behaviour of stock markets’ memories
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Applied Financial EconomicsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/rafe20
Behaviour of stock markets’ memoriesShapour Mohammadi a & Ahmad Pouyanfar aa Faculty of Management , University of Tehran , Ale Ahmad Highway, Tehran, IranPublished online: 04 Dec 2010.
To cite this article: Shapour Mohammadi & Ahmad Pouyanfar (2011) Behaviour of stock markets’ memories, Applied FinancialEconomics, 21:3, 183-194, DOI: 10.1080/09603107.2010.524620
To link to this article: http://dx.doi.org/10.1080/09603107.2010.524620
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Applied Financial Economics, 2011, 21, 183–194
Behaviour of stock markets’
memories
Shapour Mohammadi* and Ahmad Pouyanfar
Faculty of Management, University of Tehran, Ale Ahmad Highway,
Tehran, Iran
In this article, we show that the memory of markets has nonchaotic
behaviour. Its time trend is neutral and nonlinearity tests such as Brock,
Dechert, Sheinkman (BDS) rejects nonlinearity in stock markets’ memo-
ries. The estimation of fractional differencing parameters is carried out by
various methods such as Maximum Likelihood Estimation (MLE),
Nonlinear Least Squares (NLS), Hurst exponents, Gewek, Porter-
Hudak (GPH), wavelet transformation, and Whittle. Also Lyapunov
exponents are estimated by two methods of Rosenstein and Jacobian.
Results of Lyapunov exponent estimation shows memory of markets are
not chaotic. Furthermore, there are no any Autoregressive Conditional
Heteroscedasticity (ARCH) effects in memory of markets. ARCH test is
more specific than the BDS test and it may powerful test for detecting
possible ARCH effects. All of tests show memory of markets has random
behaviour.
I. Introduction
New trends in trading procedures and increases in
market information will make markets closer to the
efficient markets in future than before. So with
increasing efficiency in the stock markets, memory
of markets shortens and trading in stock markets
does not lead to abnormal profits. One of the most
common methods for measurement of markets’
memory is estimation of fractional integration
parameter (d hereafter) for the stock prices (some
researches use return instead of prices, however here
there is no difference between them, when our main
concern is behaviour of memory, not level of it).
Many research in the fields such as long memory
process and Autoregressive Fractional Integrated
Moving Average (ARFIMA) models try to estimate
memory of markets. The method of Geweke and
Porter-Hudak uses nonparametric estimation based
on periodogram for estimating of d parameter
(Geweke and Porter-Hudak, 1983). This method
gives biased estimate for ARFIMA( p, d, q) process,
whereas it is almost accurate in estimation of d for
time series with ARFIMA(0, d, 0) data generating
process. A newer version of GPH, which is based on
Schuster periodogram was introduced by Granger
and Swanson (1997). Maximum likelihood method
for ARFIMA models was introduced by Sowell
(1992a, b) in econometrics. Of course, other
researches by others have been done before in
statistics and related fields.Recently estimation of d by wavelet methods is
proposed by econometricians and statisticians as well
as other majors. Wavelet method can be used as
approximation of maximum likelihood for estimation
(Jensen, 2000). Also wavelet method by ordinary least
squares is another choice for estimation of the long
memory parameter (Jensen, 1999) and (Tkacz, 2001).
Recently, Lee (2005) uses wavelets for estimating long
memory parameter of US inflation rate.
*Corresponding author. E-mail: [email protected]
Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online � 2011 Taylor & Francis 183http://www.informaworld.com
DOI: 10.1080/09603107.2010.524620
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The rest of this article is organized as follows.
A relation between memory of market and its stock
prices is presented in the Section II. Section IIIproposes some experimental considerations
about the theoretical relation of indices and their
member stocks, where it is also simulated indices byvarious weights for members of them. Estimations of
the long memory parameter and test of nonlinearity aswell as ARCH effect are placed in Section IV. Final
section concludes this article.
II. Memory of Markets
Long memory process and ARFIMA models arefamiliar for economists from works such as
Mandelbrot and Ness (1968), Granger and Joyeux
(1980) and Hosking (1981), among the others.A typical ARFIMA( p, d, q) model can be formulated
as �ðLÞð1� LÞd ð yr � �Þ ¼ �ðLÞ"t with lag polyno-
mials, �ðLÞ ¼ 1� �1L� � � � � �pLp and �ðLÞ ¼
1þ �1Lþ � � � þ �pLp, average �, and error term
"t � Nð0, 1Þ in the time domain and equivalently
I !ð Þ ¼ �" 2�ð Þ�1 1� exp �i!ð Þ�� ���2d � exp �i!ð Þð Þ
�� ��2� � exp �i!ð Þð Þ�� ���2
in the frequency domain.For detecting trend of market memory one of the
easiest methods is estimation of fractional integrationparameters for all of stock prices in any country. Due
to specification problem and large number of com-
panies this is not possible in practice. As an examplelet we find minimum Akaike Information Criterion
(AIC) model for estimation of fractional integrationfor one company over 100 sub periods. For an
ARFIMA(5, d, 5) minimum AIC model can be find
after running exactly 3600 ¼ 6� 6� 100 regressions.Number of iterations for each regression normally is
500 and for over than 3000 stocks of New York Stock
Exchange (NYSE) (only one of stock markets in theworld) it is a tedious work. We want to do this work
for all of the countries which their data are available
internationally. Therefore, we use index of eachcountry as representative time series of each stock
exchange. Following theorem allows us to do
such work.
