Many time series in diverse fields have been found to exhibit long memory. This paper analyzes the behavior of some of the most used tests for long memory: the R/S or rescaled R/S, the GPH (Geweke and Porter-Hudak) and the DFA (Detrended Fluctuation Analysis). Some of these tests exhibit size distortions in small-samples. It is well known that the bootstrap procedure may correct this fact. In this paper, size and power for those tests, for finite samples and different distributions such as normal, uniform and log-normal are investigated. In the case of short memory process, such as AR, MA and ARCH and long memory such as ARFIMA, p-values are calculated using the post-blackening, moving block bootstrap. The Monte Carlo studies suggest that the bootstrap critical values perform better. The results are applied to financial return time series.Long memory, bootstrap, p-value, size correction, Monte Carlo

Pilar Grau-Carles

English

Research Papers in Economics

Tests of long memory, a bootstrap approach
Pilar Grau-Carles¤
Preliminary draft
Abstract
Many time series in diverse ¯elds have been found to exhibit long
memory. This paper analyzes the behaviour of some of the most used tests
of long memory: the R/S analysis, the modi¯ed R/S, the GPH (Geweke
and Porter-Hudak) test and the DFA (Detrended Fluctuation Analysis).
Some of these tests exhibit size distortions in small samples. It is well
known that the bootstrap procedure may correct this fact. In this paper,
size and power for those tests, for ¯nite samples and di®erent distributions,
such as normal, uniform and lognormal are investigated. In the case of
short memory processes such as AR, MA and ARCH and long memory
such ARFIMA, p-values are calculated using the post-blackening moving
block bootstrap. The Monte Carlo study suggests that the bootstrap
critical values perform better. The results are applied to ¯nancial return
time series.
1 Introduction
Time series with long memory appear in many contexts, for example in ¯nan-
cial economics, networks tra±c, hydrology, cardiac dynamics, meteorology, etc.
Long memory or long-range dependence is characterized by hyperbollically de-
caying autocovariance function, by spectral density that tends to in¯nity as the
frequencies tend to zero and by the self-similarity of aggregated summands.
The intensity of this phenomena can be measured either by a parameter
d, used as di®erencing parameter or by the parameter H, that is a scaling
parameter. Both parameters are related, in the case of ¯nite variance processes
by H = d + 1
2, and in the case of in¯nite variance processes by H = d + 1
®
(Taquu et al., 1998).
Several test of long-range dependence are available in the literature and
some of them are described in detail in Beran (1994). In particular the R/S
analysis, the modi¯ed R/S, the GPH (Geweke and Porter-Hudak) and the DFA
(Detrended Fluctuation Analysis) tests are investigated in this paper. As it was
¤Departamento de Econom¶ ³a Aplicada, Universidad Rey Juan Carlos, Paseo de los Ar-
tilleros s/n, 28032 Madrid, e-mail:grau@fcjs.urjc.es
1found in Cheung (1993), the modi¯ed R/S and the GPH su®er for size distor-
tion. Andersson and Gredenho® (1997) implement a bootstrap method, in order
to size adjust fractional integration tests. Bootstrap is a technique that allows
the simulation of the probability distribution of any statistic. The key idea is
to resample form the original data to create replicate data sets, from which the
empirical probability distribution of interest can be found. In the case of de-
pendent data, the resampling should preserve the temporal statistical structure
of the original series. Two are the most popular bootstrapping dependent data
approaches, the Model Based Resampling and the Moving Block Bootstrap. We
will use a strategy intermediate between them, called Post-blackening Moving
Block Bootstrap that is explained in Davison and Hinkley (1997).
2 Tests of long memory
The oldest and best-known method for detecting long memory is the R/S anal-
ysis. This method, proposed by Mandelbrot and Wallis (1968) and based on
previous hydrological analysis of Hurst (1951), allows the calculation of the self-
similarity parameter H. This parameter measures the intensity of long-range
dependence in a time series. For a time series fxtg, t = 1;:::;T we de¯ne the
range Rn
Rn =
(
max
1·i·n
i X
t=1
(xt ¡ x) ¡ min
1·i·n
i X
t=1
(xt ¡ x)
)
(1)
where x is the sample mean of the time series. If the range is rescaled with
