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Physica A 375 (2007) 159–173

www.elsevier.com/locate/physa

The sampling properties of Hurst exponent estimates

Craig Ellis�

School of Economics and Finance, University of Western Sydney, Locked Bag 1797, Penrith South DC, NSW 1797, Australia

Received 29 May 2006; received in revised form 10 August 2006

Available online 11 September 2006

Abstract

The classical rescaled adjusted range (R/S) statistic is a popular and robust tool for identifying fractal structures and

long-term dependence in time-series data. Subsequent to Mandelbrot and Wallis [Water Resour. Res. 4 (1968) 909] who

proposed the statistic be measured over several subseries contained within the whole series length, the use overlapping vs.

contiguous subseries has been a source of some debate amongst R/S theorists. This study examines the distributional

characteristics of rescaled adjusted range and Hurst exponent estimates derived using overlapping vs. contiguous subseries,

henceforth closing debate on the issue of relative bias due to either technique. Confidence intervals for the statistical

significance of the Hurst exponent are also determined.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Rescaled range; Hurst exponent; Distribution; Simulation

1. Introduction

Rescaled range (R/S) analysis has in recent times regained in popularity as a robust tool for the examinationof long-term dependence in time-series data in a range of disciplines (see Refs. [1–6]). Proposed by Hurst [7],the classical rescaled adjusted range statistic measures the standardised range of the partial sum of deviationsof a time series from its mean. Initially proposed to be estimated over the whole series length, Mandelbrot andWallis [8] developed Hurst’s methodology to incorporate ordinary least squares (OLS) regression techniquesand thus proposed the statistic be estimated over several subseries contained within the whole series length.Henceforth, the decision to employ overlapping vs. contiguous subseries in the determination of the Hurstexponent has been a source of some debate amongst R/S theorists. Beginning with the conjecture of Wallisand Matalas [9]—later supported by Ambrose et al. [10]—that contiguous subseries would yield more biasedestimates of the Hurst exponent, historic empirical research largely favoured the use of overlapping subseries(see Refs. [11–15]), yet more recent studies have promote the used of contiguous subseries (see Refs. [1,16–19]).

The relative power of rescaled range analysis vs. alternative methods of estimating the Hurst exponent forfractional Gaussian processes has been recently studied by Bassingthwaighte and Raymond [20] and Cacciaet al. [21] and for Gaussian processes by Weron [22]. This study aims to contribute to the existing literature byexamining the distributional characteristics of Hurst exponent estimates derived using overlapping

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.08.046

4620 3250; fax: +61 2 4626 6683.

ess: c.ellis@uws.edu.au.

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Notation

(R/s)n rescaled adjusted rangeCV coefficient of variationE(H) expected Hurst exponentH (observed) Hurst exponentL total number of subseriesn subseries lengthn0 minimum subseries lengthnS maximum subseries lengthN series lengthS number of different subseries lengths noN

v number of subseries starts per each n

C. Ellis / Physica A 375 (2007) 159–173160

vs. contiguous subseries, henceforth closing debate on the issue of relative bias due to the use of overlappingvs. contiguous subseries in the estimation of the classical rescaled adjusted range. Monte Carlo simulationtechniques will be used to produce Gaussian series of varying lengths, from which the distribution of classicalrescaled range estimates of the Hurst exponent will be mapped and analysed. Research in the study will alsofurther extend the work of Weron [22] to provide confidence intervals for the rescaled adjusted range derivedfrom overlapping subseries sets. The major result of the study will be to prove that the use of contiguous vs.overlapping subseries bears very little impact on mean estimates of the Hurst exponent, yet it does significantlyimpact on the standard deviation of the estimate when the series length is relatively short.

2. Subseries selection and the classical rescaled adjusted range

Proposed by Hurst [7] and modified by Mandelbrot and Wallis [8] the classical rescaled adjusted range(R/s)n is calculated as

ðR=sÞn ¼ ð1=snÞ Max1pkpn

Xk

j¼1

X j � X n

� �� Min

1pkpn

Xk

j¼1

X j � X n

� �" #, (1)

where X n is the sample mean (1/n)SjXj and sn is the series standard deviation

sn ¼ 1=nXn

j¼1

X j � X n

� �2" #0:5. (2)

The original specification of the classical rescaled adjusted range (then denoted by the exponent K) providedby Hurst [7] was such that the exponent was estimated for the whole sample length N. The procedure was latermodified by Mandelbrot and Wallis [8] to incorporate OLS regression techniques where the exponent (denotedH by Mandelbrot and Wallis) was estimated over several subseries, npN as

log ðR=sÞn ¼ aþH log ðnÞ þ �, (3)

where log(R/s)n is the logarithm of the mean rescaled range for a subseries of length n, log(n) the logarithm ofthe subseries length and H the series Hurst exponent.

The OLS procedure described in Eq. (3) remains the standard for estimating the classical rescaled adjustedrange. For a Gaussian series given long subseries lengths a known result is that the value of the exponent willtend to its asymptotic H ¼ 0.5. Yet when n is small the classical rescaled adjusted range may exhibit a varietyof transient behaviours not necessarily indicative of the limiting behaviour of the exponent H (see Ref. [24]). Inso far that OLS estimates of H are necessarily derived over subseries of varying length (both small and large),it follows that the choice of n is integral to the determination of the exponent. A significant yet largely

ARTICLE IN PRESSC. Ellis / Physica A 375 (2007) 159–173 161

unresolved question remains however as to the preferred method used to divide the sample N into subseries;namely the impact on the exponent of contiguous vs. overlapping subseries.

The use of alternative techniques for the division of the series length into subseries was first considered byWallis and Matalas [9] who deemed the manner in which the series length was divided into subseries significantto the estimation of the Hurst exponent. Two alternative techniques based on overlapping subseries wereadopted by Wallis and Matalas and were termed the ‘F-Hurst’ and ‘G-Hurst’, respectively. In order toproperly describe these, some further formal definitions must first be provided.

