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2157
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The presence of long memory in
asset returns contradicts the weak form of the market efficiency hypothesis,
which states that, conditioning on historical returns, future asset returns are
unpredictable.1
A number of studies have tested the longmemory hypothesis for stock
market returns. Using the rescaledrange (R/S) method,
 Exact

Greene and Fielitz (1977)
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report evidence of persistence in daily U.S. stock returns series. A
problem with the classical R/S method is that the distribution of its test
1 The existence of long memory in asset returns calls into question linear modeling and
invites the development of nonlinear pricing models at the theoretical level to account for
longmemory behavior.
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2580
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A
problem with the classical R/S method is that the distribution of its test
1 The existence of long memory in asset returns calls into question linear modeling and
invites the development of nonlinear pricing models at the theoretical level to account for
longmemory behavior.
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Mandelbrot (1971)
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observes that in the presence of long memory, the
arrival of new market information cannot be fully arbitraged away and martingale models of
asset prices cannot be obtained from arbitrage.
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3165
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In addition, pricing derivative securities with
martingale methods may not be appropriate if the underlying continuous stochastic processes
exhibit long memory. Statistical inferences concerning asset pricing models based on standard
testing procedures may not be appropriate in the presence of longmemory series
 Exact

(Yajima (1985)).
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statistic is not well defined and is sensitive to shortterm dependence and
heterogeneities of the underlying data generating process. These
dependencies bias the classical R/S test toward finding long memory too
frequently.
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3437
 Prefix

models based on standard
testing procedures may not be appropriate in the presence of longmemory series (Yajima (1985)). statistic is not well defined and is sensitive to shortterm dependence and
heterogeneities of the underlying data generating process. These
dependencies bias the classical R/S test toward finding long memory too
frequently.
 Exact

Lo (1991)
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developed a modified R/S method which addresses
these drawbacks of the classical R/S method. Using this variant of R/S
analysis, Lo (1991) finds no evidence to support the presence of long memory
in U.
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3594
 Prefix

These
dependencies bias the classical R/S test toward finding long memory too
frequently. Lo (1991) developed a modified R/S method which addresses
these drawbacks of the classical R/S method. Using this variant of R/S
analysis,
 Exact

Lo (1991)
 Suffix

finds no evidence to support the presence of long memory
in U.S. stock returns. Using both the modified R/S method and the spectral
regression method (described below), Cheung and Lai (1995) find no evidence
of persistence in several international stock returns series.
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3798
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Using this variant of R/S
analysis, Lo (1991) finds no evidence to support the presence of long memory
in U.S. stock returns. Using both the modified R/S method and the spectral
regression method (described below),
 Exact

Cheung and Lai (1995)
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find no evidence
of persistence in several international stock returns series. Crato (1994) reports
similar evidence for the stock returns series of the G7 countries using exact
maximum likelihood estimation.
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3912
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Using both the modified R/S method and the spectral
regression method (described below), Cheung and Lai (1995) find no evidence
of persistence in several international stock returns series.
 Exact

Crato (1994)
 Suffix

reports
similar evidence for the stock returns series of the G7 countries using exact
maximum likelihood estimation. The primary focus of these studies has been
the stochastic longmemory behavior of stock returns in major capital
markets.
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8401
 Prefix

The introduction of new financial
instruments, like warrants, options, commercial paper, etc. is currently under
way. There is no capital gains tax in Greece.
There has been limited research on the behavior of stocks traded on the
ASE.
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Papaioannou (1982, 1984)
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reports price dependencies in stock returns
for a period of at least six days. Panas (1990) provides evidence of weakform
efficiency for ten large Greek firms. Koutmos, Negakis, and Theodossiou
(1993) find that an exponential generalized ARCH model is an adequate
representation of volatility in weekly Greek stock returns.
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8520
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There has been limited research on the behavior of stocks traded on the
ASE. Papaioannou (1982, 1984) reports price dependencies in stock returns
for a period of at least six days.
 Exact