Theorem: Let dm be memory of market then
dm ¼Xn
j¼1�jf ðdj Þ,
Xn
j¼1�j ¼ 1
where dj stands for memory of ith stock prices.
Proof: First part of proof is as follows:Let
It ¼Xnj¼1
pj 0qj 0
!�1Xnj¼1
pjtqjt
be index of market, then for qit ¼ qi0 the index
can be written as It ¼Pn
j¼1 �jpjt, where �j ¼
ðPn
j¼1 pj 0qj 0Þ�1qj 0. For any time series fpjtg
Tt¼1,
pjt 2 Rþ, j ¼ 1, . . . , n, with integration parameter
dj we have Ddjpjt ¼P1
k¼1 Ckdð�1Þ
kpjt�k ¼ "jt,whereD ¼ ð1� LÞ, L is lag operator, "jt � iid ð0, �2j Þ, and
Cjd is combinations of j from d, then
#It ¼Xnj¼1
�jDdjpjt ¼
Xnj¼1
�jX1k¼0
� k� dj� �
� �dj� �
� kþ 1ð ÞLkpjt
¼Xnj¼1
�j f dj� �¼ dm ð1Þ
where �ð�Þ ¼R10 x��1e�xdx is well-known gamma
function, and # stands for an adjustment parameter
which can be determined as follows:
# ¼ I�1t
Xnj¼1
X1k¼0
�j� k� dj� �
� �dj� �
� kþ 1ð ÞLkpjt
The second part is same as proof of Chambers’ (1998)
third theorem and in this article we present only some
further explanations in the next lines.
In general, let yt ¼Pn
j¼1 wjyjt, j ¼ 1, . . . , n then ytis simple linear aggregate variable for given yjt is
ARð1Þ process and wj is weights. (Chambers, 1998,
p. 1059) When n is very large, Granger and Joyeux
(1980) specify Beta distribution for parameters of n
AR(1) processes and show that aggregate variable
will be a long memory process. Let assume that each
yjt has Wold representation
1� Bð Þdjyjt ¼
X1h¼0
jh"j,t�h j ¼ 1, . . . , n ð2Þ
where 0j ¼ 1 andP1
h¼0 jjhj51, j ¼ 1, . . . , n,
dj 2 ð�0:5, 0:5Þ and each "j,t is a white noise process
with variance �jj. The memory parameters of each yjtare different and denoted by di. Spectral densities of
yjt are given by
fj !ð Þ ¼�2j2�
1� e�i!�� ���2dj j e�i!� ��� ��2, ! 2 ��,�ð �
where
j zð Þ ¼ 1þX1h¼1
jhzh
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Chambers (1998) assumes nonzero correlationbetween "j,t and "l,t, but no serial correlation betweenthem. In other words, Eð"j,t"l,tÞ ¼ �jl andEð"j,t"l,sÞ ¼ 0. In this case whether Eð"j,t"l,tÞ 6¼ 0 ornot, aggregation by itself dose not produce longmemory in aggregate variables. However, if longmemory exists in at least one of components of ytthen long memory will exist in the aggregate variable.Following proposition is the first part of Chambers’(1998) third proposition.
Proposition: The aggregate variable yt is integratedto an order equal to the maximum order of theunderlying components in Equation 2.
Based on the proposition stock markets indices’memories are integrated of maximum order ofintegration of underlying stocks’ memories.However, when the order of integration for most ofthe stocks is almost same as each other the averagememory is the memory of the market.
In the next section, using random data we show therelation between the memory of series and memory ofan index.
III. Simulation and ExperimentalConsiderations
In this section, we generate series f"itg1000t¼1 ,
"it � Nð0, 1Þ, i ¼ 1, . . . , n with different degree ofintegration di and compose a random indexIR ¼
Pni¼1 i"it, where i is ith element of random
vector with conditionPn
i¼1 i ¼ 1. For simulationARFIMA(0, d, 0) we use a truncated autoregressionas follows1
Xlj¼0
� j� dð Þ
� �dð Þ� jþ 1ð ÞLj
!yt ¼ "t ð3Þ
As one can see from Table 1 memory of index is aproxy of market memory. Clearly, memory of index isa function of market stocks’ memory. Therefore, it isreasonable to use memory of an index as proxy ofmarket memory in various countries.
When we use equal weights for getting theoreticalmemory in general, we do not need vector andtheoretical memory is simply average of memories ofthe stocks. For example, let memory of the first stockbe 0.003 and for the second stock 0.006 and etc. theaverage of this series will be 0.1515 for the 100stocks. Also, for other values such as di ¼f0:008, 0:016, . . . , 0:8g simple average is 0.404. Thevalues under ‘theoretical memory with equal weights’in the Table 1 are calculated in his manner.