For a given series of length N let the total number of individual subseries be denoted by L. Similarly for agiven subseries of length npN let the number of subseries starting values be denoted by v. The term ‘startingvalues’ is interpreted to mean the number of times each subseries length is divided into the series length. In thecase then where n ¼ N, it follows that the number of starting values is v ¼ 1. For all other noN the number ofstarting values may be any integer value 1pvp(N�n+1). For each subseries n and all i ¼ 1; 2; . . . ; v, letindividual starts be denoted by vi and the number of different length subseries by S. Finally, let the minimumand maximum subseries lengths be denoted n0 and nS, respectively.

In the terms of the above definitions Wallis and Matalas [9] propose the F-Hurst technique requires dividingthe series length by every permissible length of subseries 5pnpN. The resulting number of different subseriesis therefore S ¼ N�4. For each different subseries length the mean value of (R/s)n is estimated from a set ofv ¼ (N�n+1) starting values and the regression of log(R/s)n vs. log(n) calculated. Given an initial start v1 ¼ 1,for all integer noN and i ¼ 2; 3; . . . ; v� 1, individual F-Hurst starts are given by

vi ¼ vi�1 þ 1 (4)

and for i ¼ v

vv ¼ ðN � nþ 1Þ. (5)

The total number of subseries L required using the F-Hurst technique was given by Wallis and Matalas as

L ¼ ðN � 3ÞðN � 4Þ=2. (6)

Demonstrating the computational intensity of the F-Hurst technique, for a series of length N ¼ 10,000 thenumber of individual subseries was shown to be L ¼ 49,965,006. In order then to provide a more tractablemethodology for series lengths N41000 Wallis and Matalas proposed that Hurst exponent estimates could beapproximated by the G-Hurst technique.

A generalisation of the F-Hurst, the G-Hurst technique limits both the number of different subseries npN

and the number of starts v per each n. Recommending a sequence of S ¼ 39 different subseries of length10pnp1000, Wallis and Matalas proposed these be evenly distributed in log space throughout the serieslength. Rather than all permissible starts, the number of G-Hurst starts per each subseries length was reducedto vp15 only. For all integer i ¼ 2; 3; . . . ; v and all noN the position of each successive start vi is the integerpart of

vi ¼ vi�1 þðN � nþ 1Þ

ðv� 1Þ. (7)

For i ¼ v, it follows that Eq. (7) reduces to Eq. (5). The significance of Eq. (7) is that starting values for all n

will be evenly distributed throughout the entire series length between, hence capturing all the regions ofbehaviour of the underlying series, e.g. regions (periods) of higher/lower volatility, or stronger/weaker trends.While rejecting the transient hypothesis with relation to the length of the transient region as a proxydeterminant of the magnitude of dependence in a series on the basis of their own computer simulations, Wallisand Matalas conclude to recommend a minimum subseries length n0 ¼ 56.

Proving the computation efficiency of the G-Hurst methodology, it can be shown that independent of boththe series length and length(s) of the different subseries, the total number of subseries reduces to

L ¼ Sv. (8)

By comparison to the F-Hurst methodology, given S ¼ 39 and vp15 the maximum total number ofG-Hurst subseries (irrespective of the length N) is only L ¼ 585.

ARTICLE IN PRESSC. Ellis / Physica A 375 (2007) 159–173162

A hybrid approach for the selection of subseries lengths advocated by Peters [23] and termed the ‘P-Hurst’technique requires the use of contiguous (nonoverlapping) starting values with subseries lengths chosen fromthe available set of integer factors of the series length N. First proposed by Mandelbrot and Wallis [24] as ameans of avoiding ‘the unnecessary repeating of information’, the use of contiguous subseries lengths was alsoinitially considered by Wallis and Matalas [9] yet ultimately rejected on the basis that the technique producedmore highly biased estimates of (R/s)n. The apparent bias due to contiguous subseries was however notempirically proven by Wallis and Matalas.

Earlier attempts to quantify mathematically the choice of contiguous subseries lengths include Davies andHarte [25] and Feder [26]. Derived using geometric progressions of declining values of n both these approachesmay be considered inferior to that developed Peters for the fact that they either generate a remainder such thatnot all starting values are indeed contiguous, and/or result in fewer individual subseries lengths S than bycalculating all the factors of N.

Individual contiguous starting values from v1 ¼ 1 to vv ¼ (N�n+1) using either of the Peters [23], Daviesand Harte [25] or Feder [26] techniques are given by

vi ¼ vi�1 þ nþ 1 (9)

and the total number of contiguous subseries by

L ¼XS

i¼1

N

ni

. (10)

Intuitively, for a given series length N the total number of individual subseries in Eq. (10) should be less thanthat for the F-Hurst technique, yet greater than the total number of required G-Hurst subseries. Relative toeither technique, the issue remains with the use of contiguous subseries that the series length N must be chosenas such to maximise the set of available factors and therefore the number of different subseries lengths S, forthe purposes of estimating the OLS regression in Eq. (3).

3. The sampling distribution of Hurst exponent estimates

Given alternative methodologies for selecting subseries for the estimation of the classical rescaled adjustedrange by OLS regression techniques it follows that the choice of methodology may too be of significantinfluence in determining the value of the Hurst exponent. Given also the question of variability in (R/s)nestimates across individual subseries, further implications of the choice between overlapping and contiguoussubseries lengths exist with respect to the significance of estimated values of the exponent.

