Ecient selection of the order of an AR(1): a unified approach without knowing the order of integratedness a

This paper suggests several information criteria to select an integrated autoregressive (AR) model without knowing the order of integratedness. When the underlying AR process is known to be stationary and of infinite order, Shibata (1980) and Ing and Wei (2005) showed that AIC is asymptotically efficient for independentand same-realization predictions, respectively. Based on the asymptotic expression for the prediction mean squared error (PMSE) of an integrated AR model developed in Ing, Sin and Yu (2010), this paper shows that AIC’s asymptotic efficiency in stationary AR(∞) processes carries over to integrated AR(∞) processes. Further, we derive the asymptotic efficiency of the two-stage information criterion of Ing (2007) in possibly integrated and infinite-order AR models, which is one of the most general order selection theories in the research field opened up by Shibata (1980). The main technical tools used in this paper are some moment bounds for the inverse of the normalized Fisher information matrix with an increasing dimension. Simulation evidence in support of our theoretical findings, as well as suggestions on the practical details on this two-stage information criterion, is also given.