Preservation of the rescaled adjusted range: 1. A reassessment of the Hurst Phenomenon

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Abstract
Previous research related to the controversial Hurst phenomenon is reviewed and evaluated. Because of the inherent statistical properties of the rescaled adjusted range (RAR) statistic it is suggested that research primarily be devoted to this statistic rather than to the various definitions of the Hurst coefficient. Simulation studies reveal that for independently distributed random variables the RAR does not significantly depend on the underlying distribution of the random variables but is a function of the sample size. For modeling correlated data the statistical attributes of a discrete fractional Gaussian noise (FGN) process are studied and also improved. An efficient maximum likelihood estimation technique is developed for the FGN model, and it is shown how the residuals of the fitted model can be calculated and then subjected to diagnostic checks. An exact simulation procedure is developed for simulating FGN in such a way that synthetic traces from the model lie outside the Brownian domain. The Akaike information criterion (AIC) is suggested as a method for choosing between a FGN and a Box‐Jenkins model. For the six annual river flow series that are considered the AIC selects the best Box‐Jenkins model in preference to the FGN process for each data set. Because Box‐Jenkins models can be shown to preserve the historical RAR, in many practical applications it may be advantageous to use a Box‐Jenkins model instead of a FGN process.

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