Bayesian analysis of vector ARFIMA processes

A general framework is presented for Bayesian inference of multivariate time series exhibiting long-range dependence. The series are modelled using a vector autoregressive fractionally integrated moving-average (VARFIMA) process, which can capture both short-term correlation structure and long-range dependence characteristics of the individual series, as well as interdependence and feedback relationships between the series. To facilitate a sampling-based Bayesian approach, the exact joint posterior density is derived for the parameters, in a form that is computationally simpler than direct evaluation of the likelihood, and a modified Gibbs sampling algorithm is used to generate samples from the complete conditional distribution associated with each parameter. The paper also shows how an approximate form of the joint posterior density may be used for long time series. The procedure is illustrated using sea surface temperatures measured at three locations along the central California coast. These series are believed to be interdependent due to similarities in local atmospheric conditions at the different locations. and previous studies have found that they exhibit 'long memory' when studied individually. The approach adopted here permits investigation of the effects on model estimation of the interdependence and feedback relationships between the series.