Long memory relationships and the aggregation of dynamic models

Abstract

By aggregating simple, possibly dependent, dynamic micro-relationships, it is shown that the aggregate series may have univariate long-memory models and obey integrated, or infinite length transfer function relationships. A long-memory time series model is one having spectrum or order ω^-2d for small frequencies ω, d>0. These models have infinite variance for $\text{d≧}\text{1}\text{2}$ but finite variance for $\text{d<}\text{1}\text{2}$. For d=1 the series that need to be differenced to achieve stationarity occur, but this case is not found to occur from aggregation. It is suggested that if series obeying such models occur in practice, from aggregation, then present techniques being used for analysis are not appropriate.