An asymptotic theory for linear model selection

In the problem of selecting a linear model to approximate the true unknown regression model, some necessary and/or sufficient conditions are established for the asymptotic validity of various model selection procedures such as Akaike's AIC, Mallows' C-p, Shibata's FPElambda, Schwarz' BIC, generalized AIC, cross-validation, and generalized cross-validation. It is found that these selection procedures can be classified into three classes according to their asymptotic behavior. Under some fairly weak conditions, the selection procedures in one class are asymptotically valid if there exist fixed-dimension correct models; the selection procedures in another class are asymptotically valid if no fixed-dimension correct model exists. The procedures in the third class are compromises of the procedures in the first two classes. Some empirical results are also presented.