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YUEN is a finance professional with over 20 years practical experience in financial accounting, financial management and investment. He is a freelance trainer and grader. He also holds a PhD(Management) with research interest in market crash and Fractal Theory. He served Qualification Program of... More
  • Method To Improve Hurst Exponent Estimation 0 comments
    Sep 24, 2013 6:51 AM

    Method to Improve Hurst Exponent Estimation

    YUEN Wai Pong Raymond

    Affiliation: Universidad Empresarial de Costa Rica, International Program

    Method to Improve Hurst Exponent Estimation

    Abstract

    This is a paper on the application of the methods to estimate the Hurst Exponent of Hang Seng Index, Hang Seng China Enterprise Index and Shanghai Composite Index. The methods employed are the Rescaled Range Analysis (Hurst, 1951) and the Geometric Method-based Analysis (Trinidad Segovia, Fernández-Martínez, Sánchez-Granero, 2012).

    The question aroused from this research is that there are some cases that the Hurst exponent estimated is out of the theoretical range or with very high value. From a theoretical ground and the practical application of Hang Seng Index, Hang Seng China Enterprise Index and Shanghai Composite Index, a method is espoused to improve the estimation of the Hurst exponent for Rescaled Range Analysis and Geometric Method-based Analysis.

    Objectives and Literature Review

    The objective of this paper is to use Rescaled Range Analysis and Geometric Method-based Analysis to estimate the Hurst Exponent for Hang Seng Index, Hang Seng China Enterprise Index and Shanghai Composite Index.

    Furthermore, a new method is proposed to improve the existing methods to estimate the Hurst Exponent.

    Literature review includes papers on the meanings of the Hurst Exponent (Hurst, 1951), methodology of Rescaled Range Analysis Method (Voss, 2013), methodology of Geometric Method-based Analysis (Trinidad Segovia, Fernández-Martínez, Sánchez-Granero, 2012), the findings of the Hurst Exponent of US stock market (Lo, 1991), and proposition to use the Hurst Exponent (Mandelbrot, 2004).

    Body of Paper

    The Hurst exponent is employed to gauge the "long term memory" of a series. This could be regarded a method to estimate the autocorrelation of the time series. The Hurst exponent is named to pay attribute to the Englishman Harold Edwin Hurst who invented the Rescaled Range Method to measure the long term autocorrelation of the time series. The time series studied by Harold Edwin Hurst is the flooding situation of the Nile River of Egypt. Using this method, Harold Edwin Hurst found that the Hurst exponent of Nile River flooding situation was 0.77.

    If the Hurst exponent above 0.5, this means that there is a high autocorrelation of the time series. If the Hurst exponent below 0.5, this means that there is a low autocorrelation of the time series. If the Hurst exponent is 0.5, this means that the time series data are independent.

    The Hurst exponent has important application for finance in the risk assessment. In the usual textbook, the basic assumption is that the price is independent or with the Hurst exponent 0.5. Applying it to the estimation of variance of time series of one year as a unit, the following formula is frequently used:

    Formula (a)

    Where

    SDy = standard deviation for one year

    SDd = standard deviation for one day

    250 is the trading days of a year

    The multiplier takes the square root of the number of days (250) because we assume the Hurst exponent is equal to 0.5, i.e. the price movement time series is independent.

    As a result, the application of the Hurst exponent is very important to risk analysis. According to Andrew Lo's paper in 1991, the Hurst exponent for the financial market is higher than 0.5. In other words, the Hurst exponent could be different from 0.5, or the time series of the prices of the financial instrument is not independent. This has huge implication on the modern financial theory because it is assumed that the financial instrument price is independent of each other. If this is not the case, the risk of the financial market may be underestimated or overestimated. However, according to Lo's research, it is underestimated.

    Because the Hurst exponent has such an important implication of investment analysis, this research is done for the estimation of the Hurst exponent for Hang Seng Index, Hang Seng China Enterprise Index and Shanghai Composite Index. The data used for Hang Seng Index are from 31 Dec 1986 to 16 Nov 2012 (6400 data points), Hang Seng China Composite Index are from 15 Jul 1993 to 20 Feb 2013 (4800 data points), and Shanghai Composite Index are from 19 Dec 1990 to 10 Jul 2009 (4800 data points).

