Computation of the autocovariances for time series with multiple long-range persistencies

Gegenbauer processes allow for flexible and convenient modeling of time series data with multiple spectral peaks, where the qualitative description of these peaks is via the concept of cyclical long-range dependence. The Gegenbauer class is extensive, including ARFIMA, seasonal ARFIMA, and GARMA processes as special cases. Model estimation is challenging for Gegenbauer processes when multiple zeros and poles occur in the spectral density, because the autocovariance function is laborious to compute. The method of splitting–essentially computing autocovariances by convolving long memory and short memory dynamics–is only tractable when a single long memory pole exists. An additive decomposition of the spectrum into a sum of spectra is proposed, where each summand has a single singularity, so that a computationally efficient splitting method can be applied to each term and then aggregated. This approach differs from handling all the poles in the spectral density at once, via an analysis of truncation error. The proposed technique allows for fast estimation of time series with multiple long-range dependences, which is illustrated numerically and through several case-studies.