Long memory in stock

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Academy of Economic Studies - Bucharest
Doctoral School of Finance and Banking
DOFIN
Long Memory in Stock Returns:
Research over Markets
Supervisor: Professor Dr. Moisă Altăr
MSc Student: Silvia Bardoş
Bucharest, July 2008
Contents
• Long memory & Motivation

• Literature review

• Steps & data used:

• Testing stationarity and long memory
• ADF & KPSS
• Hurst exponent through R/S test & Hurst exponent through wavelet
estimator
• Determining long memory by estimating fractional differencing parameter
• Geweke and Porter-Hudak test & Maximum Likelihood Estimate for an
ARFIMA process

• Conclusions
Long Memory & Motivation
• Long memory has important implications in financial markets because if it is discovered it
can be used to construct trading strategies.
• Long memory or long range dependence means that the information from “today” is not
immediately absorbed by the prices in the market and investors react with delay to any
such information.
So:
• A long memory process is a process where a past event has a decaying effect
on future events
AND
• Memory is the series property to depend on its own past realizations

• Mathematic view: Long memory processes relates to autocorrelation

• If a time series of data exhibits autocorrelation, a value from the data set xs at time ts is
correlated with another value xs+z at time ts+z. For a long memory process autocorrelation
decays over time and the decay is slower than in a stationary process (I(0) process)

• So, if a long memory process exhibits an autocorrelation function that is not consistent for a
I(1) process (a process integrated of order 1) nor for an I(0) process (a pure stationary process)
we can consider a long memory process as being the layer separating the non-stationary
process from the stationary ones – namely a fractionally integrated process.
Long Memory & Motivation
Literature review
• Evidence of long memory was first brought up by E. Hurst in 1951 when, testing the behavior of
water levels in the Nile river, he observed that the flow of the river was not random, but patterned
• Mandelbrot (1971) was among the first to consider the possibility of long range dependence in
asset returns
• Wright, J. (1999) is detecting evidence of long memory in emerging markets stock returns
(Korea, Philippines, Greece, Chile and Colombia)
• Caporale and Gil-Alana (2002), studying S&P 500 daily returns found results indicating that the
degree of dependence remains relatively constant over time, with the order of integration of stock
returns fluctuating slightly above or below zero
• Henry Olan (2002) makes a survey for finding long memory in stock returns from an
international perspective. Evidence of long memory is found in the German, Japanese, South
Korean and Taiwanese markets against UK, USA, Hong Kong, Singapore and Australia where no
sign of long memory appears.
Steps – Modeling long memory
• A series xt follows an ARFIMA (p,d,q) process if:
( )
t t
L x
d
L L c u ) ( 1 ) ( = ÷ u
( )
2
, 0 ~ o c iid
t
where Φ(L), θ(L) are the autoregressive and moving average polynomials, L is the lag, d is
the fractional differencing parameter, εt is white noise.
• For d within (0,0.5), the ARFIMA process is said to exhibit long memory or
long range positive dependence

• For d within (-0.5, 0), the process exhibits intermediate memory or long range
negative dependence

• For d within [0.5, 1) the process is mean reverting and there is no long run
impact to future values of the process

• The process is short memory for d=0 corresponding to a standard ARMA
process
Testing stationarity
• Memory is closely related to the order of integration
• In the context of non-fractionally integration is equivalent to establish whether the series is I(0) or
I(1) and the commonly used tests are ADF and KPSS

ADF
Null hypothesis: H0: d = 1 (returns series are containing a unit root)
Hassler and Wolter (1994) find that this test of unit root is not consistent against fractional alternatives so the ADF can be
inappropriate if we are trying to decide whether a set of data is fractionally integrated or not.