    Two methods are employed including the Rescaled Range Analysis and the Geometric Method-based Analysis. The results are tabulated below:

    Index Name

    Rescaled Range Analysis

    Geometric Method-based Analysis

    Hang Seng Index

    0.986 (Annex 1)

    1.554 (Annex 4)

    Hang Seng China Enterprise Index

    1.013 (Annex 2)

    1.636 (Annex 5)

    Shanghai Composite Index

    0.934 (Annex 3)

    1.891 (Annex 6)

    The results for both Rescaled Range Analysis and Geometric Method-based Analysis are left much to be desired. This is because the Hurst exponent which should be in the range of 0 to 1, the results are with 4/6 with estimates higher than 1. Even the remaining two estimates are with very high figures: 0.986 and 0.934.

    The detailed datasets are tabulated below:

    Hang Seng Index by Rescaled Range Analysis

    Count

    R/S

    LogCount

    LogR/S

    6400

    2409.575

    3.80618

    3.38194

    3200

    1353.717

    3.50515

    3.131528

    1600

    532.5561

    3.20412

    2.726365

    800

    299.2628

    2.90309

    2.476053

    400

    166.9706

    2.60206

    2.22264

    200

    79.1083

    2.30103

    1.898222

           

    Where:

    - Count is the number of days grouped

    - R/S is the Range Rescaled Figures

    - LogCount is the log of the number of days grouped

    - LogR/S is the log of the Range Rescaled Figures

    Hang Seng China Enterprise Index by Rescaled Range Analysis

    Count

    R/S

    LogCount

    LogR/S

    4800

    2023.1

    3.681241

    3.306017

    2400

    998.6664

    3.380211

    2.99942

    1200

    357.8085

    3.079181

    2.553651

    600

    216.9173

    2.778151

    2.336294

    300

    111.4978

    2.477121

    2.047266

    150

    60.92748

    2.176091

    1.784813

    Where:

    - Count is the number of days grouped

    - R/S is the Range Rescaled Figures

    - LogCount is the log of the number of days grouped

    - LogR/S is the log of the Range Rescaled Figures

    Shanghai Composite Index by Rescaled Range Analysis

    Count

    R/S

    LogCount

    LogR/S

    4800

    1472.824

    3.681241

    3.168151

    2400

    841.2938

    3.380211

    2.924948

    1200

    472.648

    3.079181

    2.674538

    600

    231.7119

    2.778151

    2.364948

    300

    113.989

    2.477121

    2.056863

    150

    60.60423

    2.176091

    1.782503

    Where:

    - Count is the number of days grouped

    - R/S is the Range Rescaled Figures

    - LogCount is the log of the number of days grouped

    - LogR/S is the log of the Range Rescaled Figures

    Hang Seng Index by Geometric Method-based Analysis*

    Count

    GM Range

    LogCount

    6400

    2.81519982

    3.80618

    3200

    1.8845504

    3.50515

    1600

    1.02132927

    3.20412

    800

    0.80916156

    2.90309

    400

    0.55922424

    2.60206

    200

    0.37918364

    2.30103

    Where:

    - Count is the number of days grouped

    - GM Range the range in geometric form

    - LogCount is the log of the number of days grouped

    * Log R/S is not required in the computation because the range itself is in geometric form.

    Hang Seng China Enterprise Index by Geometric Method-based Analysis*

    Count

    GM Range

    LogCount

    4800

    3.006895787

    3.681241

    2400

    2.26237112

    3.380211

    1200

    1.629256477

    3.079181

    600

    1.008195532

    2.778151

    300

    0.885265743

    2.477121

    150

    0.510828359

    2.176091

    Where:

    - Count is the number of days grouped

    - GM Range the range in geometric form

    - LogCount is the log of the number of days grouped

    * Log R/S is not required in the computation because the range itself is in geometric form.