Having defined alternative methods for the division of the series length N into subseries n, Wallis andMatalas [9] additionally sought to examine the characteristics of F- and G-Hurst estimates of (R/s)n for arange of finite series lengths 20pNp1000. Using Monte Carlo simulation techniques Wallis and Matalasshowed mean F-Hurst estimates to be least biased with respect to the theoretical value H ¼ 0.5, and to havethe least variance. Both relative bias and variance were shown by Wallis and Matalas to slowly diminish asN-1000. Their result suggested that convergence of the exponent to the asymptotic limit for Gaussian serieswould occur for N41000. Results pertaining to P-Hurst estimates were not reported except for the surmisethat these would be more highly biased and would induce higher variances. Though lacking empirical support,the conjecture posed by Wallis and Matalas has nonetheless been used as evidence (see Ref. [10]) in oppositionto the P-Hurst technique.

Also considering the finite sample properties of G-Hurst (R/s)n estimates, the Monte Carlo technique hasbeen employed by Aydogan and Booth [12], whose approach involved adding a dummy slope variable to theOLS regression model in Eq. (3). Pre-asymptotic behaviour, it was argued, would then be demonstrated wherethe estimated b-coefficient of the dummy was significantly greater than zero. The author’s findings seeminglyconfirmed that the OLS technique would often yield values of 0.4oHo0.6, yet may be questioned on the basisof high autocorrelations in the regression residuals producing higher than otherwise t-statistic values for theb-coefficient of the dummy variable.

ARTICLE IN PRESSC. Ellis / Physica A 375 (2007) 159–173 163

3.1. Research design

The primary objective of research in this study is to examine the sampling distribution of Hurst exponentestimates using either contiguous or overlapping subseries. In so far that the aforementioned prior studies (seeRefs. [9,12]) only considered series lengths up to N ¼ 1000 and that their analysis constituted as few as 200simulations each, the aim of this research is to provide a thorough analysis of the distribution of the Hurstexponent, by examining both longer series lengths and a greater number of simulations than has been previouslyattempted. The outcomes of the research will allow confidence intervals to be generated, using which thesignificance of Hurst exponent estimates calculated from real time-series data can be precisely measured.

The sampling distribution of G- and P-Hurst estimates of (R/s)n and H will be examined using Monte Carlosimulation techniques for simulated Gaussian series. The sampling distributions will be initially constructedfrom 1.0E+04 simulated series each. The initial analysis will specifically consider mean values of (R/s)n andH, the relative bias of the mean(s), and their standard deviation. The analysis closely follows the suggestion byWallis and Matalas [9] with respect to H that any investigation of alternative techniques for estimating theclassical rescaled adjusted range should at least address these three variables. Confidence intervals will begenerated from a further 1.0E+05 simulated series each.

Before continuing, however, a critical point of clarification must be made; namely that the asymptoticbehaviour of the classical rescaled adjusted range depends less on the overall series length than on the rangeand length of the subseries n over which (R/s)n is estimated. The series length itself is important only so far asto allow a longer maximum subseries length nS. Longer subseries, it will be noted are necessary in order tocapture the limiting behaviour of the Hurst exponent. Following from Wallis and Matalas [9], in order tominimise transience from very short subseries lengths the minimum subseries length employed in this study isn0 ¼ 56.

A further consideration in the particular case of the P-Hurst technique is the number of integer factors of N,as this constitutes a limit to the number and range of n that can be utilised given contiguous starts v. Forinstance, the composite number 10,000 has only 25 integer factors, 14 of which are greater than/equal to 56 yetthe composite 10,080 has 72 integer factors, of which 45 are greater than/equal to 56. Using this example it canbe seen that small differences in the length of the sample can have important implications with respect to theavailable factor set when using contiguous subseries.

G- and P-Hurst estimates of (R/s)n and H in this study are each initially calculated for a fixed series lengthof N ¼ 20,160; the equivalent to 80 years of daily trading data. G-Hurst subseries lengths for 56pnp1000 arethose originally proposed by Wallis and Matalas [9]. Given the larger series length in this study the set of n isextended to include log space evenly distributed values 1152pnp10,875. Also following from Wallis andMatalas, the number of starting values has been initially set to v ¼ 15 for each different subseries length ofG-Hurst (R/s)n. The total number of different G-Hurst subseries lengths from 56pnp10,875 is S ¼ 40. P-Hurst subseries lengths are chosen from the set of integer factors of the composite number 20,160 (the serieslength) from 56pnp10,080. The number of contiguous starts using the P-Hurst technique ranges fromv ¼ 306 for n ¼ 56, to v ¼ 2 for n ¼ 10,080. The total number of different P-Hurst subseries lengths from56pnp10,080 is S ¼ 37.

While G and P-Hurst estimates of (R/s)n and H both use a minimum subseries length n0 ¼ 56, theintermediate and maximum subseries lengths vary widely. In order to compensate for this variation this studyproposes a hybrid methodology. Herein termed the ‘Pg-Hurst’ technique, the new methodology isdistinguished from the P-Hurst technique by the fact that while the subseries lengths themselves are chosenfrom the set of factors of N ¼ 20,160, each Pg-Hurst (R/s)n is calculated from a fixed number of overlappingstarting values; v ¼ 15 in the present case. The novelty of this approach is that any potential influences arisingfrom the use of overlapping vs. contiguous subseries lengths may be examined independently of questionsrelating to alternative subseries lengths.

3.2. The sampling distribution of ðR=sÞn estimates

The mean and standard deviation of all G-Hurst log(R/s)n 56pnp10,875 and all P and Pg-Hurst log(R/s)n56pnp10,080 given N ¼ 20,160 is provided in Table 1. The log of the rescaled adjusted range log(R/s)n, it

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Table 1

Mean and standard deviation of log(R/s)n

n G-Hurst n P-Hurst Pg-Hurst

Mean Std. dev. Mean Std. dev. Mean Std. dev.

