App ears in Neural Networks in the Capital Markets Pro ceedings of the Third International Conference London Octob er A Refenes Y AbuMostafa J Mo o dy and A Weigend eds World Scientic London IMPROVED ESTIMATES FOR THE RESCALED RANGE AND HURST EXPONENTS John Mo o dy and Lizhong Wu Computer Science Dept Oregon Graduate Institute Portland OR Email mo o dycseogiedu lwucseogiedu ABSTRACT Rescaled Range RS analysis and Hurst Exp onents are widely used as measures of longterm memory structures in sto chastic pro cesses Our empirical studies show however that these statistics can incorrectly indicate departures from random walk b ehavior on short and intermediate time scales when very short term correlations are present A mo dication of rescaled range estimation RS analysis intended to correct bias due to shortterm dep endencies was prop osed by Lo Weshow however that Los RS statistic is itself biased and intro duces other problems including distortion of the Hurst exp onents We prop ose a new statistic RS that corrects for mean bias in the range Rbut do es not suer from the short term biases of RS or Los RS We supp ort our conclusions with exp eriments on simulated random walk and AR pro cesses and exp eriments 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 RS analysis is spurious scales by RS or Intro duction and Overview There are three widely used metho ds for longterm dep endence analysis auto correlation analysis fractional dierence mo dels Granger Joyeux Hosking and scaling law analysis including rescaled range RS analysis Hurst Hurst exp onents Hurst Mandelbrot Van Ness and drift exp onents Muller Dacorogna Olsen Pictet Schwarz Morgenegg This pap er stud ies RS analysis and Hurst exp onents whichhave b ecome recently p opular in the nance community largely due to the empirical work of Peters Compared to auto correlation analysis the advantages of RS analysis include detection of longrange dep endence in highly nongaussian time series with large skewness and kurtosis almost sure convergence for sto chastic pro cesses with innite variance and detection of nonp erio dic cycles However there are also two deciencies asso ciated with rescaled range analysis and the estimation of Hurst exp onents estimation errors exist when the time scale is very small or very large relative to the numb er of observations in the time series Mandelbrot Wallis Wallis Mata las Feder Ambrose Ancel Griths Mo o dy WuaMuller Dacorogna Pictet and the rescaled range is sensitive to shortterm de p endence McLeo d Hip el Hip el McLeo d Lo The second shortcoming will sometimes lead to completely incorrect results Lo analyzed the mean bias in the range statistic R due to shortterm dep endencies in the time series and prop osed a mo died rescaling factor S that is intended to remove or reduce these eects Wehave found however that Los 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 exp onents S N statistic for a given While Los approach fo cuses on the actual value of the R time scale of interest N Hurst and Mandelbrot test for long term dep endency by comparing the slop e of RS N curveto Our empirical results show however that Hurst exp onents standard rescaled range analysis and Los mo died rescaled range can yield incompatible results with the conventional interpretations of these statistics due to the biases contained in the R S andS statistics We prop ose 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 exp onents at short time scales The outline of this pap er is as follows In Section we will briey intro duce the rescaled range analysis and the Hurst exp onent The analysis and estimation pro cedures are then demonstrated on tickbytickinterbank foreign exchange data Through empirical comparisons we showhow seriously shortterm dep endencies in a time series can aect the rescaled range analysis In Section we explain why there is a mean bias in the range estimation and intro duce Los mo died approach Some simulation results with the mo died algorithm are shown and compared to results using the original algorithm In Section weevaluate Los mo died rescaled range analysis list and analyze the problems asso ciating with it and showhow it distorts the Hurst exp onent 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 RS and Los RS statistics In Section we conclude our pap er with a discussion RS Analysis for High Frequency FX Data Among the various approaches for quantifying correlations and deviations from gaussian b ehavior for sto chastic pro cesses several approaches have b een suggested that are based on scaling laws Unlike traditional correlation analysis these scaling law metho ds are intended to quantify structure that p ersists on a sp ectrum of time scales The Rescaled Range RS analysis and Hurst Exp onents were rst develop ed by Hurst and rened and p opularized by Mandelbrot etal in the late s and early s These b ecame p opular in nance due to the clear exp osition of the methodsinFeder and the empirical work of Peters A related approach based on the drift exp onentwas indep endently pioneered byMuller et al and scaling laws for directional change frequency have b een suggested by Guillaume In this pap er we restrict our attention to RS analysis and Hurst exp onents RS Analysis and Hurst Exponents The RS 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 p erio d changes by r the average m and biased standard deviation S of the t a returns from t t to t t N are t N X mN t r N t tt t N X r mN t S N t t N tt The partial sum of deviations of r from its mean and the range of partial sums t are then dened as t X X N t r mN t for N t tt R N t max X N t min X N t erage values The RS statistic for time scale N is simply the ratio b etween the av of R N t and S N t P R N t t RS N P S N t t a The quantity S N t conventionally used in RS analysis is an estimate of the standard deviation p that is biased downward by a factor N N The unbiased estimate of the true standard deviation N t is t N X r mN t b N t t N tt In section we present improved results using the unbiased estimate b N t Assuming that a scaling law exists for RS N we can write H RS N aN where a is a constantandH is referred to as the Hurst exp onent By estimating H we can characterize the b ehavior of time series as follows H random walk H meanreverting H meanaverting For a more detailed but readable discussion of RS analysis and Hurst exp onents see Feder RS Analysis for High Frequency FX Data High frequency interbank FX data consists of a sequence of BidAsk prices quoted byvarious rms that function as market makers While BidAsk price quotes from many market makers are displayed simultaneously by wire services such as Reuters and Telerate a single price series can b e constructed from the sequence of newly up dated quotes We are analyzing a full year of such tickbytickInterbank FX price quotes for three exchange rates the Deutschmark US Dollar rate DEMUSD the Japanese Yen US Dollar rate JPYUSD and the Deutschmark Yen DEMJPY cross rate The data were obtained from Olsen Asso ciates of Zuric h The data sample includes every tick from Octob er through Septemb er For the DEMUSD the year has ticks To study the b ehavior of returns on a sp ectrum of time scales we p erform rescaled range analysis and compute Hurst exp onents Figure shows the rescaled range anal ysis for Octob er DEMUSD Bid returns Both the original data and scrambled data were analyzed The upward shift in the curve for the scrambled data is evidence for mean reversion of the original series on all time scales measured Further results are presented in Mo o dy Wu To summarize them the b ehavior of the Hurst exp onents in the returns of DEMUSD exchange rates is qualitatively dierent from the Hurst exp onents of gaussian series and scrambled series of the returns The b ehaviors of RS N and H are more similar to those of an AR pro cess with negative co ecient To understand the nature and meaning of the apparent mean reverting b ehavior in the high frequency FX data wehave p erformed a series of investigations as describ ed in Mo o dy Wu b The question is whether the observed mean reversion over a range of intermediate times scales is due to shortterm price oscillations on time scales of a few ticks or is evidence of intrinsic dep endencies in the price movements on those intermediate time scales DEM_USD Bid_Returns Oct.92 2.2 2 1.8 1.6 Scrambled 1.4 1.2 Log10(R/S) 1 Original 0.8 0.6 0.4 0.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Log10(# of Ticks) Figure Rescaled range analysis for DEMUSD Bid returns during Octob er Between Original and Scrambled curves is a straight line with a slop e The lower slop e for the original series for time scales less than ticks suggests that mean reversion is present on short time scales One of our studies is to observe