    Shanghai Composite Index by Geometric Method-based Analysis*

    Count

    GM Range

    LogCount

    4800

    4.109771

    3.681241

    2400

    2.325881

    3.380211

    1200

    1.507624

    3.079181

    600

    1.426939

    2.778151

    300

    1.086673

    2.477121

    150

    0.884928

    2.176091

    Where:

    - Count is the number of days grouped

    - GM Range the range in geometric form

    - LogCount is the log of the number of days grouped

    * Log R/S is not required in the computation because the range itself is in geometric form.

    As a result, a new method to get round the limitation of conventional methods to estimate the Hurst exponent is proposed.

    Formula (b)

    Where

    SDy = standard deviation for total period

    SDd = standard deviation for one unit of the period

    T is the number of units of the total period

    h is the Hurst exponent

    If the Formula (b) is taken log, the resultant formula would become, Formula (c):

    Formula (c)

    There is a constant in the formula: log(SDd).

    log(SDd) is the log of the standard deviation of one unit of the period.

    As a result, to render the slope of the resultant regression more aligned with the Formula (c), we add a term for the regression which is the Count = 1, LogCount = 0, R/S = 1.4142, and LogR/S = 0.1505 (this is a constant proposed to name it the Hurst RR Constant) for the Rescaled Range Analysis and the Count = 1, the LogCount = 0, and GMRange = standard deviation of the log price in one day.

    The results for the new method are tabulated below:

    Index Name

    Rescaled Range Analysis

    Geometric Method-based Analysis

    Hang Seng Index

    0.843 (Annex 7)

    0.597 (Annex 10)

    Hang Seng China Enterprise Index

    0.841 (Annex 8)

    0.726 (Annex 11)

    Shanghai Composite Index

    0.824 (Annex 9)

    0.866 (Annex 12)

    The results are all within the theoretical boundaries of 0 to 1 in the new method.

    Conclusion

    With the employment of forced intercept in both cases of Rescaled Range Analysis and Geometric Method-based Analysis, the results of the research are much more reasonable and applicable. This new method not only raises the quality of the Hurst exponent estimates but also reduces the tedious work of long clustering process required.

    The usage of the Hurst RR Constant in the Rescaled Range Analysis case simplifies the process.

    However, future research could be on the improvement to reduce the drop in R-square in the new method as compared with that of the traditional method.

    Annex 1

    Hang Seng Index The Hurst Exponent Estimation Using Rescaled Range Analysis

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.997736

                 

    R Square

    0.995478

                 

    Adjusted R Square

    0.994347

                 

    Standard Error

    0.04186

                 

    Observations

    6

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    1.542863

    1.542863

    880.5184

    7.68E-06

         

    Residual

    4

    0.007009

    0.001752

             

    Total

    5

    1.549872

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -0.37239

    0.102931

    -3.61781

    0.022401

    -0.65817

    -0.0866

    -0.65817

    -0.0866

    LogCount

    0.986359

    0.03324

    29.67353

    7.68E-06

    0.894069

    1.078649

    0.894069

    1.078649

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 2

    Hang Seng China Enterprise Index The Hurst Exponent Estimation Using Rescaled Range Analysis

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.995222

                 

    R Square

    0.990467

                 

    Adjusted R Square

    0.988083

                 

    Standard Error

    0.062616

                 

    Observations

    6

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    1.629414

    1.629414

    415.5853

    3.42E-05

         

    Residual

    4

    0.015683

    0.003921

             

    Total

    5

    1.645097

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -0.46406

    0.147849

    -3.13874

    0.03489

    -0.87455

    -0.05356

    -0.87455

    -0.05356

    LogCount

    1.013647

    0.049723

    20.38591

    3.42E-05

    0.875594

    1.1517

    0.875594

    1.1517

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 3

    Shanghai Composite Index The Hurst Exponent Estimation Using Rescaled Range Analysis

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.998939

                 

    R Square

    0.997879

                 

    Adjusted R Square

    0.997349

                 

    Standard Error

    0.027117

                 

    Observations

    6

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    1.383808

    1.383808

    1881.935

    1.69E-06

         

    Residual

    4

    0.002941

    0.000735

             