56 0.9244 0.0255 56 0.9248 0.0054 0.9241 0.0258

63 0.9527 0.0255 63 0.9531 0.0057 0.9524 0.0257

71 0.9814 0.0257 70 0.9782 0.0060 0.9775 0.0258

80 1.0096 0.0257 80 1.0099 0.0064 1.0091 0.0257

89 1.0350 0.0256 90 1.0376 0.0068 1.0370 0.0256

100 1.0625 0.0257 96 1.0528 0.0070 1.0523 0.0256

110 1.0848 0.0258 112 1.0889 0.0075 1.0882 0.0255

125 1.1146 0.0257 126 1.1165 0.0080 1.1158 0.0253

142 1.1440 0.0258 140 1.1408 0.0084 1.1405 0.0253

158 1.1688 0.0257 160 1.1718 0.0089 1.1714 0.0254

178 1.1963 0.0256 180 1.1990 0.0095 1.1984 0.0254

224 1.2489 0.0254 224 1.2492 0.0105 1.2487 0.0254

252 1.2758 0.0255 252 1.2760 0.0111 1.2757 0.0254

282 1.3014 0.0253 280 1.2999 0.0117 1.2996 0.0254

316 1.3272 0.0251 288 1.3065 0.0119 1.3060 0.0254

354 1.3529 0.0252 315 1.3268 0.0124 1.3265 0.0252

400 1.3802 0.0252 336 1.3412 0.0128 1.3411 0.0252

450 1.4070 0.0251 360 1.3568 0.0133 1.3569 0.0250

500 1.4306 0.0249 420 1.3916 0.0143 1.3917 0.0252

563 1.4573 0.0248 480 1.4215 0.0152 1.4219 0.0252

630 1.4826 0.0248 504 1.4326 0.0156 1.4330 0.0251

710 1.5094 0.0254 560 1.4562 0.0164 1.4567 0.0251

800 1.5359 0.0269 630 1.4824 0.0174 1.4831 0.0249

890 1.5596 0.0280 672 1.4968 0.0179 1.4975 0.0250

1000 1.5856 0.0293 720 1.5122 0.0186 1.5130 0.0256

1152 1.6173 0.0307 840 1.5465 0.0200 1.5472 0.0274

1322 1.6479 0.0321 1008 1.5872 0.0219 1.5877 0.0292

1536 1.6809 0.0349 1120 1.6107 0.0230 1.6112 0.0304

1768 1.7121 0.0372 1260 1.6368 0.0244 1.6373 0.0315

2048 1.7442 0.0403 1440 1.6659 0.0260 1.6667 0.0340

2390 1.7779 0.0436 1680 1.6997 0.0280 1.7007 0.0367

2793 1.8121 0.0477 2016 1.7395 0.0306 1.7407 0.0405

3274 1.8465 0.0523 2520 1.7886 0.0341 1.7896 0.0461

3850 1.8813 0.0579 3360 1.8505 0.0392 1.8520 0.0545

4539 1.9167 0.0642 5040 1.9147 0.0478 1.9163 0.0675

5369 1.9678 0.0709 6720 1.9775 0.0550 1.9791 0.0730

6372 2.0151 0.0786 10080 2.2329 0.0673 2.2352 0.1112

7588 2.0658 0.0871

9068 2.1204 0.0966

10875 2.1792 0.1071

C. Ellis / Physica A 375 (2007) 159–173164

will be recalled constitutes the dependant variable in Eq. (3) for the estimation of the Hurst exponent. Ananalysis of the mean and standard deviation of log(R/s)n should be expected to provide information about thesampling distribution of the Hurst exponent itself.

The mean of log(R/s)n is positively related to n for each technique G-, P- and Pg-Hurst. Consistent with thefact that G-Hurst subseries lengths n are chosen to be evenly distributed in log space, mean values of G-Hurstlog(R/s)n are approximately linear with respect to n. P and Pg-Hurst log(R/s)n by contrast increasenonlinearly with n. Mean values of P- and Pg-Hurst log(R/s)n are not statistically different for subseries ofequal length. It is expected therefore that there should likewise be no statistical difference in mean values of H

using the P- and Pg-Hurst techniques; that is, that the choice of contiguous vs. overlapping subseries startsshould prove irrelevant to mean value of the Hurst exponent.

ARTICLE IN PRESSC. Ellis / Physica A 375 (2007) 159–173 165

The standard deviation of log(R/s)n increases nonlinearly with longer n for P-Hurst log(R/s)n, yet thestandard deviation of log(R/s)n presents as U-shaped for both of the G and Pg-Hurst techniques. Expressingthe standard deviation as a percentage of the mean the coefficient of variation, CV of G and Pg-Hurstlog(R/s)n is likewise U-shaped with the minimum CV at n ¼ 630 and 672 for the G- and Pg-Hurst techniques,respectively. While the standard deviation and CV of P-Hurst log(R/s)n is significantly lower than forPg-Hurst log(R/s)n for small n, the difference is not significant for longer n. The U-shaped coefficient ofvariation of Pg-Hurst log(R/s)n suggests confidence intervals for Hurst exponent estimates derived using (asmaller number of) overlapping subseries starts will be expected to converge and subsequently diverge as themaximum subseries length increases given a fixed length of N.

Differences in the standard deviation and CV of P- and Pg-Hurst log(R/s)n suggests variation in log(R/s)nand therefore H may attributed to the number of subseries starts v. In so far as the number of overlappingsubseries starts is constant for each n varying the required number of starts v for a fixed series length N willprovide information about differences in mean estimates G- and Pg-Hurst log(R/s)n.

