The approximation of long-memory processes by an ARMA model

A mean square error criterion is proposed in this paper to provide a systematic approach to approximate a long-memory time series by a short-memory ARMA(1, 1) process. Analytic expressions are derived to assess the effect of such an approximation. These results are established not only for the pure fractional noise case, but also for a general autoregressive fractional moving average long-memory time series. Performances of the ARMA(1, 1) approximation as compared to using an ARFIMA model are illustrated by both computations and an application to the Nile river series. Results derived in this paper shed light on the forecasting issue of a long-memory process.