Theoretical memory in Table 1 is calculated byTM¼ D. Where is a random vector with continuesuniform distribution normalized to aggregationcondition 1 and D ¼ d ½1 2 . . . 100�, d 2 f0:003,0:008, 0:015g. For the examination of the relationbetween index and securities’ memory, we estimatedlong memory parameters with autoregressive approx-imation method (Martin and Wilkins, 1999). For thememory of 0.15 maximum likelihood estimation haslower Root Mean Square Error (RMSE) than theautoregressive approximation. However, RMSE of
Table 1. Theoretical and estimated memory
Theoretical memory with equal weights
0.75 0.4 0.15
Method of weighting EM TM Err EM TM Err EM TM Err ML Err
URW1 0.557 0.767 0.376 0.437 0.404 �0.077 0.149 0.150 0.001 0.137 0.095URW2 0.722 0.753 0.043 0.367 0.390 0.063 0.186 0.151 �0.185 0.109 0.391URW3 0.576 0.767 0.331 0.466 0.423 �0.092 0.290 0.153 �0.472 0.167 �0.081URW4 0.676 0.723 0.068 0.345 0.391 0.136 0.208 0.153 �0.262 0.140 0.096URW5 0.655 0.753 0.150 0.393 0.394 0.003 0.144 0.151 0.044 0.147 0.036URW6 0.672 0.717 0.067 0.351 0.406 0.156 0.154 0.153 �0.006 0.158 �0.036URW7 0.691 0.763 0.104 0.268 0.411 0.531 0.145 0.148 0.017 0.142 0.040URW8 0.619 0.758 0.226 0.291 0.405 0.393 0.194 0.148 �0.240 0.180 �0.180URW9 0.713 0.784 0.099 0.317 0.390 0.230 0.139 0.148 0.065 0.157 �0.058URW10 0.682 0.742 0.087 0.499 0.427 �0.143 0.246 0.153 �0.378 0.139 0.101
Note: URW, unequal random weighting; EM, estimated memory; TM, theoretical memory (unequal); Err, error of estimationis calculated as a per cent of actual memory.
1The Matlab code for simulation of this and more general ARFIGARCH process is available from the first author on request.
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autoregressive approximation is lower for other valuesof memory parameter. These results show thatmemory of market is a weighted average of memoryof its traded securities whenmemories are close to eachother.
Although, indices of stock markets by itself areinformative and memory of the indices havevaluable information in general evaluation of thestock markets as well as index futures trading,results of this section, theorems and propositionpresented in second section show that index ofstock markets are good representatives for under-lying stocks’ memories. There are some reasons forimpossibility of estimation of memories of all ofstocks:
(1) Several thousands stocks are traded only in theUS stock markets and leading capital marketsall together have many thousands stocks thatrolling regressions of different kind for thisstocks requires huge calculations.
(2) Some stocks observations are not sufficientfor estimation the memories only for onesubsample, whereas we need at least 20observations for study behaviour of thememories.
(3) Summing up the memories for getting generalresults without theoretical or empirical consid-erations is not an easy task, whereas the indexis an aggregate indicator of stock market ingeneral.
(4) In the next section, the memories of marketsare analysed for real data of various stockmarkets.
IV. Estimation of the Memory of Markets
In this section, we use major stock markets indices fortesting chaos in the memory of markets. Indices ofthose markets that their observations are sufficientfor division to subseries are considered here. Afterdividing series to subsections, memory of eachsubseries is calculated with different methods ofestimation (GPH, Hurst exponent, MaximumLikelihood (ML), Nonlinear Least Squares (NLS),Wavelet-based Ordinary Least Square (OLS), andWhittle). In this section, world’s major stock indiceswith sufficient observations are analysed. Estimationof memories for subperiods for each index will give usthe historical series of memories. The memory seriesis tested for chaos in each stock market. For some
markets such as US we have near 20 000 observations(Nasdaq100 index in our study for example), but forsome countries the data length does not exceed than1650 daily observations (^TA100). Here, we dividedseries to k nonoverlapping subseries with 125 obser-vations for each subseries. The number 125 is almost6 months daily trades.
The results are based on NLS and Exact MaximumLikelihood (EML) estimation of memories.Specification of the models is done based on a codewhich automatically selects parameters p and q in anARIMA( p, d, q) and estimates d parameter accordingto minimum AIC criterion.2
The NLS estimation method can be stated asfollows (Chung and Baillie, 1993)
S �ð Þ ¼1
2log�2þ
1
2�2
XTt¼1
"2t
¼1
2log�2þ
1
2�2
XTt¼1
� Lð Þ� Lð Þ�1 1�Lð Þd yt��ð Þ
� �2ð4Þ
If the initial observations y0, y�1 y�2, . . . areassumed fixed, then minimizing the conditionalsum-of-squares function will be asymptotically equiv-alent to EML. Doornik and Ooms (2004) call thismethod as NLS. EML estimation uses Sowel (1992)method which is not presented here to save space.Details and recent modifications of EML are debatedin (Doornik and Ooms, 2003, 2004). We use PcGivepackage that takes to account some recent develop-ments of fractional parameter value, when we useEML and NLS estimators. Local Whittle estimationmethod is used for further precision. This method is anonparametric method in the following form
logLw �2u ,�
� �¼�
Xmj¼1
log f !j �,�2u
��� ��
1
2�
Xmj¼1
I !j
� �f !j �,�2u
��� �ð5Þ
where I(!j) stands for periodgram at jth Fourierfrequency !j ¼ n�12�j, j¼ 1 , . . . ,m and
I !j
� �¼ n�1
Xnt¼1
yt exp �it!j
� �����������2
Lieberman and Philips (2005) give some secondorder expansions such as Edgworth expansion toapproximate the Whittle function. This method canbe more efficient than the delta method. We use theexact whittle estimation instead of approximation ofit. Hurvich et al. (2005) and Shimotsu and Philips
2The code is a OX code, which calls ARFIMA package of PcGive and estimates ARFIMA ( p, d, q) models for the data.
