The autoregressive orders selected by the criterion autoregressive transfer function (CAT) of Parzen (1974), a new version, CAT$^\ast$, of CAT introduced by Parzen (1977), and the CAT$_2$ criterion of Bhansali (1985) are shown to be asymptotically efficient in the sense defined by Shibata (1980, 1981). A generalization of the penalty function considered by Shibata (1980) is introduced. The order selected by the CAT$_\alpha$ criterion of Bhansali (1985), with any fixed $\alpha > 1$, is asymptotically efficient with respect to this generalized penalty function.
"Asymptotically Efficient Selection of the Order by the Criterion Autoregressive Transfer Function." Ann. Statist. 14 (1) 315 - 325, March, 1986. https://doi.org/10.1214/aos/1176349858