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

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.

Long Memory and Volatility Dynamics in the US Dollar Exchange Rate

“…Given a covariance stationary process {x t , t = 0, ±1, … }, with autocovariance function E[(x t -Ex t )(x t-j -Ex t )] = j , according to McLeod and Hipel (1978), x t displays the property of long memory if…”

Market Efficiency, Roughness and Long Memory in PSI20 Index Returns: Wavelet and Entropy Analysis

“…A long memory process was originally defined and motivated, respectively, by McLeod and Hipel [15] and Hall [16], as a stationary process for which autocorrelations absolute values are not summable in the discrete case, that is:…”

Modeling and forecasting volatility in global food commodity prices

“…if d is 0.5 < d < 1, the series is no longer covariance stationary, but still mean reverting with the effect of a shock persist for a long period of time, and in that case, the process is said to have a long memory. given a discrete time series, y t , with autocorrelation function, ρ j , at lag j, Mcleod and hipel (1978) define long memory as a process:…”