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(2005) shows that exact local Whittle function
estimates are consistent in presence of Generalized
Autoregressive Conditional Heteroscedasticity
(GARCH) effects (Fractional Integrated GARCH
(FIGARCH) process). Also, in the case of nonsta-
tionary FIGARCH with d 2 ð0, 1Þ local Whittle
estimates are consistent. Therefore, we use this
method as one of estimation methods due to its
desirable properties.Estimation by GPH and augmented GPH are
biased in the ARFIMA( p, d, q) for higher orders of p
and q. Also, results of EML for nonstationary
process, d4 0:5, are not reliable. For these reasons
test of memories will be done based on estimations
driven from NLS method and GPH for further
examinations. At the first look, augmented GPH
method appears easy to use and powerful for
ARFIMA( p, d, q) as well as ARFIMA(0, 1, 0).
However, following equation (Martin and Wilkins,
1999) shows difficulties in practice
I !j
� �¼ �0 þ �1 ln 4 sin2
!j
2
� �h i
þ ln1þ �2 þ 2� cos!j
1þ �2 � 2� cos!j
þ �ARMA
t ð6Þ
Estimation of above equation should be done with
NLS which do not return t-statistics and SEs for
small sample in practice. Also, tapering the data can
be used for bias reduction in the GPH method.
The periodogram of tapered data will be
I !j
� �¼ 2�
Xn�1t¼0
w2t
!�1 Xn�1t¼0
wtyt exp �i!jt� ������
�����2
where
wt ¼ ð1=2Þ 1� cos 2� tþ 0:5ð Þ=nð Þ½ �
Here, we do not use either this or Robinson
modification because of negligible difference between
modified GPH and GPH methods. We use GPH
method in spite of its biasedness because the behav-
iour of memory is our concern instead of the memory
value itself.Also, we use modified R/S analysis introduced by
Lo (1991) for distinguish short and long memory of
series especially in stock market returns data. Let
y1, y2 , . . . , yT be a sample and �y ¼ 1=TP
t yt denote
the sample mean. Then the modified rescaled range,
denoted by QT, is defined as
QT ¼1
�̂T qð ÞMax1�k�T
Xkt¼1
yt � �yð Þ � Min1�k�T
Xkt¼1
yt � �yð Þ
" #,
ð7Þ
where
�̂2T qð Þ �1
T
XTt¼1
yt� �yTð Þ2
þ2
T
Xqt¼1
!t qð ÞXTs¼tþ1
ys� �yð Þ ys�t� �yð Þ
( )
¼ �̂2xþ2Xqt¼1
!t qð Þ�̂t, !t qð Þ � 1�t
1þq, q5T
ð8Þ
and �̂2x and �̂ i are the usual sample variance andautocovariance of y. Lo’s modified and classicalrescaled range differs by �̂TðqÞ in (7) instead of
S ¼X
tyt � �yð Þ
2=T� �1=2
in classical rescaled range. The difference between thetwo estimator �̂TðqÞ and S comes from autocovar-iances of {yt}. If the {yt} is a sequence with shortmemory variance of the partial sum is not sum of thevariances of the individual terms, but also includesthe weighted sum of autocovariances up to lag q.Determination of q is not easy task; however somesuggestions such as follows can be a good choice(Lo, 1991)
q ¼ kT½ �, kn �3n
2
� �1=32̂
1� ̂2
� �2=3
where [] is the well-known bracket function and ̂denotes the estimated first order autocorrelationcoefficient of the data. Also, another suggestion byAndrews (1991) for autocovariance weights in (8) is!t � 1� jt=kTj.
Another estimation method that is used frequentlyin the finance and economic area is wavelet basedestimation of the long memory parameter. A waveletfunction can be stated as
j,k tð Þ ¼ 2j=2 2jt� k� �
;
Z þ1�1
tð Þdt ¼ 0,
limt!1
tð Þ ! 0
where j, k 2 Z ¼ f0, 1, 2, . . .g are dilations andtranslation parameters respectively. Let y(t) be aL2ð<Þ real valued function, then wavelet coefficientsof y(t) are given by
cj,k ¼ 2j=2Z
y tð Þ 2jt� k� �
dt
Obviously, wavelet coefficients cj,k depend on jand k. The wavelet coefficient represents how muchinformation lost if the series y(t) is sampled less often(Jensen, 1999). For example, suppose every twoobserved values of y(t) averaged by zðt=2Þ ¼ ð yðtÞþyðtþ 1ÞÞ=2; t ¼ 0, 2, 4, . . . 2p, for getting original
Behaviour of stock markets’ memories 187
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series one should add wavelet cp,k; k ¼
0, 1, 2, . . . , 2p�1 coefficients to transformed data z(t),
i.e. yðtÞ ¼ zðtÞ þ cp,t. Therefore, cj,k is the representa-
tion of y(t) at different levels of resolution and timeperiods. For an I(d ) process y(t) with dj j5 0:5, onecan get distribution of cj,k as Nð0, �22�2jdÞ by using
autocovariance function of y(t)(Jensen, 1999). Let
R( j) be wavelet coefficient’s variance at scale j, i.e.Rð j Þ ¼ �22�2dj. After taking logarithms we obtain
lnR jð Þ ¼ ln �2 � d ln 22j
One can estimate parameter d by OLS method.
