long memory in stock

8/11/2019 Long memory in stock

1/23

Academy of Economic Studies - Bucharest Doctoral School of Finance and Banking

DOFIN

Long Mem ory in S tock Returns :Research over Markets

Supervisor : Professor Dr. Mois Alt r

MSc Student: Silvia Bardo Bucharest, July 2008


2/23

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 waveletestimator

Determining long memory by estimating fractional differencing parameter Geweke and Porter-Hudak test & Maximum Likelihood Estimate for anARFIMA process

Conclusions


3/23

Long Memory & Motivation

Long memory has important implications in financial markets because if it is discovered itcan be used to construct trading strategies.

Long memory or long range dependence means that the information from today is notimmediately 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 effecton future events

AND

Memory is the series property to depend on its own past realizations


4/23

Mathematic view: Long memory processes relates to autocorrelation

If a time series of data exhibits autocorrelation, a value from the data set x s at time t s iscorrelated with another value x s+z at time t s+z . For a long memory process autocorrelation

decays over time and the decay is slower than in a stationary process ( I(0) pro cess )

So, if a long memory process exhibits an autocorrelation function that is not consistent for aI(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


5/23

Literature review

Evidence of long memory was first brought up by E. Hurst in 1951 when, testing the behavior ofwater 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 inasset 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 thedegree 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 aninternational 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.


6/23

Steps Modeling long memory

A series x t follows an ARFIMA (p,d,q) process if:

t t L xd L L )(1)( 2,0~ iid t

where (L), (L) are the autoregressive and moving average polynomials, L is the lag, d isthe fractional differencing parameter, t is white noise.

For d w ithi n (0,0.5) , the ARFIMA process is said to exhibit long memory orlong range positive dependence

For d w ith in (-0.5, 0) , the process exhibits intermediate memory or long rangenegative dependence

For d w ith in [0.5, 1) the process is mean reverting and there is no long runimpact to future values of the process The process is short memory for d=0 corresponding to a standard ARMAprocess


7/23

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) orI(1) and the commonly used tests are ADF and KPSS

ADF

Null hypothesis: H 0: 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 beinappropriate 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


8/23

Testing stationarity

KPSSConsider x t ( t = 1, 2, , N) , as the observed return series for which we wish to teststationarity

The test decomposes the series into the sum of a random walk , a determinis t ic t rend and as ta t ionary er ror with the following linear regression model:

t t t x r t

The KPSS statistics:

T

t

t S l S T

122 1

andT

iit S

1

i 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 e t (t = 1, 2, , N ) are from theregression of x on an intercept and time trend.

Under the null hypothesis of level stationarity, the residuals e t are from a regression of x onintercept only.

Rejectio n of A DF and KPSS indic ates that the proc ess is d escrib ed by n either I(0) and I(1)pro cesses and that is pro bable better descr ibed by the fract ion al integrated al ternative (d isa no n-integer) .


9/23

Estimating long memory using R/S test

R/S test

Mandelbrot & Wallis (1969) method allows computing parameter H, which measures theintensity 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 (E m) and standard deviation (S m);

b) we subtract the sample mean Z i,m = X i,m E m for i = 1,..,m;

c) produce a time series taking form of W i,m = j,m where i = 1,, m

d) find the range R m = max{W 1,m ,., W n,m } min{ W 1,m ,., W n,m }

e) rescale the range R m by

i

j

Z 1

Sm

Rm

How do es th is proc edure rela tes to the Hurs t exponent?


10/23

Einstein discovered that the distance covered by a random variable is close related to thesquare root of time (Brownian motion)

5.0T k R , where R is the distance covered by the variable, k is a constant and T is thelength of the time.