The coefficient of variation of G-Hurst log(R/s)n for all 56pnp10,875 and all 7pvp45 is shown inFig. 1 and the mean and standard deviation of G-Hurst log(R/s)n for select n and v in Table 2. Consistentwith above-described findings for P-Hurst vs. Pg-Hurst log(R/s)n the standard deviation and CV oflog(R/s)n decreases significantly for increasing v given a range of small n. The gains from employingmore subseries starts with respect to lower standard deviation and CV however quickly diminish forlarger subseries lengths and are not significantly different to the standard deviation and CV for thedefault number of G-Hurst subseries starts v ¼ 15. The above-described U-shaped pattern of thecoefficient of variation of G and Pg-Hurst log(R/s)n is particularly evident for small v in Fig. 1 but decreasesfor larger v.

Consistent also with earlier described findings for mean P and Pg-Hurst log(R/s)n, mean G-Hurst log(R/s)nfor a given length n are not significantly different across the range of v. These findings suggest overallthat mean Hurst exponent estimates given n41000 should be invariant to the choice of overlappingvs. contiguous subseries starts, and by association largely invariant to the number of subseries starts per eachindividual n.

Table 3 shows the standard deviation of P-Hurst log(R/s)n for a range of subseries lengths 56pnp10,080and series lengths 315pNp20,160. The number of P-Hurst starts for each n it will be recalled is given byv ¼ N/n and decreases both as n-N and as N-0. Series lengths in the table are chosen such as to maximisethe set of common factors n. Estimating log(R/s)n over different series lengths therefore provides the sameinformation for contiguous subseries as does varying the required number of starts v for overlapping subseries.Compared to results presented from Table 1 for N ¼ 20,160 the impact of shorter series lengths and thereforefewer v on the standard deviations of P-Hurst log(R/s)n is evident. As the ratio of the series length to subserieslength (i.e. the number of starts v) tends to N/n ¼ 15, the standard deviation of P-Hurst log(R/s)n tends to the

Fig. 1. Coefficient of variation of G-Hurst log(R/s)n vs. v.

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Table 2

Mean and standard deviation of G-Hurst log(R/s)n vs. v

n v ¼ 7 v ¼ 15 v ¼ 30 v ¼ 45

Mean Std. dev. Mean Std. dev. Mean Std. dev. Mean Std. dev.

56 0.9235 0.0383 0.9244 0.0255 0.9246 0.0182 0.9247 0.0150

100 1.0618 0.0382 1.0625 0.0257 1.0623 0.0181 1.0625 0.0149

282 1.3009 0.0373 1.3014 0.0253 1.3023 0.0178 1.3017 0.0161

354 1.3523 0.0373 1.3529 0.0252 1.3538 0.0180 1.3532 0.0174

400 1.3799 0.0373 1.3802 0.0252 1.3811 0.0191 1.3808 0.0181

630 1.4828 0.0373 1.4826 0.0248 1.4832 0.0222 1.4828 0.0221

710 1.5092 0.0364 1.5094 0.0254 1.5099 0.0236 1.5094 0.0233

800 1.5360 0.0362 1.5359 0.0269 1.5365 0.0250 1.5362 0.0246

1000 1.5854 0.0361 1.5856 0.0293 1.5861 0.0276 1.5858 0.0274

1322 1.6472 0.0360 1.6479 0.0321 1.6476 0.0319 1.6475 0.0314

1536 1.6811 0.0370 1.6809 0.0349 1.6807 0.0343 1.6805 0.0337

1768 1.7121 0.0389 1.7121 0.0372 1.7114 0.0370 1.7115 0.0365

2048 1.7446 0.0413 1.7442 0.0403 1.7435 0.0400 1.7437 0.0397

3274 1.8467 0.0517 1.8465 0.0523 1.8457 0.0532 1.8459 0.0528

6372 2.0154 0.0780 2.0151 0.0768 2.0148 0.0785 2.0147 0.0782

7588 2.0663 0.0876 2.0658 0.0871 2.0659 0.0864 2.0657 0.0864

9068 2.1210 0.0986 2.1204 0.0966 2.1210 0.0952 2.1206 0.0955

10875 2.1801 0.1111 2.1792 0.1071 2.1805 0.1047 2.1798 0.1055

Table 3

Standard deviation of P-Hurst log(R/s)n vs. N

n/n 20,160 10,080 5040 2520 1260 630 315

56 0.0054 0.0075 0.0104 0.0148 0.0205

63 0.0057 0.0079 0.0113 0.0153 0.0218 0.0307 0.0431

70 0.0060 0.0083 0.0115 0.0167 0.0233 0.0326

84 0.0065 0.0091 0.0127 0.0180 0.0253

90 0.0068 0.0094 0.0132 0.0186 0.0260 0.0364

105 0.0073 0.0100 0.0139 0.0200 0.0276 0.0396 0.0547

126 0.0080 0.0111 0.0159 0.0222 0.0306 0.0426

140 0.0084 0.0118 0.0164 0.0232 0.0322

180 0.0095 0.0132 0.0185 0.0259 0.0366

252 0.0111 0.0156 0.0215 0.0309 0.0429

315 0.0124 0.0174 0.0247 0.0339 0.0476 0.0671

420 0.0143 0.0201 0.0282 0.0395 0.0549

504 0.0156 0.0216 0.0310 0.0430

630 0.0174 0.0245 0.0341 0.0477 0.0667

840 0.0200 0.0282 0.0389 0.0547

1008 0.0219 0.0309 0.0432

1260 0.0244 0.0342 0.0474 0.0673

1680 0.0280 0.0396 0.0553

2520 0.0341 0.0475 0.0672

5040 0.0478 0.0669

10080 0.0673

C. Ellis / Physica A 375 (2007) 159–173166

standard deviation of Pg-Hurst log(R/s)n, and for N/no15 exceeds to the standard deviation of Pg-Hurstlog(R/s)n. A should be expected the standard deviations of P-Hurst log(R/s)n furthermore increase at a fasterrate given simultaneously longer lengths n and shorter lengths N. A clear implication of this finding for thesampling distribution of Hurst exponent estimates—to be next discussed—is that contiguous subseries lengths

ARTICLE IN PRESSC. Ellis / Physica A 375 (2007) 159–173 167

should not be employed given any combination of either short N, long n, or a small number of integer factorsof N across which log(R/s)n would be calculated.

