IMPROVED ESTIMATES FOR The rescaled range and Hurst exponents

Rescaled Range R S analysis and Hurst Exponents are widely used as measures of long term memory structures in stochastic processes Our empirical studies show however that these statistics can incorrectly indicate departures from random walk behavior on short and intermediate time scales when very short term correlations are present A modi cation of rescaled range estimation R S analysis intended to correct bias due to short term dependencies was proposed by Lo We show however that Lo s R S statistic is itself biased and introduces other problems including distortion of the Hurst exponents We propose a new statistic R S that corrects for mean bias in the range R but does not su er from the short term biases of R S or Lo s R S We support our conclusions with experiments on simulated random walk and AR processes and experiments using high frequency interbank DEM USD exchange rate quotes We conclude that the DEM USD series is mildly trending on time scales of to ticks and that the mean reversion suggested on these time scales by R S or R S analysis is spurious Introduction and Overview There are three widely used methods for long term dependence analysis auto correlation analysis fractional di erence models Granger Joyeux Hosking and scaling law analysis including rescaled range R S analysis Hurst Hurst exponents Hurst Mandelbrot Van Ness and drift exponents M uller Dacorogna Olsen Pictet Schwarz Morgenegg This paper stud ies R S analysis and Hurst exponents which have become recently popular in the nance community largely due to the empirical work of Peters Compared to autocorrelation analysis the advantages of R S analysis include detection of long range dependence in highly non gaussian time series with large skewness and kurtosis almost sure convergence for stochastic processes with in nite variance and detection of nonperiodic cycles However there are also two de ciencies associated with rescaled range analysis and the estimation of Hurst exponents estimation errors exist when the time scale is very small or very large relative to the number of observations in the time series Mandelbrot Wallis Wallis Mata las Feder Ambrose Ancel Gri ths Moody Wu a M uller Dacorogna Pictet and the rescaled range is sensitive to short term de pendence McLeod Hipel Hipel McLeod Lo The second shortcoming will sometimes lead to completely incorrect results Lo analyzed the mean bias in the range statistic R due to short term dependencies in the time series and proposed a modi ed rescaling factor S that is intended to remove or reduce these e ects We have found however that Lo s statistic is itself biased and causes some new problems on short time scales while attempting to correct the mean bias of the range R including distortion of the Hurst exponents While Lo s approach focuses on the actual value of the R S N statistic for a given time scale of interest N Hurst and Mandelbrot test for long term dependency by comparing the slope of R S N curve to Our empirical results show however that Hurst exponents standard rescaled range analysis and Lo s modi ed rescaled range can yield incompatible results with the conventional interpretations of these statistics due to the biases contained in the R S and S statistics We propose a new unbiased rescaling factor S that is able to correct for the mean biases in R at all time scales without inducing new distortions of the rescaled range and Hurst exponents at short time scales The outline of this paper is as follows In Section we will brie#y introduce the rescaled range analysis and the Hurst exponent The analysis and estimation procedures are then demonstrated on tick by tick interbank foreign exchange data Through empirical comparisons we show how seriously short term dependencies in a time series can a ect the rescaled range analysis In Section we explain why there is a mean bias in the range estimation and introduce Lo s modi ed approach Some simulation results with the modi ed algorithm are shown and compared to results using the original algorithm In Section we evaluate Lo s modi ed rescaled range analysis list and analyze the problems associating with it and show how it distorts the Hurst exponent In Section we present our new unbiased rescaling factor S along with empirical results that demonstrate the improvements that it yields relative to the standard R S and Lo s R S statistics In Section we conclude our paper with a discussion R S Analysis for High Frequency FX Data Among the various approaches for quantifying correlations and deviations from gaussian behavior for stochastic processes several approaches have been suggested that are based on scaling laws Unlike traditional correlation analysis these scaling law methods are intended to quantify structure that persists on a spectrum of time scales The !Rescaled Range" R S analysis and Hurst Exponents were rst developed by Hurst and re ned and popularized by Mandelbrot et al in the late s and early s These became popular in nance due to the clear exposition of the methods in Feder and the empirical work of Peters A related approach based on the drift exponent was independently pioneered by M uller et al and scaling laws for directional change frequency have been suggested by Guillaume In this paper we restrict our attention to R S analysis and Hurst exponents R S Analysis and Hurst Exponents The R S statistic is the range of partial sums of deviations of a time series from its mean rate of change rescaled by its standard deviation Denoting a series of returns one period changes by rt the average m and biased standard deviation S of the returns from t t to t t N are a