Is BIC asymptotically efficient for minimizing prediction error if the true model is being considered? If a set of models is being compared using BIC and AIC, given the fact that the true model (the one which generated the data) is in this set (and given the other assumptions that guarantee BIC consistency in selecting the true model);
As N goes to infinity, if I understand things correctly, BIC will always pick the true model, AIC will either pick the true model or a slightly more general one with more parameters. This would mean that BIC will always pick a model that predics as well or better that the one picked by AIC.
Q1) So, can we say that as N goes to infinity, BIC will perfom better that AIC not only form an "explanation" point of view (selecting the correct model) but also from a "prediction" point of view (selecting the model that minimizes the prediction error)?
Q2) I know there are some situations where the true model is not the best for prediction, but when this happens the best should have less parameters and not more. Anyway, will this situasions happen only if N is small or are there examples of this happening as N goes to infinity?
 A: Not a definite answer but some thoughts.
Under some assumptions, AIC is asymptotically efficient regardless of whether the true model is in the set of candidate models or not. Under these assumptions, BIC cannot be asymptotically efficient because it differs from AIC to a nontrivial extent. Now if we take the assumptions that yield consistency of BIC, we do not have that AIC is asymptotically efficient (the two sets of assumptions do not coincide). Thus it seems possible that BIC is asymptotically efficient under these assumptions. However, a consistent criterion can never be efficient (Yang, 2005 as cited in Chapter 4 of Claeskens & Hjort, 2008). This suggests BIC is likely not efficient under the assumptions that make it consistent.
A theoretically solid source comparing AIC and BIC and discussing consistency and efficiency is Chapter 4 of Claeskens & Hjort "Model selection and model averaging" (2008) (a book).
References

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*Claeskens, G. and Hjort, N.L. (2008). Model Selection and Model Averaging. Cambridge University Press.

*Yang, Y. (2005). Can the strengths of AIC and BIC be shared? Biometrika, 92:937–950