3.3. The sampling distribution of Hurst exponent estimates

The distributions of Hurst exponent estimates using the G-, P- and Pg-Hurst techniques is provided inFig. 2(a)–(c), respectively. The mean and standard deviation of the sampling distribution for a subset ofG-Hurst exponent estimates is provided in Table 4 for each of 17 independent sets of overlapping subseriesranging in length from 56pnp100 to 56pnp10,875. The mean and standard deviation of the samplingdistribution for a subset of P-Hurst exponent estimates is also provided for each of 17 independent sets ofcontiguous subseries ranging in length from 56pnp96 to 56pnp10,080, and for each of 17 independent setsof overlapping subseries for the Pg-Hurst exponent. The range of N in both Fig. 2 and in Table 4 is192pNp20,160.

The distribution of Hurst exponent estimates in Fig. 2 is approximately Normal for all series lengths192pNp20,160. The distributions for longer series lengths are more tightly clustered around the meanvalue(s) of the estimates as the subseries lengths increase. Consistent with results reported by Wallis andMatalas [9], the third and fourth moments of the distributions in Fig. 2(a) reveal small levels of skewness andkurtosis for all G-Hurst subseries sets. Further there is no tendency for these levels to either diminish orincrease as the subseries length increases. Examining the third and fourth moments of the P- and Pg-Hurstdistributions similarly reveals negligible levels of skewness and kurtosis. The values of these moments are alsoneither consistently greater nor less than those of the G-Hurst distributions.

Consistent with previously described findings for log(R/s)n the distribution of P-Hurst exponent estimatesexhibits far longer tails than generated using either of the G- or Pg-Hurst techniques. Utilising a formal chi-squared test for goodness of fit, the null hypothesis that the distributions of P- and Pg-Hurst estimates aresimilar for equivalent subseries sets is strongly rejected at the 0.01 level for all except the longest series length,N ¼ 20,160 although the estimated w2 is noted to decline significantly for longer n and N as the standarddeviation of estimates using both techniques converge. The results suggest that when N is reasonably long (i.e.,N420,000), the choice between contiguous and overlapping subseries lengths has little perceivable influenceon the distribution of the estimated value of the Hurst exponent, yet the difference is highly significant forshorter N.

‘Relative bias’ is the difference in observed and expected values of the Hurst exponent, H � 0:5. Referring toTable 4 the relative bias of G-Hurst exponents decreases linearly as the log of the subseries length (log n)increases. The relative bias of P- and Pg-Hurst by contrast declines in a log-linear manner and reflects theimpact on measuring bias of choosing subseries lengths which are not even log-space distributed. Relative tothe Pg-Hurst technique, the relative bias of the P-Hurst technique is generally less for the same subserieslength, although the difference is not significant. As opposed to Wallis and Matalas [9], the relative variance ofthe P-Hurst estimates may also be seen to be lower than that for the G-Hurst technique. For subseries sets ofequivalent length, the implication of this result is that the choice between overlapping and contiguoussubseries is largely inconsequential when deriving the classical rescaled adjusted range in cases where the meanvalue of the exponent is the only variable of interest. The result is significant since it is contrary to Wallis andMatalas [9] and Ambrose et al. [10], both of whom inferred the use of contiguous subseries lengths would yieldmore highly biased exponent estimates. Independent of the underlying method employed these results confirmthe significant factor in the determination of the classical rescaled adjusted range for Gaussian series is therange and length of n over which the exponent is estimated.

The mean and standard deviation of G-Hurst exponent estimates vs. the number of starts, v is shown inFig. 3(a) and (b), respectively. Herein the dominance of the maximum subseries length nS, is again noted withrespect to both the mean and standard deviation of H. Mean values of H vs. v ¼ 7; 8; . . . ; 45 in Fig. 3(a) arenot statistically different for a given value n. As the maximum subseries length nS increases, the Hurstexponent uniformly tends to H ¼ 0.5 over the range of subseries starts v. Larger differences in the standarddeviation of the exponent over the range of v can be seen in Fig. 3(b) at very small nS, than at long nS. For56ono100 the standard deviation of the exponent ranges from a maximum of s ¼ 0.1573 for v ¼ 7 to aminimum of s ¼ 0.0649 for v ¼ 45. However for 56pnp10,875, the standard deviation of the exponent

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Fig. 2. Distributions of Hurst exponent estimates.

C. Ellis / Physica A 375 (2007) 159–173168

ranges only from 0.0508psp0.0395 as the number of subseries starts ranges from 7pvp45. The mean andstandard deviation of the exponent are furthermore nonlinear with respect to n. While the mean of theexponent is approximately linear with respect to v, the standard deviation of the exponent is nonlinear withrespect to v, though the degree of nonlinearity decreases significantly for nS41000.

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Table 4

Mean and standard deviation of Hurst exponent estimates

n G-Hurst P-Hurst Pg-Hurst

Mean Std. dev. Mean Std. dev. Mean Std. dev.