Simulations in Tkacz (2001) shows that estimateswavelet OLS for estimation of d have lower bias than
GPH and computationally is simpler than wavelet
maximum likelihood. If y(t) denotes a nonstationary
process with d 2 ð0, 1:5Þ then spectral density of
wavelet transform of y(t) at scale j around zero canbe stated as (Lee, 2005)
fj �ð Þ ¼ Cj �j j�2d ̂ �ð Þ��� ���2¼ Cj �j j
�2 d�1ð Þg2 �ð Þ as �! 0
where Cj ¼ cj=2�51 is a constant term, ̂ðtÞ is
Fourier transformation of with property j ̂ðtÞj� ¼��gð�Þ for integer �. Also properties gðt�Þ=gð�Þ ¼ 1
holds for all t, as �! 0 and 05 gð0Þ51. Sampleversion of spectral density fj ð�Þ at scale j can be
defined as follows:
I jð Þs ¼
1
2�T
X2j�1k¼0
wj kð Þ exp i�skð Þ�� ��2, s ¼ 1, 2, . . . ,m
where �s ¼ 2�s=T. Parameter m can be selected by
m ¼ T 0:5� �
or m ¼ T 0:6� �
rule. Wavelet estimation of
long memory parameter can be done by regression of
log(Is) on �2 logð�sÞ for s ¼ 1, 2, . . . ,m and adding
one to the estimated value (Lee, 2005).Results of estimations for memory parameters of
stock indices over various time intervals are too
sizable for presenting here. Over 45� 162 ¼ 7290
(45 indices over 162 subpriods) sells is needed for
results of each estimation method. Only time average
of memories for some stock markets can be reported
here.As can be seen from the Table 2 NLS, Whittle,
wavelet and GPH methods estimates are close to each
other, however, maximum likelihood and Hurst are
different from not only other methods, but also each
others. Graphical representation in Fig. 1 makes
recent debate clearer. Hurst estimation of the long
memory is done based on lag truncation of zero q ¼ 0
or classical rescaled range.The main concern of this article is behaviour of
memory in the world stock markets. Therefore, we
test chaotic behaviour for memory of each of indices
which is estimated by NLS method. Some indices
such as US markets indices have many memory
observations. This series are readily testable by
Lyaponuv exponent for chaotic behaviour, however
chaos tests in series with less observations may give
inaccurate results.With the estimated values of memory for every
market in different (nonoverlaping) time periods, one
–0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Index
Mem
ory
para
met
er
NLS Hurst Maxlike WhittleGPH Wavelet
Fig. 1. Graph of mean memory for stock markets indicesNotes: This figure plots memory values for indices estimated from various methods. The lowest line is Maxlike and the secondlowest line denotes estimates of Hurst method. The two middle lines are Wavelet and NLS estimates; and two upper linesshow estimated values of memories by Whittle and GPH methods. There is difference between the estimated values ofmemories by various methods, however, here only the behaviour of memories is important, but not the values of them.The behaviour of memory shows no trend and specific pattern generally (http://finance.yahoo.com/).
188 S. Mohammadi and A. Pouyanfar
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can make a series of memories. Here, we want testsome features as trend in memory, chaos in memoryand nonlinearity. For the examination of existence oftrend and ARMA terms in memory of market, simply
we ran a linear regression with time, AR(1) andMA(1) as independent variable
Mt ¼ �þ tþ Mt�1 þ �"t�1 þ "t ð9Þ
Estimates of above regression are reported inTable 3. There is no trend effect for most of indicesand in some of them ARMA terms are statisticallysignificant. We did not fit regression for full list ofindices because insufficient observations.
For testing chaotic behaviour in memories serieswe use Lyapunov exponents. Estimation methods inthis article are Rosenstein et al. (1993) and Gencayand Dechert (1992) methods. These methods arerobust to presence of noise and can be used for smallsamples without any correction. However, we usestate space averaging method (Schreiber, 1993) fornoise reduction to get better results. Rosensteinalgorithm can be defined by following equation
Lt ¼ T� t�m� 1ð Þ�1XT�tn¼m
log Xlþt � Xnþt
,�max ¼ dLt=dt ð10Þ
where Xn and Xl are two nearest neighbours in the mdimensional embedding space. The parameter tstands for time evolution and T is total number ofobservations. A process shows chaotic behaviour ifmaximum Lyapunov exponent �max be positive. Forestimating Lyapunov exponent it is needed that someparameters to be determined. There are differentmethods for determination of time delay in estima-tion of Lyapunov exponents. One method is auto-correlation function which selects time delay as anoptimum one, when autocorrelation of that lag fallsbelow e�1 ¼ 0:37 (Sprott, 2003, p. 314; Zeng et al.,1991, p. 3230). The second method is first minimummutual information. For determination of time delaywe use minimum mutual information instead of firstzero crossing autocorrelation. The main advantage ofmutual information (MI hereafter) is its ability tomeasure nonlinear dependence in data series. For tworandom variables of Xl and Xlþ� , MI can be definedas follows
I x0, x�ð Þ ¼
Z Zp xl, xlþ�ð Þ log
p xl, xlþ�ð Þ
p xlð Þ p xlþ�ð Þ
� �dxldxlþ�
¼ h xlð Þ þ h xlþ�ð Þ � h xl, xlþ�ð Þ ð11Þ
where h(xl) is differential entropy, which is defined by�Rpðxl Þ log pðxl Þdx. For discrete random variables
as well as time series, above definition can be writtenas (Sprott, 2003, p. 316).
