LONG-TERM MEMORY EFFECT IN STOCK PRICES ANALYSIS

ISSN
1822-6515
ISSN
1822-6515
EKONOMIKA IR VADYBA: 2009. 14






ECONOMICS & MANAGEMENT: 2009. 14
LONG-TERM MEMORY EFFECT IN STOCK PRICES ANALYSIS
Svetlana Danilenko
Vilnius Gediminas Technical University, Lithuania, svetlana.danilenko@fm.vgtu.lt
Abstract
The Hurst exponent is widely applied for time series analysis. The Hurst exponent is a statistical
measure used to classify time series. Using the Hurst parameter processes are classified into long range
dependence, antipersistence and white noise processes.
R/S analysis method is one of the few methods that evaluate the Hurst exponent. This method uses the
rescaled range statistic (R/S statistic). The R/S statistic is the range of partial sums of deviations of a time
series from its mean, rescaled by its standard deviation. A log-log plot of the R/S statistic versus the number
of points of the aggregated series should be a straight line with the slope being an estimation of the Hurst
exponent.
However, there are many methods of evaluating the Hurst exponent such as ratio variance of
residuals, the periodogram method, the Whittle method, the Abri-Veitch method, etc.
Investigation object – the Baltic sector indices. The latter represent tendencies of different sector
activity indices in the stock market.
The work concentrates on calculating the Hurst parameter, evaluated Hurst parameters of the Baltic
sector indices are given for different periods of time.
Keywords: R/S analysis, Hurst exponent, financial markets, share index.
Introduction
Long memory in exchange rates would allow investors to anticipate price movements and earn
positive average returns. Long-term memory indicates the correlation structure of a series at long lags. If a
series shows long-term memory there is persistent temporal dependence even between distant observations.
Those time series are characterized by distinct but non-periodic cyclical patterns. The presence of long-term
memory in stock returns has important implications for many 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-range dependent.
Stochastic processes with long-term dependence and methods of their identification have been
introduced first in the context of hydrology and geophysics (by Hurst, 1991, Mandelbrot & Wallis, 1969,
Mandelbrot & Ness, 1968, Mandelbrot, 1982). The Hurst exponent was originally suggested as a result of a
study of the flow of water through dams, and stems from the observation that particles 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.
Recently 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), it is now
popular for analysis of financial markets (see Lo, 1991, Corazza & Malliaris, 2002, Grech & 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 are given in
??????, 1998). It is possible to distinguish long-term non-function cycle time series and random series
through this method.
R/S analysis method is one of the most commonly used methods for the evaluation of the Hurst
exponent. However, there are many methods of evaluating the Hurst exponent such as ratio variance of
residuals, the periodogram method, the Whittle method, the Abri-Veitch method, etc.
Purpose – this paper, using Baltic stock index data, set outs to present long-term memory effect.
Investigation object – the Baltic sector indices. The latter represent tendencies of different sector
activity indices in the stock market. For analysis all stock indexes were gathered. Stock market indexes were
counted to one base. Their values in the investigational period beginning were equated to 100.
Investigational period of stock indexes is from the year 2000 till the year 2008.
The work concentrates on calculating the Hurst parameter, evaluated Hurst parameters of the Baltic
sector indices are given for different periods of time.

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Rescaled range analysis and the Hurst exponent
The application of R/S analysis to financial markets will be discussed.
Let S = (Sn) be the share index for the Baltic states, then the logarithm of the share index
n?0
h = ln Sn , n ?
1 is the proportional change in the share index or the logarithmic change.
n
Sn 1
?
We will describe the essence of R/S analysis application for the investigation of h = (hn ) sequence
n 1
?
properties.
Let us comprise H = h +...+ h , n ? 1.
n
1
n
Let us define
?
k
?
?
k
?
R = max
. (1)
H
H
min H
H
n
?
?
k
n ? ?
?
?
k
n ?
k ? n ?
n
?
k ? n ?
n
?
Let
H
k
h
n
?
where (h , h ,..., h be an empirical mean, then
? k
H
H =
h
h
is H deviation
k
n
?( ? n
i
)
1
2
n )
n
k
n
n
i =1
from its empirical mean k H . The measure R characterises the degree of dispersion of
k
?
, ?
n
n
H
H
k
n
n
k
n
n
deviations.
Let the empirical dispersion
2
n
n
n
1
1
1
2
)
S 2 =
h 2
h
h
h
(2)
n
?
?
?
k
? ?
?
? =
k
?( ? n
k
n
1
n
1
n
k =
? k=
?
k =1
and the adjusted range of the cumulative sums H , k ? n
k
R

