- 3
- Baillie, R. T. and T. Bollerslev (1994), Cointegration, Fractional Cointegration and

Total in-text references: 2- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=4797
- 7
- Cheung Y. and K. Lai (1993), A fractional cointegration analysis of purchasing power parity, Journal of Business and Economic Statistics, 11, 103-112.

Total in-text references: 4- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=16077
- Prefix
- This choice is made in light of the recommended choice by Geweke and Porter-Hudak, 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.

- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=4797
- 8
- DeGennaro, R., R. Kunkel, and J. Lee (1994), Modeling international long-term interest rates, Financial Review, 29, 577-597.

Total in-text references: 1- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=28371
- 9
- Dickey, D.A. and W.A. Fuller (1981), Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 49, 1057-1072.

Total in-text references: 1- In-text reference with the coordinate start=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 Dickey-Fuller
- Exact
- (Dickey and Fuller (1981), Fuller (1976)),
- Suffix
- 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().

- In-text reference with the coordinate start=14307
- 10
- Diebold, F. X., J. Gardeazabal and K. Yilmaz (1994), On Cointegration and Exchange

Total in-text references: 1- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=3025
- 12
- Diebold, F. X. and G. D. Rudebusch (1991), On the power of Dickey-Fuller tests against fractional alternatives, Economics Letters, 35, 155-160.

Total in-text references: 1- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=18197
- 13
- Engle, R. and C.W.J. Granger (1987), Cointegration and Error-Correction:

Total in-text references: 1- In-text reference with the coordinate start=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.

- In-text reference with the coordinate start=4740
- 15
- Fuller W. (1976), Introduction to Statistical Time Series, New York: John Wiley & Sons.

Total in-text references: 1- In-text reference with the coordinate start=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 Dickey-Fuller
- Exact
- (Dickey and Fuller (1981), Fuller (1976)),
- Suffix
- 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().

- In-text reference with the coordinate start=14307
- 16
- Geweke J. and S. Porter-Hudak (1983), The Estimation and Application of Long

Total in-text references: 2- In-text reference with the coordinate start=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 semi-parametric procedure suggested by
- Exact
- Geweke and Porter-Hudak (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

- In-text reference with the coordinate start=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 Porter-Hudak (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).

- In-text reference with the coordinate start=12071
- 18
- Granger, C. W. J. (1986), Developments in the study of cointegrated economic variables, Oxford Bulletin of Economics and Statistics, 48, 213-228. -17-

Total in-text references: 1- In-text reference with the coordinate 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.

- In-text reference with the coordinate start=8025
- 19
- Granger, C. W. J. and R. Joyeux (1980), An introduction to long-memory time series models and fractional differencing, Journal of Time Series Analysis, 1, 15-39.

Total in-text references: 2- In-text reference with the coordinate start=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

- In-text reference with the coordinate start=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 high-order correlation structure of the series.

- In-text reference with the coordinate start=4662
- 21
- Hassler, U. (1993), Regression of spectral estimators with fractionally integrated time series, Journal of Time Series Analysis, 14, 369-380.

Total in-text references: 1- In-text reference with the coordinate 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 Porter-Hudak (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.

- In-text reference with the coordinate start=12999
- 22
- Hosking, J. R. M. (1981), Fractional Differencing, Biometrika, 68, 165-176.

Total in-text references: 2- In-text reference with the coordinate start=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

- In-text reference with the coordinate 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 low-order 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.

- In-text reference with the coordinate start=4662
- 23
- Johansen, S. and K. Juselius (1990), Maximum Likelihood Estimation and Inference on Cointegration-With Application to the Demand for Money, Oxford Bulletin of Economics and Statistics, 52, 169-210.

Total in-text references: 1- In-text reference with the coordinate start=14633
- Prefix
- 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 (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.

- In-text reference with the coordinate start=14633
- 25
- Mougoue, Mbodja (1992), The term structure of interest rates as a cointegrated system: Empirical evidence from the Eurocurrency market, Journal of Financial Research, 15, 285-296.

Total in-text references: 2- In-text reference with the coordinate 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).

- In-text reference with the coordinate start=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

- In-text reference with the coordinate start=2876
- 26
- Phillips, P. C. B. (1987), Time Series Regression with a Unit Root, Econometrica, 55, 277-301. _______ and P. Perron (1988), Testing for a Unit Root in Time Series Regression, Biometrika, 75, 335-346.

Total in-text references: 1- In-text reference with the coordinate 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 Dickey-Fuller (Dickey and Fuller (1981), Fuller (1976)), Phillips-Perron
- 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().

- In-text reference with the coordinate start=14371
- 27
- Robinson, P. (1990), Time series with strong dependence, Advances in Econometrics, 6th World Congress. Cambridge: Cambridge University Press.

Total in-text references: 1- In-text reference with the coordinate 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 Porter-Hudak (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.

- In-text reference with the coordinate start=12945
- 28
- Sims, C. A. (1988), Bayesian Skepticism on Unit Root Econometrics, Journal of

Total in-text references: 1- In-text reference with the coordinate start=14499
- Prefix
- 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
- 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.

- In-text reference with the coordinate start=14499
- 30
- Stock J. H. and M. W. Watson (1988), Testing for common trends, Journal of the

Total in-text references: 1- In-text reference with the coordinate start=14633
- Prefix
- 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 (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.

- In-text reference with the coordinate start=14633