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Method To Improve Hurst Exponent Estimation 0 comments
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 Methodbased Analysis (Trinidad Segovia, FernándezMartínez, SánchezGranero, 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 Methodbased Analysis.
Objectives and Literature Review
The objective of this paper is to use Rescaled Range Analysis and Geometric Methodbased 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 Methodbased Analysis (Trinidad Segovia, FernándezMartínez, SánchezGranero, 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
SD_{y} = standard deviation for one year
SD_{d} = 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 Methodbased Analysis. The results are tabulated below:
Index Name
Rescaled Range Analysis
Geometric Methodbased 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 Methodbased 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 Methodbased 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 Methodbased 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 Methodbased 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
SD_{y} = standard deviation for total period
SD_{d} = 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(SD_{d}).
log(SD_{d}) 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 Methodbased 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 Methodbased 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 Rsquare 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
7.68E06
Residual
4
0.007009
0.001752
Total
5
1.549872
Coefficients
Standard Error
t Stat
Pvalue
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
7.68E06
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
3.42E05
Residual
4
0.015683
0.003921
Total
5
1.645097
Coefficients
Standard Error
t Stat
Pvalue
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
3.42E05
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
1.69E06
Residual
4
0.002941
0.000735
Total
5
1.38675
Coefficients
Standard Error
t Stat
Pvalue
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
1.69E06
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 Methodbased 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
Pvalue
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 Methodbased 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
Pvalue
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 Methodbased 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
Pvalue
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
1.25E06
Residual
5
0.045967
0.009193
Total
6
6.860696
Coefficients
Standard Error
t Stat
Pvalue
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
1.25E06
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
4.77E06
Residual
5
0.073126
0.014625
Total
6
6.395047
Coefficients
Standard Error
t Stat
Pvalue
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
4.77E06
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
4.11E07
Residual
5
0.026187
0.005237
Total
6
6.099436
Coefficients
Standard Error
t Stat
Pvalue
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
4.11E07
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 Methodbased 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
Pvalue
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 Methodbased 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
Pvalue
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 Methodbased 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
Pvalue
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: 770799.
Lo, A. (1991). "LongTerm Memory in Stock Market Prices". Econometrica 59: 12791313.
Mandelbrot, Benoit, and Hudson, Richard, (2004). "The (Mis)behavior of the Markets", Basic Books
Trinidad Segovia, J. E.; FernándezMartínez, M.; SánchezGranero, M. A., (2012). "A note on geometric methodbased procedures to calculate the Hurst exponent", Physica A, Volume 391, Issue 6, p. 22092214.
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/rescaledrangeanalysisamethodfordetectingpersistencerandomnessormeanreversioninfinancialmarkets/
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