n
Q ?
. (3)
n
Sn
Based on the R / S we are testing the null hypothesis ( H ) that the share index is a random walk. If
0
the null hypothesis is correct then if the value of n is large the values of R S should be proportional to
n
n
the square root of n
0,5
R
S ~ cn
,

(4)
n
n
However, in financial data analysis it usually is
H
R
S ~ cn ,

(5)
n
n
where H - Hurst parameter that is significantly different from 5
.
0 .
If we take the logarithm of Equation (5), we obtain
log(R S = log
+ ?log

(6)
n
n )
(c) H
(n)
and hence, in practice, the Hurst exponent can be calculated by plotting
(
log R S ) against log(n) and
n
n
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:
0 ? H < .
0 5 : Fractal Brown Motion. At this time, the future data of time series has the tendency
to go back a history point; therefore, it is slower than standard Brown Motion. It can be proved
that, in terms of theory, this series will return to its history point for numerous times.
H = 0.5 : Standard Brown Motion. At this time, to walk casually, the series will show Markov
chain feature.
0.5 < H ? 1 : Long-term and non-period cycle. At this time, the time series will be in a mess, its
increment will show long-term increase. Therefore, a certain range record will continue for a
relative period and form many big cycles. However, these cycles have no fixed period and it is
difficult to depend on the past data to forecast the future change.
Application of the Hurst analysis to the Baltic sector indices
The Vilnius, Riga and Tallinn stock exchanges are the only stock exchanges in their countries. Due to
the fact that the current economic and political integration goals of the Baltic states are similar, the stock
exchanges in the pursuit of reducing the differences in the stock markets of Lithuania, Latvia and Estonia

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have formed a common combined OMX Baltic states stock market. OMX uses the same classification in
both Western Europe and the Baltic states stock markets. The common index calculation methodology
promotes simplified understanding of the stock indices in both Western Europe and the Baltic states as well
as better comparability of the stock markets in these countries. The OMX Baltic states stock market index
group is comprised of the Baltic states stock market comparability, trade, all shares and sector indices.
The OMX Baltic Sector indexes are based on the Global Industry Classification Standard (GICS)
developed by Morgan Stanley Capital International Inc. (MSCI) and Standard & Poor’s. GICS is an
international classification created to meet investors’ demands for more precise, exhaustive and standardized
classification. Sector indexes show the trend of a specific sector and enable peer comparison between
companies engaged in the same sector. The indexes include all the shares listed on the Main and Secondary
lists of the Baltic exchanges (stocks of the companies where a single shareholder controls at least 90% of the
outstanding shares are not included). The indexes are calculated for each GICS sector in Euro and available
as PI and GI. The base date for the OMX Baltic Sector indexes is December 31, 1999, with a base value of
100. The index values are disseminated once after the market close (OMX Baltic index descriptions).
10 OMX Baltic sector indexes are included on the Stock Exchange: Energy, Materials, Industrials,
Consumer Discretionary, Consumer Staples, Health Care, Financials, Information Technology,
Telecommunication Services, Utilities. Investigational period of stock indexes is from the year 2000 till the
year 2008. Their development is demonstrated in Figure 1.

Figure 1. Development of the Baltic sector indices in 2000-2008
The values of share indexes’ Hurst exponents using the R/S analysis method are presented in the Table 1.
The growth of economic Baltic Sector indexes beginning with the 4th quarter of 2003 has mainly been
fuelled by the positive investor expectations relating to the coming European Union membership, its
structural support, the geographical location of the countries as well as the relative novelty of the market and
the establishment of new investment funds. Once the Baltic States have joined the European Union, the
indexes have started to rise more rapidly while the economic situation of the countries enhanced. As it is
commonly known, country‘s stock exchange is closely related to the overall state of the economy in that
country, therefore the relationship between the economic expansion and the growth of the stock market is
mutual. From mid 2007 a decrease in the indexes is noticeable. The collapse of investment banks in the US
in the fall 2008 had caused an echoing wave across the European and the Far East Stock markets as well as
slowing down the economic development of most countries. Since the beginning of economic slowdown,
further decreases in share prices are to be expected. Based on the situation discussed, the observed time