KPSS
Null hypothesis: H0: d = 0 (return series are stationary)
Lee and Schmidt (1996) find that KPSS test can be used to distinguish short memory and long memory stationary processes


Testing stationarity
KPSS
Consider xt ( t = 1, 2, …, N), as the observed return series for which we wish to test
stationarity

The test decomposes the series into the sum of a random walk, a deterministic trend and a
stationary error with the following linear regression model:
t t t
x r t | c = + +
The KPSS statistics:
( )
¿
=
T
t
t
S
l S T
1
2 2
1
and ¿
=
=
T
i
i t
S
1
c
i
c
is the residual from regressing the series against a constant or a constant and a
trend
Under the null hypothesis of trend stationary, the residuals et (t = 1, 2, …, N) are from the
regression of x on an intercept and time trend.
Under the null hypothesis of level stationarity, the residuals et are from a regression of x on
intercept only.
Rejection of ADF and KPSS indicates that the process is described by neither I(0) and I(1)
processes and that is probable better described by the fractional integrated alternative (d is
a non-integer).
Estimating long memory using R/S test
R/S test
• Mandelbrot & Wallis (1969) method allows computing parameter H, which measures the
intensity of long range dependence in a time series

Return time series of length T is divided into n sub-series of length m.

For each sub-series m = 1, ..., n, we:
a) find the mean (Em) and standard deviation (Sm);
b) we subtract the sample mean Zi,m = Xi,m − Em for i = 1,..,m;
c) produce a time series taking form of Wi,m = j,m where i = 1,…,m
d) find the range Rm = max{W1,m,…., Wn,m} – min{ W1,m,…., Wn,m}
e) rescale the range Rm by
¿
=
i
j
Z
1
Sm
Rm
How does this procedure relates to the Hurst exponent?
• Einstein discovered that the distance covered by a random variable is close related to the
square root of time (Brownian motion)
5 . 0
T k R × =
, where R is the distance covered by the variable, k is a constant and T is the
length of the time.
• Using R/S analysis, Hurst suggested that:
H
m k
S
R
× =
, where R/S is the rescaled range, m is the number of observations, k is the
constant and H is the Hurst exponent, can be applied to a bigger class of time
series (generalized Brownian motion)
• The Hurst exponent can be than found as:

log(R/S)m=log k +H log m
H value Return time series
=0.5 follow a random walk and are independent
(0,0.5)
are anti-persistent, process covers only a small
distance than in the random walk case
(0.5,1)
are persistent series, process covers a bigger
distance than a random walk (long memory)
Estimating long memory using R/S test
Hurst exponent using wavelet spectral density
• For computing the Hurst Exponent, the R wavelet estimator uses a discrete wavelet transform
then:
• averages the squares of the coefficients of the transform,
• performs a linear regression on the logarithm of the average, versus the log of the
parameter of the transform
The result provides an estimate for the Hurst exponent.

Wavelet transform behaves as a microscope that decomposes our return series into components of different
frequency so this is why we tend to consider that results obtained for H through the wavelet estimator are being
more accurate.
The GPH test (1983)
• Semi-parametric approach to obtain an estimate of the fractional differencing parameter d
based on the slope of the spectral density function around the frequency ξ=0
Periodogram (estimator of the spectral
density) of x at a frequency ξ
2
1
) (
2
1
x x e
T
t
T
t
it
÷
¿
=
ç
t
I (ξ) =
• Geweke, J. and S. Porter-Hudak(1983) proposed as an estimate of the OLS estimator of d
from the regression:
ì
ì
ç
ç e d a I + ÷ = )]
2
( ln[sin
ˆ
)] ( ln[
2
, λ= 1,…..,v
the bandwidth v is chosen such that for · ÷ T · ÷ v 0 ÷
T
v
but
Geweke and Porter-Hudak consider that the power of T has to be within (0.5,0.6). In our test we have
considered:
45 , 0
T
5 , 0
T
55 , 0
T
75 , 0
T
8 , 0
T
V =
Maximum likelihood estimates for ARFIMA model
• In the present paper we have used the MLE implemented based on the approximate maximum
likelihood algorithm of Haslett and Raftery (1989) in R. If the estimated d is significantly greater than
zero, we consider it an evidence of the presence of long-memory.