the sample standard deviation S, the R/S statistics asymptotically follows the
relation
(R=S)t / CtH (2)
The value of H is generally obtained from the linear regression over a sample
of growing temporal horizons (s = t1;t2;:::;T)
ln(R=S)s = ln(C) + H ln(s) (3)
An estimated value of H = 1=2 means that the process has no memory, but
H 6= 1=2 would mean that the process has long memory.
More recently Lo (1991) discusses the lack of robustness of the R/S statistic
in the presence of short memory or heteroskedasticity. Lo suggests the modi¯ed
rescaled range statistics, henceforth MRR, which replaces the denominator S,
the standard deviation, by a consistent estimator of the square root of the
variance of the partial sum of x.
QT =
(
max
1·i·T
i X
t=1
(xt ¡ x) ¡ min
1·i·T
i X
t=1
(xt ¡ x)
)
=sT(q) (4)
where
2sT(q) =
8
<
:
1
T
T X
i=1
(xi ¡ x)
2 +
2
T
q X
j=1
¿j(q)
0
@
T X
i=j+1
(xi ¡ x)(xi¡j ¡ x)
1
A
9
=
;
1=2
(5)
and also ¿j(q) = 1 ¡
j
q+1 with q < n
Lo derives the limiting distribution of T¡1=2QT under the null of no memory,
and shows that the modi¯ed rescaled range statistics is robust to short-range
dependence. Critical values of the distribution are tabulated in Lo (1991) table
II.
In a recent paper Teverlosky, Taqqu and Willinger (1999) show that MRR
test tends to reject the null hypothesis of no long-range dependence when the
series is in fact long dependent and that the choice of the truncation lag q is
crucial.
The other method is a semi-parametric procedure to obtain an estimate of
the fractional di®erencing parameter d. This technique, proposed by Geweke
and Porter-Hudak (1983), henceforth GPH, is based on the slope of the spectral
density function around the angular frequency w = 0. The spectral regression
is de¯ned by
lnfI (w¸)g = a + bln
n
4sin
2
³w¸
2
´o
+ n¸ ¸ = I;:::;v (6)
where I (w¸) is the periodogram of the time series at the frequencies w¸ =
2¼¸
T with ¸ = 1;:::;(T¡1)=2, T is the number of observations and v is the number
of Fourier frequencies included in the spectral regression. The least square
estimate of the slope coe±cient provides an estimation of d. The theoretical
error variance is ¼2=6 and allows the construction of the t-statistics for the
fractional di®erencing parameter d. A mayor issue on the application of this
method is the choice of v.
The last method we have test that measures the long-range dependence is
the Detrended Fluctuation Analysis (DFA) proposed by Peng et al. (1994) and
improved in Viswanathan et al. (1997). The advantage of DFA over Hurst
analysis is that avoids spurious detection of apparent long-range correlation
that is an artefact of non-stationarities. The method can be summarized as
follows. First, the integrate time series y(t0) is obtained, y(t0) =
Pt
0
T=1 x(t).
Next the integrate series is divided into non-overlapping intervals, containing
each interval m data. In each interval a least squared line is ¯tted to the data.
The y coordinate of the straight line segments is denoted by ym(t0). Next the
root of the mean square °uctuation of the integrated and detrended time series
is calculated
F(m) =
v u
u
t 1
T
T X
t0=1
[y (t0) ¡ ym(t0)]
2 (7)
3This calculation is repeated over all intervals. A linear relationship on a
double log graph of F(m) and the interval size m indicates the presence of a
power law scaling. If there is no correlation or only short correlation F(m) /
m1=2, but if there is long-range power-law correlations then F(m) / m® with
® 6= 1=2.
3 Bootstrap testing
The bootstrap technique can be used for estimation of the small-sample dis-
tribution of a statistics. Introduced by Efron (1979), the bootstrap procedure
enables correction of size distortions. For details of bootstrap tests see Davidson
and MacKinnon (1996).
The original test, designed for iid observations, fails for dependent obser-
vations. The moving block bootstrap and model based resampling methods
perform better for short range dependence. Short and long memory processes
will be examined using the post-blackened moving block bootstrap method dis-
cussed in Davison and Hinkley (1997) and studied by Srinivas and Srinivasan
(2000), which is an intermediate approach between both and appears to capture
the dependence structure of the data, even using a small number of bootstrap
replications. The idea that underlies the block resampling is that if block are
long enough, the original dependence will be preserved in the resampled series.
The procedure works as follows for a given time series fxtg;t = 1;:::;T:
