Maximum Likelihood Estimation of the Differencing Parameter for Invertible Short and Long Memory Autoregressive Integrated Moving Average Models

SUMMARY In practical applications of Box-Jenkins autoregressive integrated moving average (ARIMA) models, the number of times that the observed time series must be differenced to achieve approximate stationarity is usually determined by careful, but mostly informal, analysis of the differenced series. For many time series, some differencing seems appropriate, but taking the first or the second difference may be too strong. As an alternative, Hosking, and Granger and Joyeux proposed the use of fractional differences. For -1/2 < d < 1/2, the resulting fractional ARIMA processes are stationary. For 0 < d < 1/2, the correlations are not summable. The parameter d can be estimated, for instance by maximum likelihood. Unfortunately, estimation methods known so far have been restricted to the stationary range -1/2 < d < 1/2. In this paper, we show how any real d > -1/2 can be estimated by an approximate maximum likelihood method. We thus obtain a unified approach to fitting traditional Box-Jenkins ARIMA processes as well as stationary and non-stationary fractional ARIMA processes. A confidence interval for d can be given. Tests, such as for unit roots in the autoregressive parameter or for stationarity, follow immediately. The resulting confidence intervals for the ARMA parameters take into account the additional uncertainty due to estimation of d. A simple algorithm for calculating the estimate of d and the ARMA parameters is given. Simulations and two data examples illustrate the results.