Sammendrag
We will explore the arguments behind Akaike's information criterion (AIC) in stationary Gaussian time series models. The goal is to see if there is a similar motivation, as in the case of comparing parametric models with independence, where the AIC formula have a specific interpretation from Kullback-Leibler divergence and maximum likelihood theory.
In addition to exploring the properties of the AIC, we will extend the Focused Information Criterion (FIC) to the case of stationary Gaussian time series. The FIC differs from most of the commonly used model selection strategies in that it does not try to find an overall ``best'' model, but rather aims at finding the model that is best at solving a predefined focus or task. Examples of foci in time series models are h-step ahead prediction, certain lag of the covariance function, or threshold probabilities. We will show that the sufficient conditions needed for the derivation of the FIC are satisfied for a large class of time series models, also including the more general class of locally stationary processes. And by extending the set of valid focus functions, we will make the FIC a new and (hopefully) useful contribution to the time series community.