Step 1. Compute the tests statistics (Lo, GPH, Hurst and DFA) and obtain ^ ¿.
Step 2. "Pre-white" the time series ¯tting an AR(p) model with a suitable large
number of lags of the time series and obtain the estimated residuals et
and the centered residuals et¡¹ e. The order of the autorregresive model
will be estimated using the Schwartz's criterion.
Step 3. Resample blocks of the centered residuals from the estimated model
using the moving block bootstrap to generate B bootstrap samples of
them.
Step 4. "Post-black" the resampled centered residuals using the estimated pa-
rameters of the AR model to generate B bootstrap samples of x denoted
xb.
Step 5. For each bootstrapped sample compute the statistics ¿b.
Step 6. Compute the statistics p-value. For a two-side test is de¯ned by
p¤(^ ¿) =
1
B
B X
b=1
I(j¿bj ¸ j^ ¿j)
where I(¢) equals one if the inequality is satis¯ed and zero otherwise.
The null is rejected when the selected signi¯cance level exceeds p¤(^ ¿).
44 Monte Carlo study
For the Monte Carlo study 1000 replicated series of each model will be used.
Each series will be tested for long-memory using the tests described in section
2, for the no long-memory null hypothesis. In all the cases T +100 observations
are generated and only the T last are used to reduce the e®ect of initial values.
The block size has been set in 5.
4.1 Normal, uniform and lognormal distributions
Results of the tests when the data follows uniform, normal and lognormal distri-
butions are reported in Table 1. The simulation result shows that all the tests
perform correctly and do not show signi¯cant deviations from the nominal size.
Table 1: Rejection percentage of the nominal 5% fractional inte-
gration test when the data follows uniform, normal and lognormal
distributions of length T.
MRR GPH R/S DFA 1
Uniform
T=100 4.1 4.3 4.8 3.8
T=200 5.0 5.0 4.7 4.5
Normal
T=100 5.7 4.6 5.3 5.6
T=200 4.5 4.4 4.1 4.4
Lognormal
T=100 4.3 5.2 4.2 4.5
T=200 6.0 4.3 4.4 4.5
4.2 AR and MA speci¯cation
The following processes will be simulated
xt = Áxt¡1 + at
and
xt = at + µat¡1
where at » IID N(0;1). The AR and MA parameters are set equal to §1, §4
and §9. Table 2 reports the sensitivity of the empirical size to AR and MA
components for time series of length T=100. The simulation results suggests
that MRR and GPH for both AR and MA models perform well, but rejection
frequencies are signi¯cantly larger than the nominal signi¯cance level when µ =
1MRR: modi¯ed rescaled range test GPH: Geweke-Porter-Hudak test; R/S: Hurst test
DFA: detrended °uctuation analysis. Under the null hypothesis of no fractional integration,
the 95% con¯dence interval is 5% § 1:4%
5¡0:9. These results are slightly better than the ones obtained in Andersson
and Gedenho® (1997) using the original no parametric bootstrap. The R/S and
DFA tests show similar behavior, they are always very conservative in the sense
that they tend to reject the null less frequently than the nominal.
MRR and GPH post-blackened bootstrap give exact tests because the esti-
mated sizes of the tests coincide with the nominal, whereas R/S and DFA are
very conservative in the case of moving average and autoregressive models and
small samples.
Table 2: Rejection percentage of the nominal 5% fractional inte-
gration test when the data follows AR and MA processes of length
T=100.
MRR GPH R/S DFA
Á AR process
-0.9 4.7 4.2 0.6 0.6
-0.4 5.0 3.8 0.4 0.6
-0.1 5.1 4.8 0.3 0.4
0.1 5.3 4.5 0.9 2.7
0.4 5.3 3.7 0.6 1.4
0.9 5.9 4.8 2.4 1.5
µ MA process
-0.9 7.5 17.9 0.1 0.0
-0.4 6.3 3.8 0.1 0.1
-0.1 5.2 4.8 0.3 0.2
0.1 5.0 4.6 0.9 0.4
0.4 5.6 5.1 0.5 0.3
0.9 5.9 5.1 0.3 0.2
4.3 ARCH speci¯cation
Now, the e®ect of heteroskedasticity of ARCH type is considered. The simulated
model is
xt = ut ut=t¡1 » N(0;ht)
ht = ®0 + ®1u2
t¡1
Results in Table 3 show that MRR and GPH are quite robust, but, again
R/S and DFA tests are conservative with a very small rejection rate, so the
post-blackening bootstrap test allows the exact estimation of the long-range
dependence for small samples in the presence of heteroskedasticity.
6Table 3: Rejection percentage of the nominal 5% fractional inte-
gration test when the data follows an ARCH (1) model of length
T=100.
MRR GPH R/S DFA
® ARCH process
0.3 6.1 4.2 0.5 0.7
0.5 5.9 4.0 0.6 0.2
0.8 7.3 5.0 0.5 0.5
4.4 ARFIMA speci¯cation
Fractionally integrated white noise or ARFIMA (0,d,0) is also tested. Data are
generated following Hosking (1981):
(1 ¡ B)dxt = at
where at » IID N(0;1) and the fractional di®erencing parameter is d is set
