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1747
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, whether an ISLM based
structure or a much more elaborate framework, will contain a number of economic
variables which have been empirically identified as possessing fractional dynamics,
or elements of strong persistence, in their time series representation. The model of
fractionally integrated timeseries developed by Granger and Joyeux (1980) and
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Hosking (1981)
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allows for a fractional, as opposed to an integer, exponent in the
differencing process of the time series. This avoids the 'knifeedge' unit root
distinction while permitting a modelled series to exhibit the persistence, or 'long
memory,' which characterizes many macroeconomic timeseries.
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This avoids the 'knifeedge' unit root
distinction while permitting a modelled series to exhibit the persistence, or 'long
memory,' which characterizes many macroeconomic timeseries. For instance,
fractionally integrated output series have been identified by Diebold and Rudebusch
(1989) and
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Sowell (1992a).
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Such persistence is also evident in consumption (Diebold
and Rudebusch (1991)), interest rates (Shea (1991)), and inflation rates (Baillie,
Chung, and Tieslau (1996), Hassler and Wolters (1995), Baum, Barkoulas, and
Caglayan (1997)).
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For instance,
fractionally integrated output series have been identified by Diebold and Rudebusch
(1989) and Sowell (1992a). Such persistence is also evident in consumption (Diebold
and Rudebusch (1991)), interest rates
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(Shea (1991)), and
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inflation rates (Baillie,
Chung, and Tieslau (1996), Hassler and Wolters (1995), Baum, Barkoulas, and
Caglayan (1997)).
The importance of measures of money in a macroeconomic modelling
framework led PorterHudak (1990) to examine M1, M2 and M3 aggregates for
fractional integration.
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Such persistence is also evident in consumption (Diebold
and Rudebusch (1991)), interest rates (Shea (1991)), and inflation rates (Baillie,
Chung, and Tieslau (1996), Hassler and Wolters (1995), Baum, Barkoulas, and
Caglayan (1997)).
The importance of measures of money in a macroeconomic modelling
framework led
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PorterHudak (1990) to
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examine M1, M2 and M3 aggregates for
fractional integration. The latter study provides the motivation for this paper, in
which we extend PorterHudak's study of fractional integration in the monetary
aggregates in several important ways in order to provide comprehensive evidence
on the nature of fractional dynamic behavior in these series.
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paper, in
which we extend PorterHudak's study of fractional integration in the monetary
aggregates in several important ways in order to provide comprehensive evidence
on the nature of fractional dynamic behavior in these series. More specifically, we
1test for fractional integration, using the spectral regression method developed by
Geweke and
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PorterHudak (1983),
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in both simplesum and Divisia monetary
aggregates, monetary base, money multipliers, and velocity series. Given clear
evidence of fractional integration in the aggregates, we subsequently try to identify
which components of the monetary aggregates might be responsible for fractional
integration and therefore evaluate Granger's (1980) aggregation hypothesi
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5279
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The stochastic process
yt 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, 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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For d∈0. 5,1[) the process is mean reverting, even though it is not
covariance stationary, as there is no longrun impact of an innovation on future
values of the process.
Geweke and
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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 ξ=0.
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7005
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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 (3) provides an estimate of d.
Geweke and
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PorterHudak (1983)
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prove consistency and asymptotic normality for
d<0, while Robinson (1995) and Hassler (1993) prove consistency and asymptotic
normality for d∈0, 0. 5() in the case of Gaussian ARMA innovations in (1).
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7091
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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 (3) provides an estimate of d.
Geweke and PorterHudak (1983) prove consistency and asymptotic normality for
d<0, while
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Robinson (1995) and Hassler (1993)
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prove consistency and asymptotic
normality for d∈0, 0. 5() in the case of Gaussian ARMA innovations in (1).
Other authors have used maximum likelihood methods (i.e., the exact
maximum likelihood method proposed by Sowell (1992b)) or the approximate
frequency domain maximum likelihood method proposed by Fox and Taqqu (1986)),
which simultaneously estima
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Geweke and PorterHudak (1983) prove consistency and asymptotic normality for
d<0, while Robinson (1995) and Hassler (1993) prove consistency and asymptotic
normality for d∈0, 0. 5() in the case of Gaussian ARMA innovations in (1).