• For testing the existence of long memory we have selected indexes around the world trying to
compare return series in mature markets (US, UK, Germany, France, Japan) with emerging
markets (Romania, Poland and the BRIC countries)

• For the data series (1997 – 2008) we have first established the length as being 2 (for the wavelet
transform performed by the soft) and then we have transformed it in return series through:


• For testing and comparing we have selected mainly, daily returns
• Stationarity test were run in Eviews and long memory tests and estimation procedures were run
in R
) ln (ln * 100
1 ÷
÷ =
t t t
x x µ
n
Data used
Is there evidence of long memory in the return
time series?
S&P 500 daily return series
Null Hypothesis: SP500DAY has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=25)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -47.60859 0.0000
Test critical
values: 1% level -3.962531
5% level -3.412005
10% level -3.127909
GPH 0.45 0.5 0.55 0.75 0.8
d 0.032 -0.0906 0.082143 -0.05057 -0.03568
tstat sd (d=0) 0.258 -0.97417 0.973893 -1.25906 -1.09493
tstat asd (d=0) 0.231 -0.82632 0.935704 -1.31442 -1.12114
ARFIMA (0,d,0) mle Value
d 4.583E-05
ADF KPSS
R/S Hurst Exponent Diagnostic: 0.4834943
Wavelet estimator for H: 0.4108623
Null Hypothesis: SP500DAY is stationary
Exogenous: Constant
Bandwidth: 22 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.289248
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
FTSE100 daily return series
Null Hypothesis: FTSE100DAY has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 2 (Automatic based on SIC, MAXLAG=25)
t-Statistic Prob.*
Augmented Dickey-Fuller test
statistic -29.58352 0.0000
Test critical values: 1% level -3.962535
5% level -3.412007
10% level -3.127911
GPH\d=0 0.45 0.5 0.55 0.75 0.8
d 0.0449066 -0.024836 -0.01847 -0.06293 -0.03258
tstat sd 0.4092198 -0.2552364 -0.20357 -1.47644 -0.91725
tstat asd 0.3206718 -0.2265169 -0.21044 -1.63558 -1.02361
ARFIMA (0,d,0) mle Value
d 4.583E-05
ADF KPSS
R/S Hurst Exponent Diagnostic: 0.5587119
Wavelet estimator for H: 0.3972144
Null Hypothesis: FTSE100DAY is stationary
Exogenous: Constant
Bandwidth: 17 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.279658
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
Null Hypothesis: BETFIDAYP has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=21)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -29.22895 0.0000
Test critical
values: 1% level -3.967044
5% level -3.414212
10% level -3.129218
GPH\d=0 0.45 0.5 0.55 0.75 0.8
d 0.1112765 0.188012 0.26837 0.105296 0.11638
tstat sd 0.9531621 1.830524 2.329716 2.023603 2.708848
tstat asd 0.6536144 1.39642 2.44605 2.069451 2.717905
ARFIMA (0,d,0) mle Value
d 0.07864
BET-FI daily return series
ADF KPSS
R/S Hurst Exponent Diagnostic: 0.6177791
Wavelet estimator for H: 0.6394731
Null Hypothesis: BETFIDAYP is stationary
Exogenous: Constant
Bandwidth: 7 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.443275
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
BOVESPA daily return series
ADF KPSS
Null Hypothesis: BOVESPADAY has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=25)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -44.17009 0.0000
Test critical
values: 1% level -3.962531
5% level -3.412005
10% level -3.127909
GPH\d=0 0.45 0.5 0.55 0.75 0.8
d 0.055495 0.124207 0.138253 -0.0504179 -0.03275
tstat sd 0.4703172 1.199346 1.650381 -1.505468 -1.09286
tstat asd 0.3962822 1.132831 1.574859 -1.310459 -1.02899
ARFIMA (0,d,0) mle Value
d 0.0003773
R/S Hurst Exponent Diagnostic: 0.5681442
Wavelet estimator for H: 0.5579485
Null Hypothesis: BOVESPADAY is stationary
Exogenous: Constant
Bandwidth: 16 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.414950
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
