Modeling Long‐Memory Processes For Optimal Long‐Range Prediction

. We look at the implications of modeling observations from a fractionally differenced noise process using an approximating AR (p) model. The approximation is used because of computational difficulties in the estimation of the differencing parameter of the fractional noise model. Because the fractional noise process is long‐range dependent, we assess the applicability of the approximating autoregressive (AR) model based on its long‐range forecasting accuracy compared with that of the fractional noise model. We derive the asymptotic k‐step‐ahead prediction error for a fractional noise process modeled as an AR(p) process and compare it with the k‐step‐ahead prediction error obtained when the model for the observed series is correctly specified as a fractional noise process and the fractional differencing parameter d is either known or estimated. We also assess the validity of the asymptotic results for a finite sample size via simulation. We see that AR models can be useful for long‐range forecasting of long‐memory data, provided that consideration is given to the forecast horizon when choosing an approximating model.