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1940
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Introduction
In this study we investigate the presence of long memory and its
usefulness as a forecast generating mechanism for the U.S. monetary aggregates.
The fractional differencing model employed is the autoregressive fractionally
integrated moving average (ARFIMA) type introduced by Granger and Joyeux
(1980), Hosking (1981), and
 Exact

Geweke and PorterHudak (1983).
 Suffix

Regarding
macroeconomic time series, evidence of fractional integration has been found in
output series (Diebold and Rudebusch (1989), Sowell (1992)), consumption
(Diebold and Rudebusch (1991)), and inflation rates (Baillie, Chung, and Tieslau
(1996), Hassler and Wolters (1995), Baum, Barkoulas, and Caglayan (1999)).
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2074
 Prefix

The fractional differencing model employed is the autoregressive fractionally
integrated moving average (ARFIMA) type introduced by Granger and Joyeux
(1980), Hosking (1981), and Geweke and PorterHudak (1983). Regarding
macroeconomic time series, evidence of fractional integration has been found in
output series
 Exact

(Diebold and Rudebusch (1989), Sowell (1992)),
 Suffix

consumption
(Diebold and Rudebusch (1991)), and inflation rates (Baillie, Chung, and Tieslau
(1996), Hassler and Wolters (1995), Baum, Barkoulas, and Caglayan (1999)).
PorterHudak (1990) reported evidence of long memory in simple sum monetary
aggregates while Barkoulas, Baum, and Caglayan (1999) extended similar
findings to components of simplesum monetary aggregates, divisia monetary
indices, t
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2133
 Prefix

The fractional differencing model employed is the autoregressive fractionally
integrated moving average (ARFIMA) type introduced by Granger and Joyeux
(1980), Hosking (1981), and Geweke and PorterHudak (1983). Regarding
macroeconomic time series, evidence of fractional integration has been found in
output series (Diebold and Rudebusch (1989), Sowell (1992)), consumption
 Exact

(Diebold and Rudebusch (1991)), and
 Suffix

inflation rates (Baillie, Chung, and Tieslau
(1996), Hassler and Wolters (1995), Baum, Barkoulas, and Caglayan (1999)).
PorterHudak (1990) reported evidence of long memory in simple sum monetary
aggregates while Barkoulas, Baum, and Caglayan (1999) extended similar
findings to components of simplesum monetary aggregates, divisia monetary
indices, the monetary base, and money multipliers.
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 Start

2220
 Prefix

Regarding
macroeconomic time series, evidence of fractional integration has been found in
output series (Diebold and Rudebusch (1989), Sowell (1992)), consumption
(Diebold and Rudebusch (1991)), and inflation rates (Baillie, Chung, and Tieslau
(1996),
 Exact

Hassler and Wolters (1995),
 Suffix

Baum, Barkoulas, and Caglayan (1999)).
PorterHudak (1990) reported evidence of long memory in simple sum monetary
aggregates while Barkoulas, Baum, and Caglayan (1999) extended similar
findings to components of simplesum monetary aggregates, divisia monetary
indices, the monetary base, and money multipliers.
 (check this in PDF content)

 Start

2287
 Prefix

Regarding
macroeconomic time series, evidence of fractional integration has been found in
output series (Diebold and Rudebusch (1989), Sowell (1992)), consumption
(Diebold and Rudebusch (1991)), and inflation rates (Baillie, Chung, and Tieslau
(1996), Hassler and Wolters (1995), Baum, Barkoulas, and Caglayan (1999)).
 Exact

PorterHudak (1990)
 Suffix

reported evidence of long memory in simple sum monetary
aggregates while Barkoulas, Baum, and Caglayan (1999) extended similar
findings to components of simplesum monetary aggregates, divisia monetary
indices, the monetary base, and money multipliers.
 (check this in PDF content)