Other authors have used maximum likelihood methods (i.e., the exact
maximum likelihood method proposed by
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Sowell (1992b))
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or the approximate
frequency domain maximum likelihood method proposed by Fox and Taqqu (1986)),
which simultaneously estimate both the shortmemory and longmemory
parameters of the model.
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The unitroot hypothesis in the
growth rates of the simplesum monetary indices is decidedly rejected and evidence
of fractional dynamics with long memory features is established. If we compare the
range of these estimates to those estimated by
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PorterHudak (1990)
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over the 19591986 period, we find values that are broadly comparable.2
1 We also applied the PhillipsPerron (PP, 1988) and Kwiatkowski, Phillips, Schmidt, and Shin
(KPSS, 1992) unitroot tests to the growth rates of the monetary series.
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of the
sample period.3
3 Subsample estimates are not reported here, but are available upon request from the authors.
6Analysis of Components of the Monetary Aggregates
Given the presence of a fractional exponent in the differencing process for the
monetary aggregates, we now attempt to determine the sources of fractional
dynamics. One explanation, attributed to
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Granger (1980),
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is that a persistent process
can arise from the aggregation of constituent processes each of which has short
memory. Granger (1980) showed that if a time series ty is the sum of an infinite
number of independent firstorder Markov processes which have equal variances
and whose autoregressive parameters are drawn independently from a beta
distributio
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the Monetary Aggregates
Given the presence of a fractional exponent in the differencing process for the
monetary aggregates, we now attempt to determine the sources of fractional
dynamics. One explanation, attributed to Granger (1980), is that a persistent process
can arise from the aggregation of constituent processes each of which has short
memory.
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Granger (1980)
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showed that if a time series ty is the sum of an infinite
number of independent firstorder Markov processes which have equal variances
and whose autoregressive parameters are drawn independently from a beta
distribution with support 0,1(), then the aggregated series is asymptotically
fractionally integrated with d<0. 5.
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18909
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The time series properties behavior of the velocity of money in
the U.S. has attracted a great deal of attention in the literature given its implications
for the monetarist position. Gould and
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Nelson (1974),
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Nelson and Plosser (1982),
and Haraf (1986) conclude that money velocity contains a unit root. A similar
conclusion is reached by Serletis (1995), even after allowing for the possibility of a
9onetime break in the intercept and the slope of the trend function at an unknown
point in time.
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18960
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The time series properties behavior of the velocity of money in
the U.S. has attracted a great deal of attention in the literature given its implications
for the monetarist position. Gould and Nelson (1974), Nelson and Plosser (1982),
and
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Haraf (1986)
 Suffix

conclude that money velocity contains a unit root. A similar
conclusion is reached by Serletis (1995), even after allowing for the possibility of a
9onetime break in the intercept and the slope of the trend function at an unknown
point in time.
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19073
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The time series properties behavior of the velocity of money in
the U.S. has attracted a great deal of attention in the literature given its implications
for the monetarist position. Gould and Nelson (1974), Nelson and Plosser (1982),
and Haraf (1986) conclude that money velocity contains a unit root. A similar
conclusion is reached by
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Serletis (1995),
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even after allowing for the possibility of a
9onetime break in the intercept and the slope of the trend function at an unknown
point in time.
Table 4 reports the fractionalexponent estimates for the growth rates of both
simplesum and Divisia velocities.
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Since long memory represents nonlinear dependence in the
first moment of the distribution and hence a potentially predictable component in
the series dynamics, the possibility of improved forecasting via the estimation of an
ARFIMA model arises.
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PorterHudak (1990)
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found superior outofsample
forecasting performance of an ARFIMA model for the M1 aggregate versus a
benchmark ARIMA model. Given the substantial fractional exponent in the
differencing process in our series, similar improvements in forecasting accuracy
may be expected to result from the estimation of an appropriate ARFIMA model for
our data series.
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