The 9 reference contexts in paper John Barkoulas, Christopher F. Baum, Gurkan S. Oguz (1996) “Fractional Cointegration Analysis of Long Term International Interest Rates” / RePEc:boc:bocoec:315

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    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
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    Hosking (1981),
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    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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    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
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    Granger (1987).
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    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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    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.
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    Granger (1986)
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    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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    At the same time, the temporal effects of the autoregressive and moving average parameters, Φ(L) and Θ(L), decay exponentially, describing the low-order correlation structure of the series. Assuming that −1 2 <d<1 2 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−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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    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 Porter-Hudak (1983) prove consistency and asymptotic normality for d<0, while
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    Robinson (1990)
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    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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    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 Porter-Hudak (1983) prove consistency and asymptotic normality for d<0, while Robinson (1990) proves consistency for 0<d<1 2 .
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    Hassler (1993)
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    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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    B. Fractional Integration and Cointegration Analysis Before we proceed with our analysis, we briefly describe the results obtained by DKL. By means of Dickey-Fuller (Dickey and
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    Fuller (1981), Fuller (1976)),
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    Phillips-Perron (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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    Fractional Integration and Cointegration Analysis Before we proceed with our analysis, we briefly describe the results obtained by DKL. By means of Dickey-Fuller (Dickey and Fuller (1981), Fuller (1976)), Phillips-Perron
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    (Phillips (1987),
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    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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    By means of Dickey-Fuller (Dickey and Fuller (1981), Fuller (1976)), Phillips-Perron (Phillips (1987), Phillips and Perron (1988)), KPSS (Kwiatkowski, Phillips, Schmidt, and Shin (1992)) and Bayesian
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    (Sims (1988))
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    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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