
 Start

2876
 Prefix

This absence of cointegration implies that each
interest rate series follows its own set of fundamentals and that the development of
error correction models is not warranted, as it is not likely to improve forecasting
performance. This evidence is in contrast to that of
 Exact

Mougoue (1992)
 Suffix

for a system of
short term interest rates (who finds a cointegrating relationship) but is in agreement
with that of Diebold et al. (1994) for a system of foreign exchange rates (no evidence
of a cointegrating relationship).
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3025
 Prefix

rate series follows its own set of fundamentals and that the development of
error correction models is not warranted, as it is not likely to improve forecasting
performance. This evidence is in contrast to that of Mougoue (1992) for a system of
short term interest rates (who finds a cointegrating relationship) but is in agreement
with that of
 Exact

Diebold et al. (1994)
 Suffix

for a system of foreign exchange rates (no evidence
of a cointegrating relationship).
This study extends the existing literature by allowing deviations from
equilibrium in the system of long term international interest rates to follow a
fractionally integrated process.
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4662
 Prefix

Despite its significant
persistence in the short run, the effect of a shock to the system eventually dissipates,
so that an equilibrium relationship among the system's variables prevails in the long
run. Fractional cointegration, which uses the notion of fractional differencing
suggested by
 Exact

Granger and Joyeux (1980) and Hosking (1981),
 Suffix

was first proposed by
Engle and Granger (1987). It has been applied by Cheung and Lai (1993) and Baillie
and Bollerslev (1994) among others. The authors of the latter study found the system
of seven daily foreign exchange rates to be fractionally cointegrated, contrary to the
conclusions of the study by Diebold et al., which was based
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4740
 Prefix

significant
persistence in the short run, the effect of a shock to the system eventually dissipates,
so that an equilibrium relationship among the system's variables prevails in the long
run. Fractional cointegration, which uses the notion of fractional differencing
suggested by Granger and Joyeux (1980) and Hosking (1981), was first proposed by
 Exact

Engle and Granger (1987).
 Suffix

It has been applied by Cheung and Lai (1993) and Baillie
and Bollerslev (1994) among others. The authors of the latter study found the system
of seven daily foreign exchange rates to be fractionally cointegrated, contrary to the
conclusions of the study by Diebold et al., which was based on standard
cointegration methodology.
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4797
 Prefix

Fractional cointegration, which uses the notion of fractional differencing
suggested by Granger and Joyeux (1980) and Hosking (1981), was first proposed by
Engle and Granger (1987). It has been applied by
 Exact

Cheung and Lai (1993) and Baillie and Bollerslev (1994)
 Suffix

among others. The authors of the latter study found the system
of seven daily foreign exchange rates to be fractionally cointegrated, contrary to the
conclusions of the study by Diebold et al., which was based on standard
cointegration methodology.
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6572
 Prefix

A system of time series yt=1ty,2ty,...,nty{}is said to
be cointegrated of order I(d,b) if the linear combination tz=αty, called the error
correction term, is I(d−b) with b>0. Under the general hypothesis of cointegration
of order I(d,b) with b>0,
 Exact

Cheung and Lai (1993)
 Suffix

showed that the least squares
estimate of the cointegrating vector is consistent and converges at the rate ObT() as
opposed to the rate of OT() in standard cointegration analysis in which d=b=1.
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8025
 Prefix

The error correction term is subjected to a unit root test
in the second step to determine its order of integration. The unit root test employed
allows for a fractional exponent in the differencing process of the error correction
term.
 Exact

Granger (1986)
 Suffix

showed that the fractionally cointegrated system has an error
representation of the form
3ΨL()1−L()
d
yt=−γ1−1−L()
b
[]1−L()
d−b
zt+cL()tε(1)
where ΨL() is a matrix polynomial in the lag operator L with Ψ0() being the
identity matrix, cL() is a finite lag polynomial, and tε is a white noise error term.
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10265
 Prefix

The stochastic process y is both stationary and invertible if all roots of Φ(L)
and Θ(L) lie outside the unit circle and d<1
2
. The process is nonstationary for
d≥1
2
, as it possesses infinite variance, i.e. see
 Exact

