A generalized fractionally differencing approach in long-memory modeling

We extend the class of fractional ARIMA models to the class of fractional ARUMA models, which describe long-memory time series with long-range periodical behavior at a finite number of spectrum frequencies. The exact asymptotics of the covariance function and the spectrum at the points of peaks and zeros are given. To obtain asymptotic expansions, Gegenbauer polynomials are used. Consistent parameter estimation is discussed using Whittle's estimate. Research supported by the Alexander von Humboldt Foundation. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius; Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 1, pp. 65–81, January–March, 1995.