15 Sec 066 Danilenko Hurst

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The XIII International Conference

Applied Stochastic Models and Data Analysis(ASMDA-2009)June 30-July 3, 2009, Vilnius, LITHUANIA

ISBN 978-9955-28-463-5

L. Sakalauskas, C. Skiadas and

E. K. Zavadskas (Eds.): ASMDA-2009

Selected papers. Vilnius, 2009, pp. 329333 Institute of Mathematics and Informatics, 2009 Vilnius Gediminas Technical University, 2009

329

HURST ANALYSIS OF BALTIC SECTOR INDICES

Svetlana Danilenko

Vilnius Gediminas Technical University, Saultekio al. 11, 10223 Vilnius, LithuaniaE-mail: [email protected]

Abstract: The Hurst exponent is a statistical measure used to classify time series. Using the Hurst pa-rameter processes are classified into long-term dependence, antipersistence and white noise processes.R/S analysis method is one of the many methods that evaluate the Hurst exponent. This method uses the

rescaled range statistic (R/S statistic). The literature that focuses on particular sector indices is relativelylimited, therefore valuating whether long-term dependence exists in specific sectors of the economy isparticularly useful. Investigation object the Baltic sector indices. The latter represent tendencies of dif-

ferent sector activity indices in the stock market. The work concentrates on calculating the Hurst expo-

nent, evaluated Hurst exponents of the Baltic sector indices are given for different periods of time.

Keywords:R/S analysis, Hurst exponent, financial markets, share index.

1. Introduction

Long memory, also known as the long-term dependence property, describes the high-order correlationstructure of a series. If a time series possesses long memory, there is a persistent temporal dependence

between observations even considerably separated in time.Long memory in exchange rates would allow investors to anticipate price movements and earn posi-

tive average returns. The presence of long-term memory in stock returns has important implications formany of the paradigms in financial economics. For example, optimal consumption/savings and portfolio

decisions might become extremely sensitive to the investment period if the returns were long-term de-pendent.The evaluation of long memory of a series can be done by diverse methodologies. Among the meth-

odologies used today in the identification and quantification of long memory, the most popular ones arethe R/S classic analysis (by Hurst, 1951, Mandelbrot, 1972), the modified R/S analysis (by Lo, 1991), theestimation of the parameter of fractionary integration by spectral regression, or log-periodogram (by Ge-weke and Porter-Hudak, 1983), the log-periodgram semi-parametric estimator (by Robinson, 1995) andthe V/S analysis (by Giraitis et al., 2003). These methodologies can be used without knowing the factorswhich act in the price-generating process, taking into account only the series of returns or volatilities forthose where long memory estimations are desired.

R/S method has been developed by a hydrologist Hurst in his work (Hurst, 1951) in which he inves-tigates the level of the river Nile. Stochastic processes with long-term dependence and methods of their

identification have been introduced first in the context of hydrology and geophysics (by Mandelbrot andWallis, 1969, Mandelbrot and Ness, 1968, Mandelbrot, 1982). The Hurst exponent was originally sug-gested as a result of a study of the flow of water through dams, and stems from the observation that parti-

cles suspended in fluid move erratically, in a way commonly known as Brownian motion.Hurst proposed a method for the quantification of long-term memory which is based on estimating a

parameter for the scaling behaviour of the range of partial sums of the variable under consideration.Long ago the technique has been popularized in economics by B. B. Mandelbrot. Hurst parameter has

been used in many works. Due to its appearance in E. E. Peters works (Peters, 1991, Peters, 1994). Ineconomics and finance, long-term dependence has a long history and has remained a topic of active re-search is the study of economic and financial time series (Lo, 1991, Cutland, Kopp and Willinger, 1995,Corazza and Malliaris, 2002, Grech and Mazur, 2004, etc.).

Hurst method discloses the following properties of the statistical data: clusterisation, persistence,

short range dependence, anti-persistence, presence of periodical or non-periodical cycles, etc. (details aregiven in , 1998). It is possible to distinguish long-term non-function cycle time series and randomseries through this method.


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R/S analysis method is one of the most commonly used methods for the evaluation of the Hurst ex-ponent.

2. Measures of long-term dependence

In this paper, our measure of long-term dependence is the Hurst's exponent provided by the R/S analysis.