Granger and Joyeux (1980).
 Suffix

ARFIMA series exhibit both long term dependence and short memory. The
effect of the differencing parameter d on observations widely separated in time
decays hyperbolically as the lag increases, thus describing the highorder
correlation structure of the series.
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10860
 Prefix

At the same time, the temporal effects of the
autoregressive and moving average parameters, Φ(L) and Θ(L), decay
exponentially, describing the loworder correlation structure of the series.
Assuming that −1
2
<d<1
2
and d≠0,
 Exact

Hosking (1981)
 Suffix

showed that the correlation
function, ρ(⋅), of an ARFIMA process is proportional to 2d−1j as j→∞.
Consequently, the autocorrelations of the ARFIMA process decay hyperbolically to
zero as j→∞, which is contrary to the faster, geometric decay of a stationary ARMA
process.
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12071
 Prefix

The existence of a fractional order of integration can be determined by testing
for the statistical significance of the sample differencing parameter d, which is also
interpreted as the long memory parameter. To estimate d and perform hypothesis
5testing, we employ the semiparametric procedure suggested by
 Exact

Geweke and PorterHudak (1983).
 Suffix

They obtain an estimate of d based on the slope of the spectral
density function around the angular frequency ξ=0. More specifically, let I(ξ) be
the periodogram of y at frequency ξ defined by
I(ξ) =
1
2πT
eitξ
t=1
T
∑(yt−y)
2
.(4)
Then the spectral regression is defined by
lnI(λξ){} = 0β + 1βln sin2
ξλ
2
+ λη, λ=1,...,ν(5)
where λξ=
2π
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12847
 Prefix

T
()λ=0,...,T−1 denotes the Fourier frequencies of the sample, T is
the number of observations, and ν = g(T) << 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 (5) provides an estimate of d.
 Exact

Geweke and PorterHudak (1983)
 Suffix

prove consistency and asymptotic normality for
d<0, while Robinson (1990) proves consistency for 0<d<1
2
. Hassler (1993) proves
consistency and asymptotic normality in the case of Gaussian ARMA innovations in
(1).
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12945
 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 (5) 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 0<d<1
2
. Hassler (1993) proves
consistency and asymptotic normality in the case of Gaussian ARMA innovations in
(1). The spectral regression estimator is not 1/ 2T consistent and will converge at a
slower rate.
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12999
 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 (5) provides an estimate of d.
Geweke and PorterHudak (1983) prove consistency and asymptotic normality for
d<0, while Robinson (1990) proves consistency for 0<d<1
2
.
 Exact

Hassler (1993)
 Suffix

proves
consistency and asymptotic normality in the case of Gaussian ARMA innovations in
(1). The spectral regression estimator is not 1/ 2T consistent and will converge at a
slower rate.
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14307
 Prefix

They are obtained from the
International Financial Statistics data tape of the International Monetary Fund.
B. Fractional Integration and Cointegration Analysis
Before we proceed with our analysis, we briefly describe the results obtained
by DKL. By means of DickeyFuller
 Exact

(Dickey and Fuller (1981), Fuller (1976)),
 Suffix

PhillipsPerron (Phillips (1987), Phillips and Perron (1988)), KPSS (Kwiatkowski,
Phillips, Schmidt, and Shin (1992)) and Bayesian (Sims (1988)) unit root tests, DKL
established that each interest rate series is I1().
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14371
 Prefix

Fractional Integration and Cointegration Analysis
Before we proceed with our analysis, we briefly describe the results obtained
by DKL. By means of DickeyFuller (Dickey and Fuller (1981), Fuller (1976)), PhillipsPerron
 Exact

(Phillips (1987),
 Suffix

Phillips and Perron (1988)), KPSS (Kwiatkowski,
Phillips, Schmidt, and Shin (1992)) and Bayesian (Sims (1988)) unit root tests, DKL
established that each interest rate series is I1().
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14499
 Prefix

