Computation of the forecast coefficients for multistep prediction of long-range dependent time series

Three different linear methods, called the truncation, type-II plug-in and type-II direct, for constructing multistep forecasts of a long-range dependent time series are discussed, all three methods being based on a stochastic model fitted to the time series for characterizing its long-memory as well as short-memory components. However, while the forecast coefficients for the truncation method may be obtained from the model equation itself, those for the type-II plug-in and type-II direct methods involve the autocorrelation function of the fitted stochastic model, analytic exact expressions for which may be cumbersome to evaluate. A numerical quadrature procedure, based on the fast Fourier transform algorithm, for computing the autocorrelation function of a long-memory process, as well as the forecast coefficients for the truncation method, is suggested and the computational accuracy of the approximations is investigated for several ARFIMA models. The three methods of constructing the forecasts of a long-memory time series apply when the innovations, ϵt, of the fitted model are postulated to follow a Gaussian distribution, and, also, when they follow an infinite variance stable distribution with characteristic exponent τ, 1<τ<2. A comparison of the multistep forecasts produced by these three methods is carried out by simulating several ARFIMA models with both Gaussian and stable innovations, and two FEXP models with Gaussian innovations. In addition, their relative behaviour with an actual time series, namely, the mean temperature in England, 1659–1976, is examined.

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