  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.