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2890
 Prefix

Despite the evidence of long memory in macroeconomic series, there are
few applied studies in the literature regarding the predictive ability of ARFIMA
models.1 Such forecasting evaluation would serve as a test of model adequacy, in
discriminating among competing economic hypotheses, and be useful in guiding
policymaking decisions (see
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Fildes and Stekler (2002) and
 Suffix

references therein for
a recent review of issues regarding macroeconomic forecasting). Granger and
1 Guegman (1994) points out that, despite the fundamental interest in forecasting, very few
studies related to ARFIMA forecasts have been implemented.
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3070
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predictive ability of ARFIMA
models.1 Such forecasting evaluation would serve as a test of model adequacy, in
discriminating among competing economic hypotheses, and be useful in guiding
policymaking decisions (see Fildes and Stekler (2002) and references therein for
a recent review of issues regarding macroeconomic forecasting). Granger and
1
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Guegman (1994)
 Suffix

points out that, despite the fundamental interest in forecasting, very few
studies related to ARFIMA forecasts have been implemented.
Joyeux (1980) discuss the forecasting potential of fractional models.
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3288
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Granger and
1 Guegman (1994) points out that, despite the fundamental interest in forecasting, very few
studies related to ARFIMA forecasts have been implemented.
Joyeux (1980) discuss the forecasting potential of fractional models.
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Cheung (1993)
 Suffix

finds that ARFIMAgenerated forecasts fail to improve upon randomwalk
forecasts for foreign exchange rates. Lardic and Mignon (1996) however provide
evidence that fractional forecasts have better predictive accuracy in the short
term (relative to randomwalk and structural model forecasts) for three major
currencies.
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3409
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1 Guegman (1994) points out that, despite the fundamental interest in forecasting, very few
studies related to ARFIMA forecasts have been implemented.
Joyeux (1980) discuss the forecasting potential of fractional models. Cheung (1993) finds that ARFIMAgenerated forecasts fail to improve upon randomwalk
forecasts for foreign exchange rates.
 Exact

Lardic and Mignon (1996)
 Suffix

however provide
evidence that fractional forecasts have better predictive accuracy in the short
term (relative to randomwalk and structural model forecasts) for three major
currencies. Franses and Ooms (1997) report that ARFIMA models fail to generate
superior forecasts over competing models for the U.
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Lardic and Mignon (1996) however provide
evidence that fractional forecasts have better predictive accuracy in the short
term (relative to randomwalk and structural model forecasts) for three major
currencies.
 Exact

Franses and Ooms (1997)
 Suffix

report that ARFIMA models fail to generate
superior forecasts over competing models for the U.K. inflation rate. Through
extensive Monte Carlo analysis, Ray (1993a) and Crato and Ray (1996) find that
simple ARMA models generally outperform or provide competitive forecasts
compared to ARFIMA models.
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3792
 Prefix

Lardic and Mignon (1996) however provide
evidence that fractional forecasts have better predictive accuracy in the short
term (relative to randomwalk and structural model forecasts) for three major
currencies. Franses and Ooms (1997) report that ARFIMA models fail to generate
superior forecasts over competing models for the U.K. inflation rate. Through
extensive Monte Carlo analysis,
 Exact

Ray (1993a) and Crato and Ray (1996)
 Suffix

find that
simple ARMA models generally outperform or provide competitive forecasts
compared to ARFIMA models. On the other hand, Ray (1993b) establishes that,
by certain criteria, a fractional model provides more accurate forecasts than
benchmark models for IBM product revenues.
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3956
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Through
extensive Monte Carlo analysis, Ray (1993a) and Crato and Ray (1996) find that
simple ARMA models generally outperform or provide competitive forecasts
compared to ARFIMA models. On the other hand,
 Exact

Ray (1993b)
 Suffix

establishes that,
by certain criteria, a fractional model provides more accurate forecasts than
benchmark models for IBM product revenues. Barkoulas and Baum (1997) show
that longmemory forecasts result in dramatic improvements in forecasting
accuracy, especially over longer horizons, relative to rival models for several
Eurocurrency deposit rates.
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4105
 Prefix

