On Estimating Conditional Mean‐Squared Prediction Error in Autoregressive Models

. Zhang and Shaman considered the problem of estimating the conditional mean‐squared prediciton error (CMSPE) for a Gaussian autoregressive (AR) process. They used the final prediction error (FPE) of Akaike to estimate CMSPE and proposed that FPE's effectiveness be judged by its asymptotic correlation with CMSPE. However, as pointed out by Kabaila and He, the derivation of this correlation by Zhang and Shaman is incomplete, and the performance of FPE in estimating CMSPE is also poor in Kabaila and He's simulation study. Kabaila and He further proposed an alternative estimator of CMSPE, V, in the stationary AR(1) model. They reported that V has a larger normalized correlation with CMSPE through Monte Carlo simulation results. In this paper, we propose a generalization of V, V˜, in the higher‐order AR model, and obtain the asymptotic correlation of FPE and V˜ with CMSPE. We show that the limit of the normalized correlation of V˜ with CMSPE is larger than that of FPE with CMSPE, and hence Kabaila and He's finding is justified theoretically. In addition, the performances of the above estimators of CMSPE are re‐examined in terms of mean‐squared errors (MSE). Our main conclusion is that from the MSE point of view, V˜ is the best choice among a family of asymptotically unbiased estimators of CMSPE including FPE and V˜ as its special cases.