By means of DickeyFuller (Dickey and Fuller (1981), Fuller (1976)), PhillipsPerron (Phillips (1987), Phillips and Perron (1988)), KPSS (Kwiatkowski,
Phillips, Schmidt, and Shin (1992)) and Bayesian
 Exact

(Sims (1988))
 Suffix

unit root tests, DKL
established that each interest rate series is I1(). They then applied the Johansen
(Johansen and Juselius (1990)) and Stock & Watson (1988) cointegration methods and
established the absence of any common nonstationary components in the system of
long term interest rates.
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14633
 Prefix

By means of DickeyFuller (Dickey and Fuller (1981), Fuller (1976)), PhillipsPerron (Phillips (1987), Phillips and Perron (1988)), KPSS (Kwiatkowski,
Phillips, Schmidt, and Shin (1992)) and Bayesian (Sims (1988)) unit root tests, DKL
established that each interest rate series is I1(). They then applied the Johansen
 Exact

(Johansen and Juselius (1990)) and Stock & Watson (1988)
 Suffix

cointegration methods and
established the absence of any common nonstationary components in the system of
long term interest rates. We have reproduced the results obtained by DKL.
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16077
 Prefix

This choice is made in light of
the recommended choice by Geweke and PorterHudak, based on forecasting
experiments, and the test performance of simulation experiments conducted by
 Exact

Cheung and Lai (1993).
 Suffix

Table 1 reports the empirical estimates for the fractional
differencing parameter ̃d=1−d as well as the test results regarding its statistical
significance based on the GPH test.
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18197
 Prefix

In this case, the error correction term follows a fractionally integrated
process and the system's variables form a fractionally cointegrated system. It must
be noted that, by simulation methods,
 Exact

Diebold and Rudebusch (1991)
 Suffix

found that
standard unit root tests have very low power against fractional alternatives. Cheung
and Lai (1993) found similar evidence when the unit root null hypothesis for the
error correction term is tested against fractional alternatives.
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18330
 Prefix

In this case, the error correction term follows a fractionally integrated
process and the system's variables form a fractionally cointegrated system. It must
be noted that, by simulation methods, Diebold and Rudebusch (1991) found that
standard unit root tests have very low power against fractional alternatives.
 Exact

Cheung and Lai (1993)
 Suffix

found similar evidence when the unit root null hypothesis for the
error correction term is tested against fractional alternatives.
The hypothesis of fractional cointegration requires testing for fractional
integration in the error correction term.
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28371
 Prefix

Conclusions and Implications
We reexamined the long run dynamics of a system of long term interest rates
of the U.S., Canada, Germany, the U.K., and Japan by allowing deviations from
equilibrium to follow a fractionally integrated process (fractional cointegration).
Contrary to previous evidence by
 Exact

DeGennaro et al. (1994),
 Suffix

which was based on
cointegration tests allowing for only integer orders of integration in the error
13correction term, we find that the long term interest rates form a fractionally
cointegrated system.
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29424
 Prefix

of the long run behavior
of the various subsystems of interest rates identifies a strong comovement between
the Canadian and U.S. interest rates, which appears to be robust with respect to the
dimension of the system. This is hardly surprising considering the strong ties
between the two economies. Our evidence is consistent with that of
 Exact

Baillie and Bollerslev (1994)
 Suffix

that a fractional cointegrating relationship exists among daily
exchange rates for seven major currencies.
Our findings have several implications concerning modeling and forecasting
of long term interest rates.
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31042
 Prefix

The impact of long memory is, by definition, spread over
a lengthy period, so that any improvement in forecasting accuracy may only be
apparent in the very long run.
14Future research should extend the framework of fractional cointegration to a
system of international short term interest rates. Even though
 Exact

Mougoue (1992)
 Suffix

found
that such a system exhibits cointegration, utilizing a more appropriate VAR
specification in the Johansen procedure might overturn his original conclusion.
Also, the possibility that nonlinear cointegration might exist should also be
addressed.
15Notes
1 Some authors refer to a process as a long memory process for all d≠0.
2 A high R2 in the cointegrating r
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