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The Lo's modified rescaled adjusted range test (R/S test) (Lo, 1991), GPH test (Geweke and Porter-Hudak, 1983) and two approximate maximum likelihood estimation methods, i.e., Whittle's estimator (W-MLE) and another one implemented in S-Plus (S-MLE) based on the algorithm of Haslett and Raftery (1989) are evaluated through intensive Monte Carlo simulations for detecting the existence of long- memory. It is shown that it is difficult to find an appropri- ate lag q for Lo's test for different short-memory autore- gressive (AR) and fractionally integrated autoregressive and moving average (ARFIMA) processes, which makes the use of Lo's test very tricky. In general, the GPH test outperforms the Lo's test, but for cases where a strong short-range de- pendence exists (e.g., AR(1) processes with =0.95 or even 0.99), the GPH test gets useless, even for time series of large data size. On the other hand, the estimates of dgiven by S- MLE and W-MLE seem to give a good indication of whether or not the long-memory is present. The simulation results show that data size has a significant impact on the power of all the four methods because the availability of larger sam- ples allows one to inspect the asymptotical properties bet- ter. Generally, the power of Lo's test and GPH test increases with increasing data size, and the estimates of d with GPH method, S-MLE method and W-MLE method converge with increasing data size. If no large enough data set is available, we should be aware of the possible bias of the estimates. The four methods are applied to daily average discharge series recorded at 31 gauging stations with different drainage areas in eight river basins in Europe, Canada and USA to detect the existence of long-memory. The results show that the presence of long-memory in 29 daily series is confirmed by at least three methods, whereas the other two series are indicated to be long-memory processes with two methods. The intensity of long-memory in daily streamflow processes has only a very weak positive relationship with the scale of watershed.

Detecting long-memory: Monte Carlo simulations and application to daily streamflow processes