I �ð Þ ¼XNi¼1
XNJ¼1
pij �ð Þ ln pij �ð Þ � 2XNi¼1
pi ln pi ð12Þ
Table 2. Mean of memory for stock markets
Method
Index Whittle GPH NLS Wavelet Hurst Maxlike
^DJI 1.025 1.006 0.758 0.695 0.459 0.122^DJT 1.069 1.040 0.885 0.705 0.463 0.195^DJA 0.974 0.995 0.813 0.663 0.456 0.137^NYA 1.003 1.024 0.678 0.535 0.447 0.119^NUS 0.961 0.892 0.761 0.514 0.432 �0.027^NIN 1.008 0.963 0.618 0.610 0.427 0.078^NWL 0.998 0.913 0.781 0.513 0.424 0.000^NTM 0.997 1.112 0.966 0.637 0.431 �0.011^IXIC 1.032 1.055 0.841 0.684 0.465 0.172^IXQ 0.973 1.010 0.680 0.696 0.470 0.109^NDX 0.974 1.028 0.837 0.656 0.464 0.126^IXF 0.988 0.966 0.784 0.730 0.467 0.062^NBI 0.966 1.031 0.858 0.783 0.463 0.126^GSPC 1.006 1.019 0.806 0.657 0.464 0.113^OEX 0.943 0.939 0.847 0.658 0.446 0.043^MID 0.967 0.943 0.714 0.615 0.446 0.194^SML 1.043 1.009 0.766 0.699 0.449 0.045^SPSUPX 0.941 0.880 0.301 0.537 0.435 0.015^MERV 1.039 1.123 0.783 0.745 0.478 0.156^BVSP 1.001 1.062 0.929 0.733 0.471 0.140^GSPTSE 1.009 1.033 0.857 0.739 0.477 0.192^GSPC 1.006 1.019 0.806 0.657 0.464 0.113^AORD 1.034 1.000 0.791 0.723 0.473 0.168^SSEC 0.996 0.985 0.799 0.560 0.466 0.094^HSI 1.030 1.006 0.942 0.704 0.464 0.270^BSESN 1.122 1.090 0.914 0.790 0.456 0.222^JKSE 1.049 1.045 0.856 0.736 0.476 0.229^KLSE 1.074 1.139 0.861 0.796 0.460 0.201^N225 0.985 0.946 0.847 0.675 0.463 0.109^NZ50 1.144 0.947 0.703 0.874 0.503 0.403^STI 1.013 0.966 0.789 0.682 0.446 0.201^KS11 0.969 1.030 0.936 0.636 0.457 0.132^TWII 1.037 1.051 0.974 0.570 0.454 0.153^ATX 1.060 1.020 0.835 0.744 0.478 0.185^BFX 1.031 1.021 0.893 0.699 0.468 0.220^FCHI 0.977 0.984 0.783 0.600 0.436 0.086^GDAXI 0.982 0.933 0.839 0.720 0.477 0.108^AEX 1.002 0.979 0.968 0.641 0.438 0.074^OSEAX 1.072 0.980 1.039 0.765 0.473 0.111^MIBTEL 1.052 0.962 0.916 0.691 0.470 0.138^SMSI 1.020 0.889 0.660 0.504 0.425 �0.069^SSMI 0.969 0.992 0.764 0.683 0.462 0.126^FTSE 0.983 0.952 0.685 0.679 0.442 0.137^CCSI 1.062 1.037 0.730 0.716 0.455 0.124^TA100 0.98 0.95 0.92 0.57 0.45 0.09
Notes: Memories of indices are estimated by differentmethods GPH, Hurst exponent, Maxlike, NLSA, Whittle,and Wavelet. For estimating series of memories, for eachindex the whole series of data is divided to some subserieswith 125 observations, then estimation methods wereapplied on subseries.
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Jacobian method based on artificial neural net-
works approximation, which is used here, is based on
Gencay and Dechert (1992). For some explanations
on this method let f : Rn! R
n be a dynamical system
with trajectory ytþ1 ¼ f ð ytÞ, t ¼ 1, 2, . . . . For this
system there are n Lyapunov exponents which are
measures of the average rate of divergence and
convergence of trajectories. Lyapunov exponents
can be defined as
�j ¼ limt!1
t�1 ln Df t� �
y0�
, � 2 VjnVjþ1,Vj � Rn
Df t� �
y0¼ Dfð Þyt Dfð Þyt�1 . . . Dfð Þy0 ð13Þ
where Vj, j ¼ 1, 2, . . . , n are nested subspaces. With
this formulation Lyapunov exponents can be calcu-
lated by evaluating the Jacobian of the function f
along a trajectory {yt}. In the case of dealing with
known functional forms calculation of Jacobian is
almost straightforward task. However, for time series
data there is no functional form and without an
estimated function getting Jacobian is not possible.
For estimating functional forms from time series
there is many methods. One of the best methods in
approximation of functions is Artificial Neural
Networks (ANN). This method is very powerful,
such that it is called universal approximator accord-
ing to Hornik et al. (1989, 1990). This theorem states
that an ANN with sufficient hidden unites, a flexible
functional form and properly adjusted parameters
can approximate any function f : Rn! R arbitrarily
well in useful spaces of functions (Kuan and
White, 1994).
We use a code which is written in Matlab for test ofchaos based on Rosenstein algorithm. The codeconsiders mutual information for choosing delays.For further examination we use a programme titledNetle 3.01 which is written by C.-M. Kuan, T. Liu,and R. Gencay. This programme uses neural net-works for estimation Lyapunov exponents.
The reported estimates of Lyapunov exponents inTable 4 show negative exponents for long time seriesand positive exponents for short time series.The estimate of Lyaponv exponent needs moredata than that is available at present. However,the results of Jacobian methods such as Gencay andDechert (1992) are more precise and robust to noisesthan the direct methods such as Rosenstein et al.(1993) and Kantz (1994). As noted in Lai and Chen(2003) the robustness of Jacobian-based methods isachieved in cost of computational intensity. In thisarticle, we rely on results of Kuan and Liu (1995)method when there is inconsistency between thismethod and that of Rosenstein et al. (1993).Rosenstein method is applicable for short timeseries with 100–1000 data points. As one can seefrom the results in Table 4 for the indices withobservations larger than 100 data points Lyapunovexponents have negative signs that are consistentwith results of Jacobian method. Graphical analysisand BDS test confirm nonchaotic behaviour of stockmarket memories. Also, method of Kuan and Liu(1995) works well for short time series.