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frame has been divided into several intervals. The Hurst exponents have been estimated for the whole
observation period as well as the separate intervals.
The Hurst parameters of the Baltic sector indices are calculated with the help of R/S analysis. They 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 countries’ share
indices time series is within the following interval 0.5 < H ? 1 which shows that there is a week correlation in
them.
In order to validate the stability of the value of H, the original time sequence has been disarranged and
the value of H recalculated. This time value H is supposed to be distinctively close to 0.5. After computing,
the result proves that the value of H is significant.
Table 1. The results of Hurst analysis

OMX Baltic sector 2000.01.01- 2000.01.01- 2004.05.01- 2004.05.01- 2007.07.01-

indices
2008.12.31 2004.04.30 2008.12.31 2007.06.30 2008.12.31

Number of
2317 1125 1192 815 377
observations

Energy
0,650 0,650 0,666 0,671 0,579
Materials
0,665 0,663 0,643 0,639 0,575

Industrials
0,713 0,708 0,736 0,692 0,695
Consumer

0,658 0,650 0,690 0,651 0,670
Discretionary
Consumer Staples
0,616 0,598 0,674 0,625 0,657

Health Care
0,557 0,553 0,601 0,575 0,638

Financials
0,615 0,607 0,684 0,634 0,658
Information
0,653 0,635 0,684 0,634 0,618
Technology

Telecommunication
0,651 0,655 0,640 0,646 0,673
Services

Utilities
0,584 0,607 0,613 0,589 0,537

The R/S analysis method is one of the methods of the Hurst exponent evaluation. However there are
many methods of evaluating the Hurst exponent with the most commonly used being the following (Beran,
1994): Ratio variance of residuals, Periodogram method, Whittle method, Abri-Veitch method.
Table 2. Evaluation of Hurst exponent using different methods
Ratio
OMX Baltic sector
Periodogram
Whittle
Abri-Veitch
variance of
indices
method
method
method
residuals
Energy 0,632
0,388
0,639
0,697
Materials 0,695
0,622
0,55
0,583
Industrials 0,751
0,739
0,624
0,647
Consumer
Discretionary
0,590 0,662 0,567 0,594
Consumer Staples
0,597
0,586
0,53
0,574
Health Care
0,600
0,455
0,500
0,520
Financials 0,577
0,582
0,589
0,640
Information
Technology 0,706
0,718
0,541
0,560
Telecommunication
Services 0,663
0,525
0,573
0,635
Utilities 0,512
0,346
0,500
0,518

The share indices’ Hurst exponents evaluated using the above methods are presented in the following
Table 2. Investigational period is 2000.01.01-2008.12.31.

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ECONOMICS & MANAGEMENT: 2009. 14
Conclusion
Hurst exponent is widely applied for the time series analysis. When little information is held of the
investigated system, the Hurst exponent helps to classify the time series. R/S analysis method is one of the
few methods that evaluate the Hurst exponent. This method uses the rescaled range statistic (R/S statistic).
The Hurst exponent of financial data is usually bigger than 0.5 . This is supported by the share indices’
Hurst exponents findings for the Baltic sector indices, that shows existence of long-term dependence in
analysed series.
The research of existence of long-term dependence of Baltic sector indices shows that Hurst exponent
of Industrial sector indices is largest in all analysed periods. In Industrial sector exists stronger long-term
dependence than in other sectors. The least Hurst exponent is in Utilities and Health Care sectors that shows
weak long-term dependence between these sectors. R/S method and other methods used in article, like Ratio
variance of residuals, Periodogram, Whittle, Abri-Veitch, verify these result.
The existence of long-range dependence indicates that the Baltic sector indices exhibit dynamics that
are not consistent with the random walk behaviour. Participants in the Baltic equity markets should consider
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 beneficial if pricing
models included an assumption about the long-term dependence.
References
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