On unified model selection for stationary and nonstationary short- and long-memory autoregressive processes

The question of model choice for the class of stationary and nonstationary, fractional and nonfractional autoregressive processes is considered. This class is defined by the property that the dth difference, for -1/2 < d < infinity, is a stationary autoregressive process of order-p(o) < infinity. A version of the Akaike information criterion, AIC, for determining an appropriate autoregressive order when d and the autoregressive parameters are estimated simultaneously by a maximum likelihood procedure (Beran, 1995) is derived and shown to be of the same general form as for a stationary autoregressive process, but with d treated as an additional estimated parameter. Moreover, as in the stationary case, this criterion is shown not to provide a consistent estimator of p(o). The corresponding versions of the BIC Of Schwarz (1978) and the HIC of Hannan & Quinn (1979) are shown to yield consistent estimators of p(o). The results provide a unified treatment of fractional and nonfractional, stationary and integrated nonstationary autoregressive models.