Through
extensive Monte Carlo analysis, Ray (1993a) and Crato and Ray (1996) find that
simple ARMA models generally outperform or provide competitive forecasts
compared to ARFIMA models. On the other hand, Ray (1993b) establishes that,
by certain criteria, a fractional model provides more accurate forecasts than
benchmark models for IBM product revenues.
 Exact

Barkoulas and Baum (1997)
 Suffix

show
that longmemory forecasts result in dramatic improvements in forecasting
accuracy, especially over longer horizons, relative to rival models for several
Eurocurrency deposit rates.
In this paper we investigate the ability of ARFIMAbased forecasts to
outperform benchmark linear forecasts on an outofsample basis for seasonally
adjusted U.
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6776
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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 said to exhibit longmemory behavior for ()1,0∈d. For d∈0.5,1[),
yt is nonstationary (having an infinite variance) but it is mean reverting.
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Robinson (1995a)
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proposes a Gaussian semiparametric estimator, GS
hereafter, of the selfsimilarity parameter H. Assume that the spectral density of
the time series, denoted by ()⋅f, behaves as
ξξ
H
fG
12
()~
− as
ξ→0+(2)
for ()∞∈,0G and ()1,0∈H.
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Given the GS estimates of d, we approximate the shortrun series
dynamics by fitting an AR model to the fractionally differenced series using BoxJenkins methods.2,3 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
2 All fractional forecasts are based on the GS estimates of the longmemory parameter.
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We forecast the monetary series by casting the
fitted fractionalAR model in infinite autoregressive form, truncating the infinite
autoregression at the beginning of the sample (thus setting data points before the
sample period equal to zero), and applying Wold's chain rule. A similar
procedure was followed by
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Diebold and Lindner (1996) to
 Suffix

forecast the real
interest rate and Ray (1993b) to forecast IBM product revenues.
The long memory forecasts are compared to those generated by a linear
AR model.4 Observations corresponding to the sample period starting in 1991:1
until the end of the sample are our test set (postprediction interval).5 We
consider 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, 21, and 24months
ahe
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12940
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We forecast the monetary series by casting the
fitted fractionalAR model in infinite autoregressive form, truncating the infinite
autoregression at the beginning of the sample (thus setting data points before the
sample period equal to zero), and applying Wold's chain rule. A similar
procedure was followed by Diebold and Lindner (1996) to forecast the real
interest rate and
 Exact

Ray (1993b) to
 Suffix

forecast IBM product revenues.
The long memory forecasts are compared to those generated by a linear
AR model.4 Observations corresponding to the sample period starting in 1991:1
until the end of the sample are our test set (postprediction interval).5 We
consider 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 18, 21, and 24months
ahead forecasting horizons.
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14572
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to the forecasting
4 In specifying the lag length for the linear AR model, we follow the same procedure as for the
choice of the shortmemory (AR) order of the fractional model.
5 We maintain a validation set of adequate size in order to effectively compare the outofsample
accuracy of competing forecasts for all prediction horizons.
6 See
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Lardic and Mignon (1996)
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for arguments on the comparative performance of fractional
forecasts over short and longerterm horizons.
horizon. However, the AR forecasts dominate the fractional forecasts for the
simple sum M3 aggregate.
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Even though the fractional forecasts result in
rather sizeable forecasting improvements over the benchmark forecasts, such
superiority is not generally statistically significant.
We also employ the forecasting encompassing testing approach for our
competing forecasts suggested by
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Clements and Hendry (1998).
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Let 1=ARFIMA
model, 2=AR model, =Eiforecast error for model 2,1=i, and =Dthe difference
between the forecasts from the two models. The forecast encompassing test is
based on running two regressions: the first involves regressing the forecast error
from the ARFIMA model on the difference of forecasts, i.e., ε+β+α=tttDE,111,1,
and the second involves the regression ε+β+α=tttDE,222,2.
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