We use BDS test for test of independence. Brocket al. (1996) introduced a test for independence intime series data. This test is a statistical test which has
Table 3. Short memory and trend of long memory parameters
Index Constant Trend AR(1) MA(1) F�p value DW
AORD 0.815* �0.000 0.226 �0.027 0.663 1.995DJA 0.945** �0.006** 0.426** �0.960** 0.003 1.768DJI 0.722** 0.000 0.656 �0.629 0.644 2.004DJT 0.908** �0.000 0.784** �0.695* 0.326 2.034FTSE 0.709** �0.001 �0.086 0.030 0.984 1.994GSPC 0.920** �0.002 0.042 �0.166 0.268 2.003HSI 1.004** �0.001 0.828 �1.320** 0.001 2.119IXIC 0.806** 0.003 0.621** �0.949** 0.116 1.728N225 0.859** �0.000 0.717** �1.353** 0.000 2.133NDX 0.776** 0.003 �0.191 0.093 0.821 1.999OEX 0.786** 0.003** 0.472** �0.982** 0.002 2.241STI 0.682** 0.006** 0.532* �0.997** 0.026 2.005
Notes: For the examination of possible patterns in the behaviour of memories we used a linear time seriesmodel. The model is a short memory model with a trend regressor(Trend), an Autoregressiveregressor(AR(1)), and a moving average regressor(MA(1)). In the table, F�p value stands for probabilityof F and DW is Durbin Watson statistic.* and ** denote significance at 5 and 1% levels, respectively.
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Table
4.Lyapunovexponents
formarketsmem
ories
Index
AORD
DJI
DJT
DJA
FTSE
GSPC
HSI
IXIC
N225
NDX
OEX
STI
Rosensteinmethod
Lambda
0.785
�0.693
�0.704
0.716
0.608
�0.697
0.743
0.396
0.606
0.574
0.538
0.648
Obs.
43
154
154
50
42
112
38
42
43
40
47
35
Kuan,LiandGencay
method(selectedwith
minum
SIC
)
Lambda
�9.373
�19.376
�2.173�3.331�4.765
�3.456�1.339�8.234�0.891�15.557�1.966�0.795
Obs.
42
153
153
49
41
111
37
41
42
39
46
34
Lambda1
0.253
�2.635
�0.946�8.183�0.227
–0.224
–�0.088
0.087
–1.139
Lambda2�2.324
�4.180
�1.509�9.548�1.068
–�0.843
–�0.763
�0.478
–�1.042
Obs.
41
152
152
48
40
110
36
40
41
38
45
33
Lambda1�0.447
�0.929
�0.239
0.261
0.555
0.050
0.779
0.538
0.689
0.469
–1.224
Lambda2�0.576
�1.981
�0.509�0.676�1.056
�0.871�0.262
0.418
0.006
�0.103
–�1.508
Lambda3�1.083
�2.833
�0.628�2.127�2.416
�2.287�1.307
0.000�1.009
�0.705
–�3.344
Obs.
40
151
151
47
39
109
35
39
40
37
–37
Notes:Obs.,number
ofobservations;SIC
,Schwarz
Inform
ationCriterion.
Forsomeofindices
thereissufficientnumber
ofmem
ory
value,
soLyapunovexponent,Lambda,canbecalculated.Lyapunovexponentiscalculatedbytw
omethodsof
Rosensteinet
al.;andKuan,LiandGencay.In
theRosensteinet
al.method,only
largestLyapunovexponentcanbeestimated,butKuan,LiandGencaygives
spectrum
of
exponents.Here,
weestimatedLyaponuvexponents
withthreeassumptions(1
Lambda,2Lambdasand3lambdas)
forKuan,LiandGencaymethod.
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application in various problems such as chaos and
serial independence. The BDS statistic can be
defined as
T m, rð Þ ¼ limN!1
ffiffiffinp
C m, "ð Þ � C 1, "ð Þm
½ �=� m, "ð Þ ð14Þ
where m is dimension, " denotes radius of a ball that
contains pairs of points and n is number of
observations. When a time series white noise
(iid: independently identically distributed)
Cðm, "Þ ¼ Cð1, "Þm and � m, "ð Þ=ffiffiffinp
is SD.The results in Table 5 show that there is not
nonlinear dependence in series of market memories.
