Aggregation of a Seasonal Long Memory model

Abstract

In this paper we investigate the implications of temporal aggregating on the properties of a seasonal long memory process. We consider two types of temporal aggregation - skip sampling as well as average sampling. We allow for both stationary and non-stationary processes at potentially several frequencies. We extend the existing literature in several ways. We allow for long memory at several frequencies and, thus, are able to analyze their interaction in the aggregated process. Further, by defining the process according to the truncated Type II definition, our approach encompasses both stationary as well non-stationary processes and, thus, does not require prior knowledge on the case and further allows for the joint presence of both. We further apply a recent methodology of writing the process in its vector season representation which enables us to directly deduce to which frequencies in the aggregated series the poles in the disaggregated series are mapped. Specifically, systematic sampling, unlike average sampling, can impact on the non-seasonal memory properties, insofar as the systematically sampled series can have long memory at frequency zero, even though the disaggregated series does not have. On the other hand, average sampling can lead to an aggregated series without long memory even though the disaggregated series has long memory. By simulations, we illustrate the mapping of the frequencies implied by both types of temporal aggregation and analyze the estimation of the memory in the aggregated series by applying a semi-parametric exact local Whittle estimator for seasonal long memory time series.