equal to §0:45, §0:25 and §0:05. Results in table 4 show that MRR and GPH
have good power except for d = ¡0:05 that leads to small rejection frequencies.
The R/S and DFA tests have good power for persistent processes, or positive
d parameter, but the performance of these tests is really poor for antipersitent
series with a very low rejection rates.
Table 4: Rejection percentage of the nominal 5% fractional inte-
gration test when the data follows an ARFIMA (0,d,0) processes
of length T=100.
MRR GPH R/S DFA
d AR process
-0.45 7.4 9.0 2.6 3.8
-0.25 7.7 7.9 3.4 4.7
-0.05 6.0 4.0 5.3 7.2
0.05 6.8 4.0 7.5 8.3
0.25 14.3 8.5 21.8 12.4
0.45 22.6 14.4 23.0 19.7
5 Application to ¯nancial data
Finally all four test are used to test long memory in time series of returns, abso-
lute returns and squared returns of two mayor daily stock indices, the Standard&
Poors 500 and the Dow Jones Average. Two samples of each index are analyzed,
one year of daily data, from 01/02/2003 to 31/12/2003 yielding 252 observations
and a longer series of ¯ve years of data, form 01/02/1999 to 31/12/2003, with
a total of 1256 observations. The returns are calculated using the logarithmic
7di®erenced data
rt = (ln(xt) ¡ ln(xt¡1)) ¤ 100
A common ¯nding in much of the empirical literature is that returns them-
selves contain little serial correlation, however, the absolute returns and their
power transformation present long memory, as discussed in Granger and Ding
(1993).
Results of the bootstrap p-values are given in Table 5 and 6.
Table 5: Point estimation and p-value results for test of long mem-
ory for the SP500.
MRR GPH R/S DFA
SP500 T=251
rt 1.1968 -0.0615 0.5984 0.3989
p-value 0.4030 0.6870 0.3940 0.7880
jrtj 0.3967 0.5746 0.6750 0.4723
p-value 0.0000 0.0010 0.2130 0.6579
r2
t 0.6814 0.5943 0.6432 0.5481
p-value 0.0010 0.0000 0.3360 0.3032
SP500 T=1255
rt 1.5529 0.0850 0.5131 0.4426
p-value 0.1010 0.2980 0.9000 0.8560
jrtj 0.4778 0.4699 0.7908 0.7499
p-value 0.0040 0.0000 0.0000 0.0000
r2
t 0.8106 0.3973 0.7744 0.7523
p-value 0.0340 0.0000 0.0000 0.0000
Table 6: Point estimation and p-value results for test of long mem-
ory for the Dow Jones.
MRR GPH R/S DFA
Dow Jones T=251
rt 1.2619 0.0016 0.6068 0.4318
p-value 0.2540 0.9350 0.4330 0.7090
jrtj 0.4219 0.4093 0.7057 0.5150
p-value 0.0000 0.0080 0.1940 0.6910
r2
t 0.7257 0.4574 0.6761 0.5407
p-value 0.0040 0.0060 0.3090 0.2910
Dow Jones T=1255
rt 1.3249 0.1205 0.5668 0.4858
p-value 0.2940 0.1380 0.5120 0.5510
jrtj 0.6348 0.4683 0.8027 0.7726
p-value 0.0000 0.0000 0.0000 0.0000
r2
t 1.0789 0.3062 0.7710 0.7544
8p-value 0.0050 0.0040 0.0020 0.0050
For the shorter time series of both indexes, the MRR and GPH found long
memory in absolute returns and squared returns series, but the R/S and DFA
do not allow the rejection of the null hypothesis of no long memory.
In the case of the series that covers ¯ve years, all of the tests ¯nd no memory
in returns and long memory in absolute and squared returns.
6 Conclusions
Post-blackening moving blocks bootstrap provides an e®ective method to im-
prove the test for long memory, especially for MRR and GPH tests and short
time series.
R/S and DFA tests can also bene¯ce from the bootstrap approach. They
should not be used to test long memory in short time series, but when applied
to longer series, p-values that allow hypothesis testing can be obtained.
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10

A critical look at Lo modi¯ed R/S statistic

An introduction to Bootstrap,

Bootstrap methods: Another Look at the Jackknife,

Bootstrap testing for fractional integration. Working papers

E.(1997) Quanti¯cation of DNA Patchiness using correlation measures,

Fractional di®erencing,

Long term memory in stock market prices,

Long term storage capacity of reservoirs,

Methods and their application. Cambridge:

Noah, Joseph and operational hydrology,

Post-blackening approach for modeling dependent annual stream°ows,

practical guide to heavy tails: Statistical techniques and applications, Birkhauser,

Statistics for Long Memory Processes, Chapman and Hall. London

Tests for fractional integration: A Monte Carlo Investigation,

The estimation and Appication of Long Memory Time Series Models,

The size distortion of bootstrap tests,

Test for long memory processes. A bootstrap approach

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Test for long memory processes. A bootstrap approach

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