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.
"Order Selection in Nonstationary Autoregressive Models." Ann. Statist. 12 (4) 1425 - 1433, December, 1984. https://doi.org/10.1214/aos/1176346801