    Total

    5

    1.38675

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -0.24044

    0.064028

    -3.75528

    0.019856

    -0.41821

    -0.06267

    -0.41821

    -0.06267

    LogCount

    0.934134

    0.021533

    43.38127

    1.69E-06

    0.874348

    0.99392

    0.874348

    0.99392

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 4

    Hang Seng Index The Hurst Exponent Estimation Using Geometric Method-based Analysis

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.939691

                 

    R Square

    0.883019

                 

    Adjusted R Square

    0.853774

                 

    Standard Error

    0.356037

                 

    Observations

    6

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    3.827412

    3.827412

    30.19358

    0.005346

         

    Residual

    4

    0.50705

    0.126762

             

    Total

    5

    4.334462

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -3.49914

    0.875485

    -3.9968

    0.016173

    -5.92987

    -1.0684

    -5.92987

    -1.0684

    LogCount

    1.553545

    0.282726

    5.494869

    0.005346

    0.76857

    2.338519

    0.76857

    2.338519

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 5

    Hang Seng China Enterprise Index The Hurst Exponent Estimation Using Geometric Method-based Analysis

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.976218

                 

    R Square

    0.953001

                 

    Adjusted R Square

    0.941251

                 

    Standard Error

    0.228704

                 

    Observations

    6

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    4.242378

    4.242378

    81.10771

    0.000842

         

    Residual

    4

    0.209222

    0.052305

             

    Total

    5

    4.4516

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -3.23964

    0.540014

    -5.99918

    0.003884

    -4.73896

    -1.74032

    -4.73896

    -1.74032

    LogCount

    1.635595

    0.181612

    9.005982

    0.000842

    1.131359

    2.139831

    1.131359

    2.139831

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 6

    Shanghai Composite Index The Hurst Exponent Estimation Using Geometric Method-based Analysis

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.891546

                 

    R Square

    0.794853

                 

    Adjusted R Square

    0.743567

                 

    Standard Error

    0.604859

                 

    Observations

    6

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    5.6701

    5.6701

    15.49825

    0.017006

         

    Residual

    4

    1.463417

    0.365854

             

    Total

    5

    7.133517

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -3.64749

    1.428188

    -2.55393

    0.063042

    -7.61277

    0.3178

    -7.61277

    0.3178

    LogCount

    1.890891

    0.480314

    3.936782

    0.017006

    0.557326

    3.224457

    0.557326

    3.224457

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 7

    Hang Seng Index The Hurst Exponent Estimation Using Rescaled Range Analysis (With Forced Intercept)

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.996644

                 

    R Square

    0.9933

                 

    Adjusted R Square

    0.99196

                 

    Standard Error

    0.095882

                 

    Observations

    7

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    6.814729

    6.814729

    741.2631

    1.25E-06

         

    Residual

    5

    0.045967

    0.009193

             

    Total

    6

    6.860696

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    0.07616

    0.088819

    0.857477

    0.430355

    -0.15216

    0.304475

    -0.15216

    0.304475

    LogCount

    0.843492

    0.030981

    27.22615

    1.25E-06

    0.763853

    0.923131

    0.763853

    0.923131

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 8

    Hang Seng China Enterprise Index The Hurst Exponent Estimation Using Rescaled Range Analysis (With Forced Intercept)

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.994266

                 

    R Square

    0.988565

                 

    Adjusted R Square

    0.986278

                 

    Standard Error

    0.120935

                 

    Observations

    7

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    6.321921

    6.321921

    432.263

    4.77E-06

         

    Residual

    5

    0.073126

    0.014625

             

    Total

    6

    6.395047

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    0.057047

    0.111359

    0.51228

    0.630273

    -0.22921

    0.343305

    -0.22921

    0.343305

    LogCount

    0.841034

    0.040452

    20.79093

    4.77E-06

    0.737049

    0.945019

    0.737049

    0.945019

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 9

    Shanghai Composite Index The Hurst Exponent Estimation Using Rescaled Range Analysis (With Forced Intercept)

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.997851

                 

    R Square

    0.995707

                 

    Adjusted R Square

    0.994848

                 

    Standard Error

    0.07237

                 

