Aggregation and systematic sampling of periodic ARMA processes

The aim of this work is to investigate the effects of temporal aggregation and systematic sampling on periodic autoregressive moving average (PARMA) time series. Firstly, it is shown that the class of weak PARMA processes, i.e. with uncorrelated but possibly dependent errors, is closed under a particular class of linear transformations that include both temporal aggregation and systematic sampling. This extends a similar result for autoregressive moving average processes; see [Wei, W.W.S., 2006. Time Series Analysis: Univariate and Multivariate Methods, second ed. Addison-Wesley, New York (Chapter 20)] for a review on the subject. Secondly, the properties of the noise of the transformed process are investigated. A sufficient condition is given under which aggregation and systematic sampling of a strong PARMA process, i.e. with independent errors, give rise in general to a weak PARMA process. Under that condition, the noise of the transformed process is neither strong nor a martingale difference. This result points out that the assumption of strong PARMA should not be used without careful considerations when analyzing aggregated time series that naturally occur in many scientific fields. The sufficient condition for non-independent errors is illustrated with the PARMA(1,1) model. A simulation study underlines the practical relevance of our findings and the importance of taking into account the dependence of the errors when fitting a PARMA model to an aggregated time series.