Information criteria for selecting possibly misspecified parametric models

We consider penalized likelihood criteria for selecting models of dependent processes. The models may be strictly nested, overlapping or nonnested, linear or nonlinear, and correctly specified or misspecified. We provide sufficient conditions on the penalty to guarantee the selection, with probability one (or with probability approaching one), of the model attaining the lower average Kullback-Leibler Information Criterion (KLIC) or, when both have the same KLIC, the more parsimonious model. As special cases, our results describe the Akaike, Schwarz, and Hannan-Quinn information criteria. As examples, we consider selection of ARMAX-GARCH and STAR models.