Small-sample Autocorrelation Structure for Long-memory Time Series

Abstract

This paper provides results on the behaviour of the sample autocorrelation function for an ARUMA model—that is, an ARIMA process in which the differencing operator (1 − B)^d is replaced by a more general real polynomial, U _d (B), having each of its zeros anywhere on the unit circle (subject to the restraint that complex zeros occur in conjugate pairs). Various simulated series are considered, and it is found empirically that the observed sample autocorrelations are in close agreement with appropriate ratios of limiting serial covariance expectations. This agreement is better than that with the theoretical autocorrelations, in the nearly non-stationary case, or the limits of the theoretical autocorrelations (as the homogeneous non-stationarity boundary is approached), in the non-stationary case. We also note some striking features and distinctions in the serial correlation behaviour of processes, according to the position of the zero (or zeros) of U _d on the unit circle, and demonstrate the generally considerable difference between the serial correlations of a non-stationary ARUMA model and a nearly non-stationary approximation to it, obtained by replacing the U _d (B) with U _d (αB), where α is a little (but appreciable bit) less than unity.