Jeffreys' and g priors, on the model parameters, we derive the marginal posterior distributions of the TSAR model parameters. In particular, we show that the marginal posteriors are multivariate t and gamma distributions for the TSAR model coefficients vector and precision, respectively. Using the marginal posterior distribution of the TSAR model coefficients vector, we present an identification procedure for the TSAR models based on a sequence of t-test of significance. We evaluate the accuracy of the proposed Bayesian identification procedure by conducting an extensive simulation study, followed by a real application to hourly electricity load datasets in six European countries.