  Most long memory forecasting studies assume that the memory is generated by
the fractional difference operator. We argue that the most cited theoretical
arguments for the presence of long memory do not imply the fractional
difference operator, and assess the performance of the autoregressive
fractionally integrated moving average $(ARFIMA)$ model when forecasting series
with long memory generated by nonfractional processes. We find that high-order
autoregressive $(AR)$ models produce similar or superior forecast performance
than $ARFIMA$ models at short horizons. Nonetheless, as the forecast horizon
increases, the $ARFIMA$ models tend to dominate in forecast performance. Hence,
$ARFIMA$ models are well suited for forecasts of long memory processes
regardless of the long memory generating mechanism, particularly for medium and
long forecast horizons. Additionally, we analyse the forecasting performance of
the heterogeneous autoregressive ($HAR$) model which imposes restrictions on
high-order $AR$ models. We find that the structure imposed by the $HAR$ model
produces better long horizon forecasts than $AR$ models of the same order, at
the price of inferior short horizon forecasts in some cases. Our results have
implications for, among others, Climate Econometrics and Financial Econometrics
models dealing with long memory series at different forecast horizons. We show
in an example that while a short memory autoregressive moving average $(ARMA)$
model gives the best performance when forecasting the Realized Variance of the
S\&P 500 up to a month ahead, the $ARFIMA$ model gives the best performance for
longer forecast horizons.