Using R/S analysis, Hurst suggested that:

H mk S

R , where R/S is the rescaled range, m is the number of observations, k is theconstant and H is the Hurst exponent, can be applied to a bigger class of timeseries (generalized Brownian motion)

The Hurst exponent can be than found as:log (R/S)m= lo g k + H lo g m

H value Return t ime series

= 0.5 follow a random walk and are independent

(0,0.5)are anti-persistent, process covers only a smalldistance than in the random walk case

(0.5,1)are persistent series, process covers a biggerdistance than a random walk (long memory)

Estimating long memory using R/S test


11/23

Hurst exponent using wavelet spectral density

For computing the Hurst Exponent, the R wavelet estimator uses a discrete wavelet transformthen:

averages the squares of the coefficients of the transform, performs a linear regression on the logarithm of the average, versus the log of theparameter of the transform

The result provides an estimate for the Hurst exponent.

Wavele t t ransform behaves as a m icroscope tha t decomposes our re turn ser ies in to com ponents of d i fferen t

f requency so th is i s w hy w e tend to cons ider tha t resu l t s ob ta ined for H throu gh the w avele t es t imator a re be ing

mo re accurate.


12/23

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 =0

Periodogram (est imator of the spectraldens i ty ) of x at a frequency

2

1

)(2

1 x xe

T t

T

t

it

I () =

Geweke, J. and S. Porter-Hudak(1983) proposed as an estimate of the OLS estimator of d from the regression:

ed a I )]

2(ln[sin

)](ln[ 2 , = 1,..,v

the bandwidth v is chosen such that for T v 0T

vbut

Geweke and Porter-Hudak con sider that th e pow er of T has to be w ithin (0.5,0.6). In o ur test w e havecons idered:

45,0T 5,0T 55,0T 75,0T 8,0T V =


13/23

Maximum likelihood estimates for ARFIMA model

In the present paper we have used the MLE implemented based on the approximate maximumlikelihood 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.


14/23

For testing the existence of long memory we have selected indexes around the world trying tocompare 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 wavelettransform 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 runin R

)ln(ln*100 1t t t x x

n

Data used


15/23

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 criticalvalues: 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: 0.4834943Wavelet 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


16/23

FTSE100 daily return series

Null Hypothesis: FTSE100DAY has a unit root



t-Statistic Prob.*

Augmented Dickey-Fuller teststatistic -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


d 4.583E-05

ADF KPSS


Null Hypothesis: FTSE100DAY is stationary

Exogenous: Constant


LM-Stat.


Asymptotic critical values*: 1% level 0.739000

5% level 0.463000

10% level 0.347000


17/23

Null Hypothesis: BETFIDAYP has a unit root



t-Statistic Prob.*



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


d 0.07864

BET-FI daily return series

ADF KPSS


Null Hypothesis: BETFIDAYP is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000


18/23

BOVESPA daily return series

ADF KPSS

Null Hypothesis: BOVESPADAY has a unit root



t-Statistic Prob.*



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


d 0.0003773


Null Hypothesis: BOVESPADAY is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000


19/23

RTS daily return series

ADF KPSS

Null Hypothesis: RTSDAY has a unit root



t-Statistic Prob.*



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


LM-Stat.



5% level 0.463000

10% level 0.347000


20/23

SENSEX daily return series

ADF KPSS

Null Hypothesis: SENSEXDAY has a unit root



t-Statistic Prob.*


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


Null Hypothesis: SENSEXDAY is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000


21/23

Hang Seng daily return series

ADF KPSS

Null Hypothesis: HANGSENGDAY has a unit root



t-Statistic Prob.*



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


d 4.583E-05

R/S Hurst Exponent Diagnostic: 0.528084

Wavelet estimator for H: 0.495059

Null Hypothesis: HANGSENGDAY is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000


22/23

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


23/23

Conclusions

Using a range of test and estimation procedures we have investigated whether stock returnsexhibit long memory

Our results come to increase a bit the idea that emerging markets have a weak form of longmemory as resulted in case of Russia and India or a stronger form like discovered in case ofRomania (BET-FI), China and Brazil. Mature markets, in which we include US & UK amongGermany, 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 intoaccount their returns and second because they can be used in case of portfolio diversification asemerging market countries have low correlation with mature markets.

long memory in stock

Documents