Testing fractionally integrated time series

Abstract

A generalization of the autoregressive integrated moving average (ARIMA(p,d,q)) model is the autoregressive fractionally integrated moving average (ARFIMA(p, d, q)) model in which d is allowed to take non-integer values. The continuum of values of d represents differences in the degree of long-run dependency of the time series. This paper is concerned with testing whether a time series is integer integrated or fractionally integrated. Without loss of generality, the typical case of testing d = 1 against d > 1 or d < 1 is considered. A locally best invariant test, based on King and Hillier (1985), is constructed for the simple ARFIMA(O,d,O) case. The test is then modified to test the general ARFIMA(p,d,q) series. The power of these tests is investigated using the Monte Carlo method. Also investigated is the power of a test based on Geweke and Porter-Hudak (1983)'s method, and an augmented Dickey-Fuller test. The invariant tests proposed here can also be used to test different integer values of d, such as d = 2 against d = 1.