192 0.5424 0.0738 0.5383 0.1512 0.5469 0.1132

320 0.5365 0.0518 0.5430 0.1248 0.5421 0.0731

560 0.5344 0.0465 0.5371 0.1012 0.5369 0.0524

720 0.5334 0.0442 0.5180 0.0918 0.5344 0.0467

840 0.5316 0.0401 0.5157 0.0866 0.5338 0.0446

1008 0.5300 0.0365 0.5120 0.0810 0.5324 0.0407

1260 0.5293 0.0349 0.5086 0.0741 0.5312 0.0375

1440 0.5286 0.0336 0.5047 0.0710 0.5300 0.0351

1680 0.5270 0.0337 0.5033 0.0666 0.5294 0.0340

2016 0.5264 0.0331 0.5028 0.0626 0.5285 0.0328

2240 0.5256 0.0344 0.5019 0.0596 0.5278 0.0320

2520 0.5242 0.0337 0.5010 0.0570 0.5270 0.0314

3360 0.5235 0.0336 0.4994 0.0516 0.5254 0.0322

4032 0.5210 0.0354 0.4991 0.0481 0.5245 0.0320

6720 0.5175 0.0381 0.5006 0.0398 0.5223 0.0321

10080 0.5153 0.0401 0.5003 0.0337 0.5192 0.0323

20160 0.5141 0.0412 0.5011 0.0260 0.5150 0.0321

C. Ellis / Physica A 375 (2007) 159–173 169

3.4. Confidence intervals for Hurst exponent estimates

The tendency to normality in the distributions of Hurst exponents estimates as already described providessufficient evidence for the application of a confidence interval approach to testing the significance of theexponent estimates. A common approach to the statistical significance of the Hurst exponent (see Refs.[27–30]) is the significance measure

H � EðHÞ

½VarEðHÞ�0:5. (11)

Essentially, a form of t-test for the difference between a single observed result (the observed value of H) andits expected value E(H), the significance measure uses the central limit theorem (CLT) to replace the standarddeviation of the sampling distribution of the mean s=

ffiffiffiffiffiNp

in the denominator of Eq. (11) with its estimateVar EðHÞ ¼ 1=

ffiffiffiffiffiNp

(see Ref. [31]). The result of Eq. (11) is the equivalent z-statistic for the required level ofconfidence; that is a significance measure in excess of 1.96 would indicate with 90% confidence that theobserved Hurst exponent is statistically different from its expected value. In the absence of a bootstrap orMonte Carlo mean for the Hurst exponent, the Anis and Lloyd [32] expected rescaled range may be used toproxy the variable E(H) in Eq. (11). In so far, however, that the Anis and Lloyd expected rescaled adjustedrange is calculated by formulae and hence has no standard deviation, some measure of the standard error ofthe estimate is still required to determine the significance of the difference between the observed and expectedHurst exponent. An important question remains though as to the validity of the estimate Var EðHÞ ¼ 1=

ffiffiffiffiffiNp

in Eq. (11) and the consequent likelihood of Type II error via the incorrect classification of a short-termdependant process as a long-term dependant process when using the significance measure.

Prior analysis in this study has suggested the possibility of that confidence bounds will become divergent forlonger lengths of n and therefore N. To test this hypothesised result Monte Carlo simulation techniques areused to produce a further 1.0E+05 simulated Gaussian series each ranging in length 100pNp20,000. Usingthe G-Hurst technique of overlapping subseries for 56pnp10,875 values of (R/s)n and H are recorded and90%, 95%, and 99% upper and lower confidence bounds estimated over a range of n. Simulated confidencebounds and confidence bounds calculated from the estimate Var EðHÞ ¼ 1=

ffiffiffiffiffiNp

in Eq. (11) are shown inFig. 4(a) and (b), respectively. Simulated upper and lower confidence bounds are additionally provided inTable 5.

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Fig. 3. Mean and standard deviation of G-Hurst exponent estimates vs. v: (a) mean G-Hurst exponent vs. n and v, (b) standard deviation

of G-Hurst exponent vs. n and v.

C. Ellis / Physica A 375 (2007) 159–173170

Relative to confidence bounds via the CLT estimate Var EðHÞ ¼ 1=ffiffiffiffiffiNp

in Fig. 4(b), simulated 90%,95%, and 99% confidence bounds in Fig. 4(a) provide a very close fit for series lengths defined by therange 100pnSp1000 (log n ¼ 2.0–log n ¼ 3.0). The simulated findings are consistent with those byPeters [31] using contiguous subseries lengths in the range 10pnp50, yet contrast sharply withWeron [22] who shows that confidence bounds using the P-Hurst technique are significantly larger (furtherfrom the mean) than the CLT estimate Var EðHÞ ¼ 1=

ffiffiffiffiffiNp

given nSp1000. P-Hurst confidence boundsprovided by Weron [22] are additionally larger than the CLT estimated bounds for subseries as long asnSp65,536.

A marked departure from the CLT estimated confidence bounds in Fig. 4(b) is observed beyond nS ¼ 1000as simulated confidence bounds diverge from the expected value of the exponent E(H). At the 95% level forinstance, the upper (lower) confidence bound estimated via Eq. (11) given 56pnp1152 is 0.5841 (0.4686). Thisrepresents an error of approximately 70.0071 relative to the simulated confidence bounds in Table 5,equivalent to a 10.9% higher probability of Type II error than for the simulated confidence bounds. Given56pnp10,875 upper and lower confidence bounds estimated via Eq. (11) are 0.5329 and 0.4953, respectively,and the relative error to the simulated confidence bounds extends to approximately 70.0620, equivalent to a76.8% higher probability of Type II error. While the upper and lower confidence bounds reported herein arenot as large as those reported by Weron [22] using contiguous subseries, this findings nonetheless confirm thetendency for the Significance measure in Eq. (11) to too often reject the null hypothesis of no long-termdependence.

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Fig. 4. Confidence intervals of G-Hurst exponents: (a) Simulated 90%, 95%, and 99% confidence bounds, (b) 90%, 95%, and 99%

confidence bounds estimated via CLT estimate Var EðHÞ ¼ 1=ffiffiffiffiffiNp

.

