Information criteria for multistep ahead predictions

We propose an information criterion for multistep ahead predictions. It is also used for extrapolations. For the derivation, we consider multistep ahead predictions under local misspecification. In the prediction, we show that Bayesian predictive distributions asymptotically have smaller Kullback--Leibler risks than plug-in predictive distributions. From the results, we construct an information criterion for multistep ahead predictions by using an asymptotically unbiased estimator of the Kullback--Leibler risk of Bayesian predictive distributions. We show the effectiveness of the proposed information criterion throughout the numerical experiments.