AbstractIn this paper, it is shown that the performance of various frequency-domain estimators of the memory parameter can be boosted by the inclusion of non-Fourier frequencies in addition to the regular Fourier frequencies. A fast two-stage algorithm for the efficient computation of the amplitudes at these additional frequencies is presented. In the first stage, the naïve sine and cosine transforms are computed with a modified version of the Fast Fourier Transform. In the second stage, these transforms are amended by taking the violation of the standard orthogonality conditions into account. A considerable number of auxiliary quantities, which are required in the second stage, do not depend on the data and therefore only need to be computed once. The superior performance (in terms of root-mean-square error) of the estimators based also on non-Fourier frequencies is demonstrated by extensive simulations. Finally, the empirical results obtained by applying these estimators to financial high-frequency data show that significant long-range dependence is present only in the absolute intraday returns but not in the signed intraday returns.
Fast computation of discrete invariants associated to a differential rational mapping
