Long-memory process and aggregation of AR(1) stochastic processes: A new characterization

TL;DR

This paper introduces a new way to characterize the long-memory properties of aggregated AR(1) processes, revealing that the infinite autoregressive process has a unit root when long memory is present.

Contribution

It provides a novel characterization of the autoregressive coefficients in aggregated AR(1) processes with long memory, highlighting the presence of a unit root in the infinite AR process.

Findings

Infinite AR process has a unit root with long memory

New characterization of autoregressive coefficients

Examples using well-known density functions

Abstract

Contemporaneous aggregation of individual AR(1) random processes might lead to different properties of the limit aggregated time series, in particular, long memory (Granger, 1980). We provide a new characterization of the series of autoregressive coefficients, which is defined from the Wold representation of the limit of the aggregate stochastic process, in the presence of long-memory features. Especially the infinite autoregressive stochastic process defined by the almost sure representation of the aggregate process has a unit root in the presence of the long-memory property. Finally we discuss some examples using some well-known probability density functions of the autoregressive random parameter in the aggregation literature. JEL Classification Code: C2, C13.