Extended Fractional Gaussian Noise and Simple ARFIMA Approximations

Extended fractional Gaussian noise (eFGN) is the limiting structure of long memory time series aggregates. We propose a flexible class of low-order ARFIMA (0, d, q) models that closely approximates eFGN. Such ARFIMA approximation and a metric to measure precision can be easily obtained from the eigenvector and eigenvalue of an aggregation matrix of dimension q+1, constructed by utilizing the invariant property. A comparison to Man and Tiao's (2006) ARFIMA (0, d, dI) approximation that uses fixed MA order is also made. In practice, our result suggests that when aggregated long enough, many long memory time series aggregates will tend to follow a low-order ARFIMA model with pretty stable MA structure determined by d. This makes simple ARFIMA models appealing for modeling long memory time series aggregates.