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. Copyright © 2001 John Wiley & Sons, Ltd.     

The Effect of Estimating Parameters on Long‐Term Forecasts for Cointegrated Systems

This paper concerns Long‐term forecasts for cointegrated processes. First, it considers the case where the parameters of the model are known. The paper analytically shows that neither cointegration nor integration constraint matters in Long‐term forecasts. It is an alternative implication of Long‐term forecasts for cointegrated processes, extending the results of previous influential studies. The appropriate Mote Carlo experiment supports our analytical result. Secondly, and more importantly, it considers the case where the parameters of the model are estimated. The paper shows that accuracy of the estimation of the drift term is crucial in Long‐term forecasts. Namely, the relative accuracy of various Long‐term forecasts depends upon the relative magnitude of variances of estimators of the drift term. It further experimentally shows that in finite samples the univariate ARIMA forecast, whose drift term is estimated by the simple time average of differenced data, is better than the cointegrated system forecast, whose parameters are estimated by the well‐known Johansen's ML method. Based upon finite sample experiments, it recommends the univariate ARIMA forecast rather than the conventional cointegrated system forecast in finite samples for its practical usefulness and robustness against model misspecifications. Copyright © 2011 John Wiley & Sons, Ltd.