    Observations

    7

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    6.073249

    6.073249

    1159.587

    4.11E-07

         

    Residual

    5

    0.026187

    0.005237

             

    Total

    6

    6.099436

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    0.091056

    0.06664

    1.366388

    0.230059

    -0.08025

    0.26236

    -0.08025

    0.26236

    LogCount

    0.824327

    0.024207

    34.05271

    4.11E-07

    0.7621

    0.886554

    0.7621

    0.886554

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 10

    Hang Seng Index The Hurst Exponent Estimation Using Geometric Method-based Analysis (With Forced Intercept)

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.776918

                 

    R Square

    0.603602

                 

    Adjusted R Square

    0.524322

                 

    Standard Error

    0.67002

                 

    Observations

    7

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    3.41794

    3.41794

    7.613577

    0.039869

         

    Residual

    5

    2.244635

    0.448927

             

    Total

    6

    5.662574

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -0.49658

    0.620659

    -0.80008

    0.459973

    -2.09203

    1.098881

    -2.09203

    1.098881

    LogCount

    0.597364

    0.216493

    2.759271

    0.039869

    0.04085

    1.153877

    0.04085

    1.153877

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 11

    Hang Seng China Enterprise Index The Hurst Exponent Estimation Using Geometric Method-based Analysis (With Forced Intercept)

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.850154

                 

    R Square

    0.722762

                 

    Adjusted R Square

    0.667315

                 

    Standard Error

    0.600901

                 

    Observations

    7

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    4.706724

    4.706724

    13.03507

    0.015374

         

    Residual

    5

    1.805408

    0.361082

             

    Total

    6

    6.512131

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -0.4927

    0.553323

    -0.89045

    0.414015

    -1.91507

    0.929658

    -1.91507

    0.929658

    LogCount

    0.725685

    0.200998

    3.610411

    0.015374

    0.209004

    1.242367

    0.209004

    1.242367

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Annex 12

    Shanghai Composite Index The Hurst Exponent Estimation Using Geometric Method-based Analysis (With Forced Intercept)

     

     

    SUMMARY OUTPUT

                 
                     

    Regression Statistics

                 

    Multiple R

    0.811193

                 

    R Square

    0.658034

                 

    Adjusted R Square

    0.58964

                 

    Standard Error

    0.83508

                 

    Observations

    7

                 
                     

    ANOVA

                   
     

    df

    SS

    MS

    F

    Significance F

         

    Regression

    1

    6.709507

    6.709507

    9.62132

    0.026798

         

    Residual

    5

    3.486791

    0.697358

             

    Total

    6

    10.1963

               
                     
     

    Coefficients

    Standard Error

    t Stat

    P-value

    Lower 95%

    Upper 95%

    Lower 95.0%

    Upper 95.0%

    Intercept

    -0.55473

    0.76896

    -0.7214

    0.502974

    -2.53141

    1.421944

    -2.53141

    1.421944

    LogCount

    0.866432

    0.27933

    3.101825

    0.026798

    0.148392

    1.584471

    0.148392

    1.584471

    (click to enlarge)

    (click to enlarge)

    (click to enlarge)

    Reference

    Hurst, H. E. (1951). "Long term storage capacity of reservoirs". Trans. Am. Soc. Eng. 116: 770-799.

    Lo, A. (1991). "Long-Term Memory in Stock Market Prices". Econometrica 59: 1279-1313.

    Mandelbrot, Benoit, and Hudson, Richard, (2004). "The (Mis)behavior of the Markets", Basic Books

    Trinidad Segovia, J. E.; Fernández-Martínez, M.; Sánchez-Granero, M. A., (2012). "A note on geometric method-based procedures to calculate the Hurst exponent", Physica A, Volume 391, Issue 6, p. 2209-2214.

    Voss, Jason, 2013. "Rescaled Range Analysis: A Method for Detecting Persistence, Randomness, or Mean Reversion in Financial Markets", CFA Blog,

    URL: http://blogs.cfainstitute.org/investor/2013/01/30/rescaled-range-analysis-a-method-for-detecting-persistence-randomness-or-mean-reversion-in-financial-markets/

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