The application of R/S analysis to financial markets will be discussed.Let ( )

0=

nnSS be the share index for the Baltic states, then the logarithm of the share index

1n,ln1

=

n

n

n

S

Sh is the proportional change in the share index or the logarithmic change.

We will describe the essence of R/S analysis application for the investigation of ( )1

=nn

hh sequence

properties.

Let us comprise 1,...1

++= nhhHnn

.

Let us define

=

nk

nknk

nkn

Hn

kHH

n

kHR minmax . (1)

Letn

Hhn

n where ( )nhhh ,...,, 21 be an empirical mean, then ( )=

=

k

i

ninkhhH

n

kH

1

isk

H devia-

tion from its empirical meann

Hn

k. The measure

nR characterises the degree of dispersion of

nkHn

kH

nk , deviations.

Let the empirical dispersion

( )===

=

=

n

k

nk

n

k

k

n

k

knhh

nh

nh

nS

1

2

2

11

22 111 (2)

and the adjusted range of the cumulative sums nkHk

,

n

n

n

S

RQ . (3)

Based on the SR / we are testing the null hypothesis (0

H ) that the share index is a random walk. If

the null hypothesis is correct then if the value of n is large the values ofnn

SR should be proportional

to the square root of n 5,0

~ cnSRnn

, (4)

However, in financial data analysis it usually isH

nncnSR ~ , (5)

whereH

- Hurst parameter that is significantly different from5.0

.If we take the logarithm of Equation (5), we obtain

( ) ( ) ( )nHcSRnn

logloglog += (6)

and hence, in practice, the Hurst exponent can be calculated by plotting ( )nn

SRlog against ( )nlog and

estimating the slope over a judiciously chosen linear region by OLS.The Hurst Exponents of time series is between 0 and 1. The time series can show different features:

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3. Overview of the Baltic sector indices

10 OMX Baltic sector indices are included on the Stock Exchange: Energy, Materials, Industrials, Con-sumer Discretionary, Consumer Staples, Health Care, Financials, Information Technology, Telecommu-nication Services, Utilities. Investigational period of stock indices is from the year 2000 till the year 2008.

The growth of economic Baltic Sector indices beginning with the 4th quarter of 2003 has mainlybeen fuelled by the positive investor expectations relating to the coming European Union membership, itsstructural support, the geographical location of the countries as well as the relative novelty of the marketand the establishment of new investment funds. Once the Baltic States have joined the European Union,the indices have started to rise more rapidly while the economic situation of the countries enhanced.

Based on the situation discussed, the observed time frame has been divided into several intervals. TheHurst exponents have been estimated for the whole observation period as well as the separate intervals.

4. Empirical results

The R/S analysis method is one of the methods of the Hurst exponent evaluation. However there aremany methods of evaluating the Hurst exponent with the most commonly used being the following

(Beran, 1994): Ratio variance of residuals, Periodogram method, Whittle method, Abry-Veitch method.Only the methods of R/S analysis and Ratio variance of residuals are good enough to estimate the

Hurst exponent (Belov, Kabainskas, Sakalauskas, 2006).The values of share indices Hurst exponents using the R/S analysis method are presented in the Ta-

ble 1, using Ratio variance of residuals in the Table 2. The correlation coefficient (CC) for the Hurst ex-ponent illustrates the adequacy of estimation.

Table 1. The results of Hurst analysis using the R/S analysis method

H CC, % H CC, % H CC, %

Energy 0,650 99,95 0,650 99,84 0,666 99,79

Materials 0,665 99,81 0,663 99,70 0,643 99,86

Industrials 0,713 99,78 0,708 99,66 0,736 99,60Consumer Discretionary 0,658 99,91 0,650 99,74 0,690 99,84

Consumer Staples 0,616 99,91 0,598 99,92 0,674 99,85

Health Care 0,557 99,61 0,553 99,82 0,601 99,92

Financials 0,615 99,88 0,607 99,54 0,684 99,88

Indformation Technology 0,653 99,96 0,635 99,97 0,684 99,72

Telecommunication Services 0,651 99,92 0,655 99,87 0,640 99,95

Utilities 0,584 99,79 0,607 99,90 0,613 99,92

Number of observations 2317 1125 1192

Table 2. The results of Hurst analysis using Ratio variance of residuals

2000.01.01-2008.12.31 2000.01.01-2004.04.30 2004.05.01-2008.12.31OMX Baltic sector indices

