Upload
selvaemimasaronrose
View
218
Download
0
Embed Size (px)
Citation preview
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
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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.
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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) .
8/11/2019 Long memory in stock
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?
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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.
8/11/2019 Long memory in stock
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 =
8/11/2019 Long memory in stock
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.
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
16/23
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 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
ARFIMA (0,d,0) mle Value
d 4.583E-05
ADF KPSS
R/S Hurst Exponent Diagnostic: 0.5587119Wavelet 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
8/11/2019 Long memory in stock
17/23
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 criticalvalues: 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: 0.6177791Wavelet 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
8/11/2019 Long memory in stock
18/23
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 criticalvalues: 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: 0.5681442Wavelet 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
8/11/2019 Long memory in stock
19/23
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 criticalvalues: 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
8/11/2019 Long memory in stock
20/23
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: 0.568345Wavelet 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
8/11/2019 Long memory in stock
21/23
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 criticalvalues: 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
8/11/2019 Long memory in stock
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
8/11/2019 Long memory in stock
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.