
 Start

4697
 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 Granger and Joyeux (1980) and
 Exact

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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 Start

4751
 Prefix

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
Engle and
 Exact

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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 Start

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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 Start

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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 Start

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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 Start

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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 Start

14321
 Prefix

B. 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
 Exact

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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 Start

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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 Start

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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