Asymptotic Post-Selection Inference for Akaike's Information Criterion

Ignoring the model selection step in inference after selection is harmful. This paper studies the asymptotic distribution of estimators after model selection using the Akaike information criterion. First, we consider the classical setting in which a true model exists and is included in the candidate set of models. We exploit the overselection property of this criterion in the construction of a selection region, and obtain the asymptotic distribution of estimators and linear combinations thereof conditional on the selected model. The limiting distribution depends on the set of competitive models and on the smallest overparameterized model. Second, we relax the assumption about the existence of a true model, and obtain uniform asymptotic results. We use simulation to study the resulting postselection distributions and to calculate confidence regions for the model parameters. We apply the method to data.