Panas (1990)
 Suffix

provides evidence of weakform
efficiency for ten large Greek firms. Koutmos, Negakis, and Theodossiou
(1993) find that an exponential generalized ARCH model is an adequate
representation of volatility in weekly Greek stock returns.
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8944
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Koutmos, Negakis, and Theodossiou
(1993) find that an exponential generalized ARCH model is an adequate
representation of volatility in weekly Greek stock returns. The intertemporal
relation between the U.S. and Greek stock markets is analyzed in
Theodossiou, Koutmos, and Negakis (1993).
 Exact

Barkoulas and Travlos (1996)
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test
whether Greek stock returns are characterized by deterministic nonlinear
structure (chaos).
In this paper, we test for the presence of fractional dynamics, or long
memory, in the returns series for the Greek stock market.
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11283
 Prefix

The stochastic process ty is both stationary and invertible if all roots of
Φ(L) and Θ(L) lie outside the unit circle and d<0. 5. The process is
nonstationary for d≥0. 5, as it possesses infinite variance, i.e. see
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Granger and Joyeux (1980).
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Assuming that d∈0, 0.5() and d≠0, Hosking (1981) showed that
the correlation function, ρ(⋅), of an ARFIMA process is proportional to 2d−1k
as k→∞. Consequently, the autocorrelations of the ARFIMA process decay
hyperbolically to zero as k→∞ which is contrary to the faster, geometric
decay of a stationary ARMA process.
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11344
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The stochastic process ty is both stationary and invertible if all roots of
Φ(L) and Θ(L) lie outside the unit circle and d<0. 5. The process is
nonstationary for d≥0. 5, as it possesses infinite variance, i.e. see Granger and Joyeux (1980). Assuming that d∈0, 0.5() and d≠0,
 Exact

Hosking (1981)
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showed that
the correlation function, ρ(⋅), of an ARFIMA process is proportional to 2d−1k
as k→∞. Consequently, the autocorrelations of the ARFIMA process decay
hyperbolically to zero as k→∞ which is contrary to the faster, geometric
decay of a stationary ARMA process.
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12196
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dependence.2 The process exhibits intermediate memory, or
longrange negative dependence for d∈−0. 5, 0() and short memory for d=0,
corresponding to a stationary and invertible ARMA model. For d∈0. 5, 1[) the
process is mean reverting, even though it is not covariance stationary, as there
is no long run impact of an innovation to future values of the process.
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Geweke and PorterHudak (1983)
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suggested a semiparametric
procedure to obtain an estimate of the fractional differencing parameter d
based on the slope of the spectral density function around the angular
frequency ξ=0.
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13138
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λξ=
2πλ
T
()λ=0,...,T−1 denotes the Fourier frequencies of the sample,
T is the number of observations, and ν = gT() << T is the number of Fourier
frequencies included in the spectral regression.
Assuming that
T→∞
limgT()=∞,
T→∞
lim
gT()
T
=0, and
T→∞
lim
lnT()2
gT()
=0,
the negative of the slope coefficient in (4) provides an estimate of d.
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Geweke and PorterHudak (1983)
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prove consistency and asymptotic normality for
d<0, while Robinson (1990) proves consistency for d∈0, 0.5(). Hassler (1993)
proves consistency and asymptotic normality in the case of Gaussian
innovations in (1).
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13235
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Assuming that
T→∞
limgT()=∞,
T→∞
lim
gT()
T
=0, and
T→∞
lim
lnT()2
gT()
=0,
the negative of the slope coefficient in (4) provides an estimate of d. Geweke and PorterHudak (1983) prove consistency and asymptotic normality for
d<0, while
 Exact

Robinson (1990)
 Suffix

proves consistency for d∈0, 0.5(). Hassler (1993)
proves consistency and asymptotic normality in the case of Gaussian
innovations in (1). The spectral regression estimator is not 1/ 2T consistent as it
converges at a slower rate.
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13292
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Assuming that
T→∞
limgT()=∞,
T→∞
lim
gT()
T
=0, and
T→∞
lim
lnT()2
gT()
=0,
the negative of the slope coefficient in (4) provides an estimate of d. Geweke and PorterHudak (1983) prove consistency and asymptotic normality for
d<0, while Robinson (1990) proves consistency for d∈0, 0.5().
 Exact

