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2315
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The presence of fractional structure in asset prices raises issues
regarding theoretical and econometric modeling of asset prices, statistical
testing of pricing models, and pricing efficiency and rationality. Applications
of longmemory analysis include
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Greene and Fielitz (1977), Lo (1991), and Barkoulas and Baum (1996)
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for U.S. stock prices; Cheung (1993a) for spot
2exchange rates; and Fang, Lai, and Lai (1994), and Barkoulas, Labys, and
Onochie (1996) for futures prices. The overall evidence suggests that stochastic
long memory is absent in stock market returns but it may be a feature of some
spot and futures foreign currency rates.
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2405
 Prefix

The presence of fractional structure in asset prices raises issues
regarding theoretical and econometric modeling of asset prices, statistical
testing of pricing models, and pricing efficiency and rationality. Applications
of longmemory analysis include Greene and Fielitz (1977), Lo (1991), and Barkoulas and Baum (1996) for U.S. stock prices;
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Cheung (1993a)
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for spot
2exchange rates; and Fang, Lai, and Lai (1994), and Barkoulas, Labys, and
Onochie (1996) for futures prices. The overall evidence suggests that stochastic
long memory is absent in stock market returns but it may be a feature of some
spot and futures foreign currency rates.
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5191
 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, (⋅), 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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5249
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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,
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Hosking (1981)
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showed that the correlation function, (⋅), 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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6051
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The process exhibits short memory for d=0,
corresponding to stationary and invertible ARMA modeling. 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 on future values of the process.
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Geweke and PorterHudak (1983)
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suggest 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 =0.
More specifically, let I() be the periodogram of y at frequency defined by
I() =
1
2T
eit
t=1
T
∑(yt−y )
2
.(3)
Then the spectral regression is defined by
+ , =1,...,(4)
ln
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6836
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1lnsin
2
2
where =
2
T
(=0,...,T−1) denotes the Fourier frequencies of the sample,
T is the number of observations, and = 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 OLS estimate 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 ARMA innovations in (1).
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6924
 Prefix

Assuming that
T→∞
limgT()=∞,
T→∞
lim
gT()
T
=0, and
T→∞
lim
lnT()2
gT()
=0,
the negative of the OLS estimate of the slope coefficient in (4) provides an
estimate of d. Geweke and PorterHudak (1983) prove consistency and
asymptotic normality for d<0, while
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Robinson (1990)
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proves consistency for
d∈0,0.5(). Hassler (1993) proves consistency and asymptotic normality in the
case of Gaussian ARMA innovations in (1). The spectral regression estimator
5is not 1/2T consistent and will converge at a slower rate.
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6973
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Assuming that
T→∞
limgT()=∞,
T→∞
lim
gT()
T
=0, and
T→∞
lim
lnT()2
gT()
=0,
the negative of the OLS estimate 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().
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Hassler (1993)
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proves consistency and asymptotic normality in the
case of Gaussian ARMA innovations in (1). The spectral regression estimator
5is not 1/2T consistent and will converge at a slower rate. The theoretical
asymptotic variance of the spectral regression error term is known to be
2
6.
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11327
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In the frequency domain, long memory is indicated
by the fact that the spectral density becomes unbounded as the frequency
approaches zero; the series has power at low frequencies.
The evidence of fractional structure in these returns series may not be
robust to nonstationarities in the mean and shortterm dependencies.
Through extensive Monte Carlo simulations,
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Cheung (1993b)
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shows that the
spectral regression test is robust to moderate ARMA components, ARCH
effects, and shifts in the variance. However, possible biases of the spectral
regression test against the no long memory null hypothesis may be caused by
infrequent shifts in the mean of the process and large AR parameters (0.7 and
higher), both of which bias the test toward detecting long memory.
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13495
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Experiment
The discovery of fractional orders of integration suggests possibilities
for constructing nonlinear econometric models for improved price
forecasting performance, especially over longer forecasting horizons. An
ARFIMA process incorporates this specific nonlinearity and represents a
flexible and parsimonious way to model both the short and longterm
dynamical properties of the series.
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Granger and Joyeux (1980)
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discuss the
forecasting potential of such nonlinear models and Geweke and PorterHudak (1983) confirm this by showing that ARFIMA models provide more
reliable outofsample forecasts than do traditional procedures.
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14911
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The AR orders are 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 ddifferenced series inherit the Tconsistency of the semiparametric estimate
of d. We forecast the Eurocurrency deposit rates 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.
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15288
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We forecast the Eurocurrency deposit rates 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.
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Ray (1993)
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uses a similar procedure to forecast IBM product revenues.
The longmemory forecasts are compared to those generated by two
standard linear models: an autoregressive model (AR), described earlier, and
a randomwalkwithdrift model (RW).
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19435
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This evidence accentuates the usefulness of longmemory
models as forecast generating mechanisms for some Eurocurrency returns
series, and casts doubt on the hypothesis of the weak form of market
efficiency for longer horizons. It also contrasts with the failure of ARFIMA
models to improve on the random walk model in outofsample forecasts of
foreign exchange rates
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(Cheung (1993a)).
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V. Conclusions
Using the spectral regression method, we find significant evidence of
longterm stochastic memory in the returns series (yield changes) of threeand sixmonth Eurodeposits denominated in German marks, Swiss francs,
and Japanese yen, as well as threemonth Eurodeposits denominated in
Canadian dollars.
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20620
 Prefix

We have established the practical usefulness of developing longmemory models for some Eurocurrency returns series. These results could
potentially be improved in future research via estimation of ARFIMA
models based on maximum likelihood methods (e.g.
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Sowell (1992).
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These
procedures avoid the twostage estimation process followed in this paper by
allowing for the simultaneous estimation of the long and short memory
components of the series. Given the sample size of our series, however,
implementing these procedures will be very computationally burdensome, as
closedform solutions for these onestage estimators do not exist.
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