H CC, % H CC, % H CC, %

Energy 0,632 99,87 0,649 99,16 0,973 99,66

Materials 0,695 99,39 0,753 99,39 0,607 99,55

Industrials 0,751 99,63 0,754 99,17 0,792 99,68

Consumer Discretionary 0,590 99,51 0,608 99,85 0,772 99,17

Consumer Staples 0,597 99,95 0,536 99,71 0,636 99,31

Health Care 0,600 99,70 0,481 99,73 0,675 99,49

Financials 0,577 99,48 0,561 98,50 0,706 99,87

Indformation Technology 0,706 99,65 0,651 99,65 0,691 99,63

Telecommunication Services 0,663 99,85 0,636 99,53 0,641 99,12Utilities 0,512 99,15 0,800 99,88 0,644 99,94

Number of observations 2317 1125 1192


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Fig. 1. Hurst Exponent estimates by the methods of R/S analysis and Variance of residuals for all series

The Hurst exponents are obtained using SELFIS a programme that is freely distributed for common

use (The SELFIS Tool).The obtained values of the Hurst parameters indicate that the Hurst parameter for all sector indices

time series is within the following interval 15.0 < H which shows that there is a week correlation in

them.In the first figure we can see the relationship between the two Hurst Exponent estimates, calculated

using two different methods.In order to validate the stability of the value ofH, the original time sequence has been disarranged

and the value ofH recalculated. This time value H is supposed to be distinctively close to 0.5. After com-puting, the result proves that the value ofH is significant.

5. Conclusion

This work contains Hurst exponents calculated for 10 of the Baltic sector indices for the whole period inquestion as well as separate sections. The estimation has been performed using two methods: R/S analysis

and the Ratio variance of residuals. The Hurst exponent of financial data is usually bigger than 5.0 . This

is supported by the share indices Hurst exponents findings for the Baltic sector indices.The existence of long-term dependence indicates that the Baltic sector indices exhibit dynamics thatare not consistent with the random walk behaviour. Participants in the Baltic equity markets should con-sider the long-term movements when determining the dynamics of their investment assets. The results

have an important bearing on the pricing of equity derivatives in the Baltic markets. It may be beneficialif pricing models included an assumption about the long-term dependence.

It is widely accepted that stable models implying long-term dependence are quite popular for use inthe modern financial engineering (Samorodnitsky and Taqqu, 2000, Kabainskas, Rachev, Sakalauskas,Sun, Belovas, 2009). Further research could be performed on the composition of such models for Baltic

sector indices as well as their analysis.

References

Belov, I., Kabainskas, A., Sakalauskas, L. 2006. A study models of stock markets. Information Technology andControl 35 (1):3456.

Beran, J. 1994. Statistics for Long-memory Processes. New York, Chapman and Hall.

Corazza, M. and Malliaris, A.G. 2002. Multi-Fractality in Foreign Currency Markets, Multinational Finance Jour-nal6(2): 6598.

Cutland, N.J., Kopp, P.E., Willinger, W. 1995. Stock price returns and the Joseph effect: a fractional version of theBlack-Scholes model. Progress in probability 36: 327351.

Geweke, J. and Porter-Hudak, S. 1983. The estimation and application of long memory time series models.Journal

of Time Series Analysis 4: 221238.

Giraitis, L, Kokoszka, P., Leipus, R., Teyssiere, G. (2003). Rescaled variance and related testes for long memory in

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Grech, D. and Mazur, Z. 2004. Can one make any crash prediction in finance using the local Hurst exponent idea?Physica A: Statistical Mechanics and its Applications 336: 133145.


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Hurst, H. E. 1951. Long-term strong capacity of reservoirs, Transactions of the American society of civil engineers116: 770799.

Kabainskas, A., Rachev, S., Sakalauskas, L, Sun, I., Belovas, I. 2009. Alfa-stable paradigm in financial markets.Journal of Computational Analysis and Applications 11(3): 642688.

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15 Sec 066 Danilenko Hurst