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Any dynamic macroeconomic model, 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
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Granger and Joyeux (1980) and Hosking (1981)
 Suffix

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
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Diebold and Rudebusch (1989) and 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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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 Sowell (1992a). Such persistence is also evident in consumption
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(Diebold and Rudebusch (1991)),
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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 PorterHudak (1990) to examine M1, M2 and M3 aggregates for
fractional integration.
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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
 Exact

(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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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 (Shea (1991)), and inflation rates (Baillie,
Chung, and Tieslau (1996),
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Hassler and Wolters (1995),
 Suffix

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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2574
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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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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. More specifically, we
1test for fractional integration, using the spectral regression method developed by
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Geweke and 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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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
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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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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,
 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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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 longrun 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 ξ=0.
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of the
sample λξ
πλ
==λ
−
2
1
1
T
,...,, T is the number of observations, and ν = gT() <<
T2
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 (3) 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 (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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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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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
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Fox and Taqqu (1986)),
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which simultaneously estimate both the shortmemory and longmemory
parameters of the model. These estimation methods are computationally
burdensome, rely on the correct specification of the highfrequency (ARMA)
structure to obtain consistent parameter estimates, the final ARFIMA specification
chosen generally varies across different selection
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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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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
distribution with support 0,1(), then the aggregated series is asymptotically
fractionally integrated with d<0. 5.
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Granger and Ding (1996)
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extended the
aggregation argument to mixtures of Ijd() processes for a range of distributions for
dj; they also showed that other data generating mechanisms, like timevarying
coefficient models and possibly nonlinear models, can have the longmemory
property.
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However, a more
disaggregated data set may be needed in order to fully address the aggregation
argument.
8Analysis of Divisia Indices and Velocity Series
We subsequently test for a fractional integration order in an alternative set of
monetary aggregates: the Divisia indices
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(Thornton and Yue (1992)).
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The Divisia
monetary aggregates were proposed by Barnett et al. (1984) as superior to simplesum
aggregates which “implicitly view distant substitutes for money as perfect substitutes
for currency.” (1984, p.1051) Barnett et al. found that the Divisia aggregates
performed considerably better in terms of causality tests, tests of the structural
stability of money demand f
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However, a more
disaggregated data set may be needed in order to fully address the aggregation
argument.
8Analysis of Divisia Indices and Velocity Series
We subsequently test for a fractional integration order in an alternative set of
monetary aggregates: the Divisia indices (Thornton and Yue (1992)). The Divisia
monetary aggregates were proposed by
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Barnett et al. (1984)
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as superior to simplesum
aggregates which “implicitly view distant substitutes for money as perfect substitutes
for currency.” (1984, p.1051) Barnett et al. found that the Divisia aggregates
performed considerably better in terms of causality tests, tests of the structural
stability of money demand functions, and forecasting.
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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.
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Gould and Nelson (1974), Nelson and Plosser (1982), and Haraf (1986)
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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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Although
not every component of the simplesum aggregates exhibits long memory, the
overall evidence is substantial and robust in support of fractional monetary
dynamics with longmemory features. Our findings of fractional integration orders
between one and two (and statistically distinguishable from one and two) is contrary
to the conclusion reached by
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King et al. (1991) and Friedman and Kuttner (1992)
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that
nominal money balances are I2() processes. A shock to the growth rate of the
monetary series displays significant persistence, but it eventually dissipates. The
money velocity series are best characterized as unitroot processes.
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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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The nonlinear relations arising in this context deserve
further scrutiny. Finally, care must be exercised in interpreting results from
regressions involving the growth rates of monetary series.
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Tsay and Chung (1995)
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have shown the existence of spurious effects in regressions involving two
11independent long memory fractionally integrated processes whose orders of
integration sum up to a value greater than 0.5.
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monetary
series would be likely to trigger these effects.
12Appendix
All data series are seasonally adjusted, monthly observations obtained from
the Federal Reserve Bank of St Louis' FRED database, which contains series
originally published by the Board of Governors of the Federal Reserve System. The
Divisia aggregates series were originally published in
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Thornton and Yue (1992).
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For
the Divisia M1, M2, M3, and L series, the sample period is 1960:1 to 1992:12. The
sample period is 1959:1 to 1995:10 for the following series: simplesum M1, M2, M3,
L, currency in circulation, demand deposits, total checkable deposits, small time
deposits at commercial banks, small time deposits at thrift institutions, savings
deposits at comm
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