Gibbs sampling for Bayesian estimation of triple seasonal autoregressive models

In last few decades some researchers have extended existing time series models to fit high frequency time series characterized by exhibiting multiple seasonalities. In the Bayesian framework, most of the work in this direction is only at the level of double seasonality. Therefore, in this paper we have the objective to fill part of this gap by extending autoregressive models to fit time series that have three layers of seasonality, i.e., triple seasonal autoregressive (TSAR) models, and introducing the Bayesian estimation for these models using the Gibbs sampling algorithm. In order to achieve this objective, we first assume the model errors are normally distributed and employ the normal-gamma and g priors for the model parameters. We then derive the full conditional posterior distributions to be multivariate normal for the model coefficients and inverse gamma for the model variance. Using these closed-form conditional posterior distributions, we propose the Gibbs sampling algorithm to approximate empirically the joint posterior of the TSAR model parameters, enabling us easily to carry out the Bayesian estimation of TSAR models. By employing the g prior for the model parameters, we execute an extensive simulation study to evaluate the efficiency of the proposed Bayesian estimation of TSAR models, and we then apply our work to real-world hourly electricity load time series datasets in six European countries.