Asymptotically efficient order selection in nonstationary AR processes

In this paper we investigate the issue of asymptotic efficiency in nonstationary AR(infinity) processes. Since the inverse of the autocovariance matrix of the underlying process cannot be evaluated due to the fact that the matrix is singular, traditional methods and techniques (Karagrigoriou 1995, 1997; Bhansali 1996) cannot be applied. Here we attempt to reduce the nonstationary case to a stationary one so that known results can then be applied to the reduced process. Asymptotic results regarding the overestimation of the order of an AR process with several unit roots are presented and the asymptotic efficiency of the order selected is established in the case where d (d > 0) unit roots are present.