Hassler (1993)
 Suffix

proves consistency and asymptotic normality in the case of Gaussian
innovations in (1). The spectral regression estimator is not 1/ 2T consistent as it
converges at a slower rate. The theoretical variance of the error term in the
spectral regression is known to be
π2
6
.
3.
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13885
 Prefix

Data and Empirical Estimates
The data set consists of weekly stock returns based on the closing prices
of a valueweighted index comprised of the thirty most heavily traded stocks
(during the period 19881990) on the Athens Stock Exchange (ASE30)
developed by
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Travlos (1992).
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The sample period spans 01/07/1981 to
12/27/1990 for a total of 521 weekly observations. The period 01/07/1981 to
10/11/1989 is used for insample estimation with the remaining observations
used for outofsample forecasting.
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20404
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The nonlinear model construction suggested is that of an
ARFIMA process, which represents a flexible and parsimonious way to model
both the short and longterm dynamic properties of the series. Granger and
4 Through extensive Monte Carlo simulations,
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Cheung (1993) and
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Agiakloglou, Newbold,
and Wohar (1993) found the spectral regression test to be biased toward finding long memory
()d>0 in the presence of infrequent shifts in the mean of the process and large AR parameters
(0.7 and higher).
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21009
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We investigated the potential presence of these biasinducing data features in
the ASE30 returns series and found that neither a shift in mean nor strong shortterm dynamics
are responsible for detecting long memory in the Greek stock market.
10
Joyeux (1980) have discussed the forecasting potential of such nonlinear
models and
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Geweke and PorterHudak (1983)
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have confirmed this by
showing that ARFIMA models provide more reliable outofsample forecasts
than do traditional procedures. The possibility of speculative profits due to
superior longmemory forecasts would cast serious doubt on the basic tenet
of market efficiency: unpredictability of future returns.
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22462
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selected on the basis of statistical
significance of the coefficient estimates and Q statistics for serial dependence
(the AR order chosen in each case is given in subsequent tables). A question
arises as to the asymptotic properties of the AR parameter estimates in the
second stage. Conditioning on the d estimate obtained in the first stage,
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Wright (1995)
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shows that the ARp() fitted by the YuleWalker procedure to
the d differenced series inherit the δTconsistency of the semiparametric
estimate of d.
The Greek stock returns series is forecast by casting the fitted
fractionalAR model in infinite autoregressive form, truncating the infinite
autoregression at the beginning of the sample, and applying Wol
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22931
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The Greek stock returns series is forecast by casting the fitted
fractionalAR model in infinite autoregressive form, truncating the infinite
autoregression at the beginning of the sample, and applying Wold's chain
rule. A similar procedure was followed by
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Ray (1993) to
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forecast IBM product
revenues and Diebold and Lindner (1996) to forecast the real interest rate. The
11
longmemory forecasts are compared to those obtained by estimating two
standard linear models: a random walk (RW) model, as suggested by the
market efficiency hypothesis in its weak form, and an autoregressive (AR)
model fit to the ASE30 returns
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22985
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The Greek stock returns series is forecast by casting the fitted
fractionalAR model in infinite autoregressive form, truncating the infinite
autoregression at the beginning of the sample, and applying Wold's chain
rule. A similar procedure was followed by Ray (1993) to forecast IBM product
revenues and
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Diebold and Lindner (1996) to
 Suffix

forecast the real interest rate. The
11
longmemory forecasts are compared to those obtained by estimating two
standard linear models: a random walk (RW) model, as suggested by the
market efficiency hypothesis in its weak form, and an autoregressive (AR)
model fit to the ASE30 returns series according to the Akaike information
criterion (AIC).
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