The 14 reference contexts in paper John Barkoulas, Christopher F. Baum (1996) “Fractional Differencing Modeling and Forecasting of Eurocurrency Deposit Rates” / RePEc:boc:bocoec:317

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The presence of fractional structure in asset prices raises issues regarding theoretical and econometric modeling of asset prices, statistical testing of pricing models, and pricing efficiency and rationality. Applications of long-memory analysis include
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Greene and Fielitz (1977), Lo (1991), and Barkoulas and Baum (1996)
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for U.S. stock prices; Cheung (1993a) for spot -2exchange rates; and Fang, Lai, and Lai (1994), and Barkoulas, Labys, and Onochie (1996) for futures prices. The overall evidence suggests that stochastic long memory is absent in stock market returns but it may be a feature of some spot and futures foreign currency rates.
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The presence of fractional structure in asset prices raises issues regarding theoretical and econometric modeling of asset prices, statistical testing of pricing models, and pricing efficiency and rationality. Applications of long-memory analysis include Greene and Fielitz (1977), Lo (1991), and Barkoulas and Baum (1996) for U.S. stock prices;
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Cheung (1993a)
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for spot -2exchange rates; and Fang, Lai, and Lai (1994), and Barkoulas, Labys, and Onochie (1996) for futures prices. The overall evidence suggests that stochastic long memory is absent in stock market returns but it may be a feature of some spot and futures foreign currency rates.
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The stochastic process ty 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, (⋅), 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 stochastic process ty 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,
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Hosking (1981)
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showed that the correlation function, (⋅), 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 long run impact of an innovation on future values of the process.
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Geweke and Porter-Hudak (1983)
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suggest a semi-parametric procedure to obtain an estimate of the fractional differencing parameter d based on the slope of the spectral density function around the angular frequency =0. More specifically, let I() be the periodogram of y at frequency defined by I() = 1 2T eit t=1 T ∑(yt−y ) 2 .(3) Then the spectral regression is defined by            + , =1,...,(4) ln
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1lnsin 2 2  where = 2 T (=0,...,T−1) denotes the Fourier frequencies of the sample, T is the number of observations, and = gT() << 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 (4) provides an estimate of d.
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Geweke and Porter-Hudak (1983)
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prove consistency and asymptotic normality for d<0, while Robinson (1990) proves consistency for d∈0,0.5(). Hassler (1993) proves consistency and asymptotic normality 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 (4) 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 d∈0,0.5(). Hassler (1993) proves consistency and asymptotic normality in the case of Gaussian ARMA innovations in (1). The spectral regression estimator -5is 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 (4) provides an estimate of d. Geweke and Porter-Hudak (1983) prove consistency and asymptotic normality for d<0, while Robinson (1990) proves consistency for d∈0,0.5().
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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 -5is not 1/2T consistent and will converge at a slower rate. The theoretical asymptotic variance of the spectral regression error term is known to be 2 6.
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In the frequency domain, long memory is indicated by the fact that the spectral density becomes unbounded as the frequency approaches zero; the series has power at low frequencies. The evidence of fractional structure in these returns series may not be robust to nonstationarities in the mean and short-term dependencies. Through extensive Monte Carlo simulations,
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Cheung (1993b)
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shows that the spectral regression test is robust to moderate ARMA components, ARCH effects, and shifts in the variance. However, possible biases of the spectral regression test against the no long memory null hypothesis may be caused by infrequent shifts in the mean of the process and large AR parameters (0.7 and higher), both of which bias the test toward detecting long memory.
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Experiment The discovery of fractional orders of integration suggests possibilities for constructing nonlinear econometric models for improved price forecasting performance, especially over longer forecasting horizons. An ARFIMA process incorporates this specific nonlinearity and represents a flexible and parsimonious way to model both the short- and long-term dynamical properties of the series.
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Granger and Joyeux (1980)
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discuss the forecasting potential of such nonlinear models and Geweke and PorterHudak (1983) confirm this by showing that ARFIMA models provide more reliable out-of-sample forecasts than do traditional procedures.
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The AR orders are selected on the basis of statistical significance of the coefficient estimates and Q statistics for serial dependence (the AR order chosen in each case is given in subsequent tables). A question arises as to the asymptotic properties of the AR parameter estimates in the second stage. Conditioning on the d estimate obtained in the first stage,
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Wright (1995)
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shows that the ARp() fitted by the Yule-Walker procedure to the ddifferenced series inherit the T-consistency of the semiparametric estimate of d. We forecast the Eurocurrency deposit rates by casting the fitted fractional-AR model in infinite autoregressive form, truncating the infinite autoregression at the beginning of the sample, and applying Wold's chain rule.
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We forecast the Eurocurrency deposit rates by casting the fitted fractional-AR model in infinite autoregressive form, truncating the infinite autoregression at the beginning of the sample, and applying Wold's chain rule.
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Ray (1993)
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uses a similar procedure to forecast IBM product revenues. The long-memory forecasts are compared to those generated by two standard linear models: an autoregressive model (AR), described earlier, and a random-walk-with-drift model (RW).
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This evidence accentuates the usefulness of long-memory models as forecast generating mechanisms for some Eurocurrency returns series, and casts doubt on the hypothesis of the weak form of market efficiency for longer horizons. It also contrasts with the failure of ARFIMA models to improve on the random walk model in out-of-sample forecasts of foreign exchange rates
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(Cheung (1993a)).
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V. Conclusions Using the spectral regression method, we find significant evidence of long-term stochastic memory in the returns series (yield changes) of threeand six-month Eurodeposits denominated in German marks, Swiss francs, and Japanese yen, as well as three-month Eurodeposits denominated in Canadian dollars.
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We have established the practical usefulness of developing longmemory models for some Eurocurrency returns series. These results could potentially be improved in future research via estimation of ARFIMA models based on maximum likelihood methods (e.g.
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Sowell (1992).
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These procedures avoid the two-stage estimation process followed in this paper by allowing for the simultaneous estimation of the long and short memory components of the series. Given the sample size of our series, however, implementing these procedures will be very computationally burdensome, as closed-form solutions for these one-stage estimators do not exist.
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