Asymptotic Equivalence between Information Criteria and Accumulated Prediction Errors in Misspecified Autoregressive Time Series

We investigate the predictive ability of the accumulated prediction error (APE) of Rissanen in an infinite-order autoregressive (AR(∞)) model. Since there are infinitely many parameters in the model, all finite-order AR models are misspecified. We first show that APE is asymptotically equivalent to Bayesian information criterion (BIC) and is not asymptotically efficient in the misspecified case. To rectify this difficulty, a modification of APE, APEδ, is proposed. Instead of accumulating squares of sequential prediction errors from the beginning, APEδ is obtained by accumulating squares of sequential prediction errors from stage nδ, where n is the sample size and 0 < δ < 1 may depend on n. Under certain regularity conditions, we show that APEδ is asymptotically efficient in the sense that the mean-squared prediction error (MSPE) of the AR model with order selected by APEδ can ultimately achieve the best compromise between model complexity and the goodness of fit. Based on this result, we further show that APEδ is asymptotically equivalent to Akaike’s information criterion (AIC) in an AR(∞) model. This is a somewhat interesting discovery because the proposed modification totally changes the nature of APE from a BIC-like criterion to an AIC-like criterion. An extensive simulation study is given to illustrate this theoretical result. We surprisingly found from the simulation results that APEδ has uniformly better finite-sample performance than both AIC and BIC. Finally, a suitable choice of δ is suggested for n between 100 and 500.