Therefore, our statement which stock markets’ mem-
ories are not chaotic and do not show any trend is
valid based on this test.Also for further analysis of memory behaviour
we test Autoregressive Conditional Heteroscedastic
(ARCH) effects in the parameters values. These types
of models are introduced in economics by Engel
(1982) for modelling UK inflation rate series. Also,
there is huge literature about ARCH models in
economics and financed. In finance volatility estima-
tion, option pricing, calculation of the value at risk
are some examples. Extension of the ARCH to the
generalized form is done by Bollerslev (1986) which is
called GARCH in the literature. Let "t ¼ yt � �� xtbe a simple classical regression and "t ¼ �t�t. Where
{vt} is a sequence of iid random variables with mean 0
and variance 1. Then "t follows a GARCHð p, qÞ
pattern if
�2t ¼ �0 þXpi¼1
�i�2t�i þ
Xqj¼1
j"2t�j ð15Þ
This models declines to an ARCHð pÞ if
�i ¼ 0; i ¼ 1, . . . , p. Here, we want to test only
ARCH effects and estimate the model in the form
of �2t ¼ �0 þPp
j¼1 j"2t�j. For the test of the ARCH,
one can use TR2 � �2p statistics where R2 is coefficient
determination of the following regression
e2t ¼ �0 þXp
j¼1 je
2t�j
Also, et is residual of the main regression
yt ¼ �þ xt þ "t. In the case of memory main
regression is only yt ¼ � and test of ARCH is done
based on demeaned values of {yt}. The lag value p is
supposed to be 1 in the all of models. Following table
shows results of the ARCH model for some indices’
memoriesThe lag value is selected by l¼ [T1/4], where l is lag
value and T stands for total number of observation
and ½� bracket function. For further examination
other values for lag is tried but for saving space are
not reported. Lag values f1, 2, 3, 4, 5g is used for
testing ARCH effect test, however results are same as
results of lag value by l¼ [T1/4]. Other criteria for lag
selection such as l¼ [T1/3] and l¼ [T1/2]is proposed
(Mills, 1999, p. 120) also Tsay (2005) suggests
Partial Autocorrelation Function (PACF) for order
determination of the ARCH models. All of this
methods gives lag values such that l 2 f1, 2, 3, 4, 5g.
Also, we used PACF for lag determination of ARCH
model and in some cases proposed lags by this criteria
was higher than pervious criterion, however the
ARCH test with that lags also cannot reject null
hypothesis.Results of ARCH test in Table 6 show there is no
ARCH effects for almost all of indices at 5% level.
Therefore, memories of markets have not any vola-
tility clustering property. Only in the case of DJT and
OEX, ARCH effects are significant in 5% level of
first type error.
Table 5. BDS test on memory of markets
Dimension
Index 2 3 4 5 6
AORD BDS 0.024 0.025 �0.025 �0.017 �0.004p 0.239 0.351 0.731 0.932 0.779
DJA BDS �0.018 �0.035 �0.036 �0.040 �0.039p 0.448 0.372 0.490 0.441 0.468
DJI BDS 0.000 0.000 0.006 0.008 0.003p 0.911 0.896 0.630 0.565 0.713
DJT BDS 0.015 0.021 0.023 0.025 0.020p 0.155 0.191 0.232 0.229 0.279
FTSE BDS �0.003 0.004 0.012 0.017 0.023p 0.965 0.708 0.516 0.436 0.316
GSPC BDS �0.013 �0.016 �0.012 �0.015 �0.004p 0.334 0.534 0.755 0.688 0.946
HSI BDS �0.005 �0.014 �0.027 �0.043 �0.060p 0.958 0.840 0.744 0.636 0.503
IXIC BDS �0.027 �0.038 �0.026 �0.011 �0.008p 0.189 0.351 0.732 0.926 0.842
N225 BDS �0.005 �0.012 �0.021 �0.069 �0.072p 0.801 0.742 0.852 0.275 0.356
NDX BDS �0.005 �0.018 �0.015 �0.022 �0.013p 0.998 0.793 0.972 0.877 0.932
OEX BDS 0.029 0.050 0.079 0.083 0.075p 0.151 0.138 0.079 0.092 0.116
STI BDS �0.033 �0.049 �0.105 �0.106 �0.081p 0.205 0.306 0.068 0.097 0.225
Notes: BDS, Brock, Dechert, Scheinkman and LeBaron teststatistic; p, Bootstrapped p-values.The BDS test is used for detecting of possible nonlineardependence in the series of memories. Dimension 2–6 isconsidered for embedding dimension of state space mem-ories’ dynamics.
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V. Conclusion
Level of memory in stock markets and shortening of
it may be helpful in detecting the market efficiency
and its tendency to the efficiency. Our results and
most of memory estimation can not give more useful
information about level of memory and level of
efficiency because estimation bias in almost all of
ARFIMA estimation methods. However, this arti-
cle’s findings on trend and behaviour of memories are
reliable and can be used for examination of tendency
to efficiency in world stock markets. Trend and short
memory in series of long memory parameters are not
confirmed by regression in most of cases and we can
conclude that there is no significant tendency to
efficiency in the world markets. Also, Lyapunov
exponents detect nonchatic behaviour for memory of
markets when there is sufficient number of observa-
tions. Empirical results based on BDS test show there
is no chaotic behaviour in memory of markets and it
is really white noise. For further test we used ARCH
test fore specific nonlinearity and volatility in
memory of market. Results of ARCH test are in the
line with BDS test. Based of the results one can
conclude that there is no trend, chaotic behaviour,
nonlinearity as well as ARCH effects in the memory
of markets.Our results show no tendency in market memory
behaviour; therefore investors can use traditional
technical and fundamental trading rules to predict
markets such as before. Little variation in memory
parameters across markets show that there are not
much differences in the markets efficiency levels; so
there is some profitability opportunities in the mar-
kets that can be exploited by international diversifi-
cation. There is a recommendation for capitalmarkets policy makers that are concerned about
markets efficiency: financial markets need fast and
exact reaction to information to shortening market
efficiency and increase in market efficiency. Most of
tests of market efficiency are test of inefficiency tests;because they test predictive ability of some specific
models not all of possible models, whereas market
memory behaviour can be a test of efficiency changes;
therefore future studies can focus to market memory
as an indicator of efficiency.
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