RTS daily return series
ADF KPSS
Null Hypothesis: RTSDAY has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=25)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -42.96666 0.0000
Test critical
values: 1% level -3.962531
5% level -3.412005
10% level -3.127909
GPH\d=0 0.45 0.5 0.55 0.75 0.8
d -0.1386964 -0.11365 -0.05167 -0.0030198 0.028219
tstat sd -1.185368 -1.29636 -0.67683 -0.0828427 0.909179
tstat asd -0.990412 -1.03654 -0.58861 -0.0784892 0.886579
ARFIMA (0,d,0) mle\d=0 Value
d 0.03032
R/S Hurst Exponent Diagnostic: 0.543887
Wavelet estimator for H: 0.531688
Null Hypothesis: RTSDAY is stationary
Exogenous: Constant
Bandwidth: 1 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.083395
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
SENSEX daily return series
ADF KPSS
Null Hypothesis: SENSEXDAY has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=25)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -41.41383 0.0000
Test critical values: 1% level -3.962531
5% level -3.412005
10% level -3.127909
GPH\d=0 0.45 0.5 0.55 0.75 0.8
d 0.0759082 0.009728 0.031772 -0.0129993 0.013071
tstat sd 0.5596446 0.078911 0.34705 -0.3247642 0.402457
tstat asd 0.5420498 0.088728 0.361916 -0.3378765 0.410681
ARFIMA (0,d,0) mle\d=0 Value
d 0.04682
R/S Hurst Exponent Diagnostic: 0.568345
Wavelet estimator for H: 0.525448
Null Hypothesis: SENSEXDAY is stationary
Exogenous: Constant
Bandwidth: 10 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.509953
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
Hang Seng daily return series
ADF KPSS
Null Hypothesis: HANGSENGDAY has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic based on SIC, MAXLAG=25)
t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -45.92523 0.0000
Test critical
values: 1% level -3.962531
5% level -3.412005
10% level -3.127909
GPH\d=0 0.45 0.5 0.55 0.75 0.8
d 0.131589 0.051335 0.039096 0.0278982 0.019198
tstat sd 0.896207 0.469499 0.48214 0.7254383 0.60606
tstat asd 0.939662 0.468203 0.445347 0.7251295 0.603159
ARFIMA (0,d,0) mle Value
d 4.583E-05
R/S Hurst Exponent Diagnostic: 0.528084
Wavelet estimator for H: 0.495059
Null Hypothesis: HANGSENGDAY is stationary
Exogenous: Constant
Bandwidth: 5 (Newey-West using Bartlett kernel)
LM-Stat.
Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.499391
Asymptotic critical values*: 1% level 0.739000
5% level 0.463000
10% level 0.347000
Index daily H value via R/S
S&P 500 0.48
FTSE100 0.5587
CAC40 0.4738
DAX 0.5189
NIKKEI 225 0.5045
WIG 0.593
BET 0.4232
BET C 0.6306
BET FI 0.6187
BOVESPA 0.5681
RTS 0.5439
SENSEX 0.5683
HANG SENG 0.5281
Index daily H value via Wavelet estimator
S&P 500 0.4109
FTSE100 0.3072
CAC40 0.4161
DAX 0.4957
NIKKEI 225 0.4927
WIG 0.4786
BET 0.5337
BET C 0.5894
BET FI 0.6395
BOVESPA 0.5579
RTS 0.5317
SENSEX 0.5254
HANG SENG 0.4951
Comparison between indices - Hurst
BRIC
Conclusions
• Using a range of test and estimation procedures we have investigated whether stock returns
exhibit long memory

• Our results come to increase a bit the idea that emerging markets have a weak form of long
memory as resulted in case of Russia and India or a stronger form like discovered in case of
Romania (BET-FI), China and Brazil. Mature markets, in which we include US & UK among
Germany, France show mixed evidence

• We have tested for long memory the return series for BRIC countries indices

Why?

Because it is important to see is there is some kind of correlation between distant observations
in these markets as emerging markets are of great interest to potential investors first taking into
account their returns and second because they can be used in case of portfolio diversification as
emerging market countries have low correlation with mature markets.

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