C. Ellis / Physica A 375 (2007) 159–173 171

As may be evident from close analysis of Fig. 4(a) and (b), the deviation in simulated confidence boundsfrom confidence bounds estimated via Eq. (11) is linear with respect to log n for n41000. The significance ofthis result is that the relative error in Eq. (11) for n41000 may be precisely modelled for other lengths of n

than those included in Table 5. Simulated upper and lower confidence intervals for all n41000 may beprecisely obtained by

EðHÞ � KðSn � a1 þ a2 log nSÞ, (12)

where E(H) is the expected value of the Hurst exponent, K is the z-statistic for the required level of confidence,Sn ¼ 1

� ffiffiffiffiffinSp

, a1 and a2 are coefficients, and log nS is the log of the maximum subseries length nS. For the 90%level of confidence a1 ¼ 0.1356 and a2 ¼ 0.0467. For the 95% level of confidence a1 ¼ 0.1356–0.025 anda2 ¼ 0.0467+0.009. Finally, for the 99% level of confidence a1 ¼ 0.1356–0.025(2) and a2 ¼ 0.0467+0.009(2).For subseries lengths np1000 a1 ¼ a2 ¼ 0 and Eq. (12) reduces to produce confidence intervals equivalent tothe simulated confidence intervals in Fig. 4(a) and those obtained by Eq. (11).

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Table 5

Upper and lower confidence bounds of G-Hurst exponent estimates

56pnp 90% 95% 99%

Upper Lower Upper Lower Upper Lower

100 0.7262 0.3705 0.7603 0.3364 0.8269 0.2698

110 0.7092 0.3849 0.7402 0.3539 0.8009 0.2932

125 0.6902 0.4010 0.7179 0.3733 0.7720 0.3191

142 0.6750 0.4125 0.7001 0.3874 0.7492 0.3382

158 0.6638 0.4209 0.6870 0.3977 0.7325 0.3522

178 0.6535 0.4287 0.6750 0.4072 0.7171 0.3651

224 0.6375 0.4409 0.6563 0.4221 0.6931 0.3853

252 0.6284 0.4471 0.6457 0.4298 0.6797 0.3959

282 0.6217 0.4514 0.6380 0.4350 0.6699 0.4031

316 0.6160 0.4548 0.6315 0.4394 0.6617 0.4092

354 0.6110 0.4578 0.6256 0.4432 0.6543 0.4145

400 0.6061 0.4606 0.6201 0.4467 0.6473 0.4194

450 0.6016 0.4633 0.6149 0.4501 0.6408 0.4241

500 0.5976 0.4656 0.6102 0.4530 0.6349 0.4283

563 0.5937 0.4678 0.6058 0.4558 0.6294 0.4322

630 0.5902 0.4699 0.6017 0.4584 0.6242 0.4359

710 0.5868 0.4718 0.5978 0.4608 0.6193 0.4393

800 0.5838 0.4733 0.5944 0.4627 0.6151 0.4420

890 0.5843 0.4711 0.5952 0.4603 0.6164 0.4391

1000 0.5825 0.4716 0.5931 0.4610 0.6138 0.4403

1152 0.5808 0.4720 0.5912 0.4616 0.6115 0.4412

1322 0.5822 0.4690 0.5930 0.4582 0.6142 0.4370

1536 0.5808 0.4690 0.5915 0.4583 0.6124 0.4374

1768 0.5797 0.4687 0.5903 0.4581 0.6111 0.4373

2048 0.5787 0.4682 0.5893 0.4577 0.6100 0.4370

2390 0.5779 0.4675 0.5885 0.4569 0.6092 0.4362

2793 0.5773 0.4665 0.5879 0.4559 0.6087 0.4352

3274 0.5792 0.4629 0.5903 0.4517 0.6121 0.4300

3850 0.5790 0.4614 0.5903 0.4501 0.6123 0.4281

4539 0.5791 0.4595 0.5906 0.4481 0.6130 0.4257

5369 0.5796 0.4572 0.5914 0.4455 0.6143 0.4226

6372 0.5802 0.4547 0.5922 0.4427 0.6157 0.4193

7588 0.5807 0.4521 0.5930 0.4398 0.6171 0.4157

9068 0.5813 0.4493 0.5940 0.4366 0.6187 0.4119

10875 0.5819 0.4464 0.5949 0.4334 0.6202 0.4080

C. Ellis / Physica A 375 (2007) 159–173172

4. Concluding remarks

Rescaled range (R/S) analysis has in recent times regained in popularity as a robust tool for the examinationof long-term dependence in time-series data. Subsequent to Mandelbrot and Wallis [8] who developed theoriginal Hurst [7] methodology to incorporate OLS regression techniques, the use overlapping vs. contiguoussubseries in the determination of the Hurst exponent has been a source of some debate amongst R/S theorists.

Following the conjecture of Wallis and Matalas [9] that contiguous subseries provide more biased estimatesof the Hurst exponent, historic empirical research has largely favoured the use of overlapping subseries. Sincethe nature of this alleged bias has not been previously described nor proven, a major contribution of this studyhas been to examine the distributional characteristics of Hurst exponent estimates derived using overlappingand contiguous subseries. Given relatively long subseries and series lengths, results attributable to overlappingvs. contiguous subseries were shown to be comparable when the sole variable of concern was the mean valueof the exponent. Given a range of shorter series lengths however, overlapping subseries are clearly preferred asthe use of contiguous subseries when the series length is short necessitates both a smaller number of different

ARTICLE IN PRESSC. Ellis / Physica A 375 (2007) 159–173 173

subseries, S and shorter length subseries; either of which will result in more highly variable Hurst exponentestimates.

While these results provide some support for the conjecture of Wallis and Matalas [9]—previously upheldby Ambrose et al. [10]—that contiguous subseries indeed yield more highly biased estimates of the Hurstexponent, the increasing availability of very long time-series data sets largely negates their argument.Irrespective however of whether overlapping or contiguous subseries are employed, this study concurs withWeron [22] that confidence bounds derived from the CLT can too often result in the incorrect classification ofa short-term-dependant process as a long-term dependant process.

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