Order Selection in Nonstationary Autoregressive Models

In Hannan (1980), some limiting properties of the order selection criteria, AIC, BIC, and $\phi(p, q)$ for modeling stationary time series were derived. In this paper, we generalize these properties to the case in which the underlying process follows a nonstationary autoregressive model. We show that BIC and $\phi(p, 0)$ are weakly consistent. For the AIC, we prove that the asymptotic distribution given by Shibata (1976) for the stationary autoregressive models continues to hold.