The predictive capability of a modification of Rissanen's accumulated prediction error (APE) criterion, APE$_{δ_n}$, is investigated in infinite-order autoregressive (AR($\infty$)) models. Instead of accumulating squares of sequential prediction errors from the beginning, APE$_{δ_n}$ is obtained by summing these squared errors from stage $nδ_n$, where $n$ is the sample size and $1/n\leq δ_n\leq 1-(1/n)$ may depend on $n$. Under certain regularity conditions, an asymptotic expression is derived for the mean-squared prediction error (MSPE) of an AR predictor with order determined by APE$_{δ_n}$. This expression shows that the prediction performance of APE$_{δ_n}$ can vary dramatically depending on the choice of $δ_n$. Another interesting finding is that when $δ_n$ approaches 1 at a certain rate, APE$_{δ_n}$ can achieve asymptotic efficiency in most practical situations. An asymptotic equivalence between APE$_{δ_n}$ and an information criterion with a suitable penalty term is also established from the MSPE point of view. This offers new perspectives for understanding the information and prediction-based model selection criteria. Finally, we provide the first asymptotic efficiency result for the case when the underlying AR($\infty$) model is allowed to degenerate to a finite autoregression.