Multivariate periodic ARMA(1,1) processes

The main objective of this paper is to investigate the properties of multivariate periodic autoregressive moving average (ARMA)(1,1) processes. Such properties include the covariance structure, the parameter space, and estimation. The parameter space of such periodic processes is derived by aggregation (for instance, monthly models will aggregate to annual models). In particular, the parameter space and estimation are analyzed for contemporaneous ARMA (CARMA) models. It is shown that in general, the aggregation of a multivariate periodic ARMA(1,1) model leads to a stationary multivariate ARMA(1,1) model. However, not always a periodic CARMA(1,1) model aggregates into a stationary CARMA(1,1) model. Furthermore, it is shown that in estimating the aggregated model parameters, the method based on the periodic model and corresponding parameters, is superior to that based on the aggregated series.