A maximization of the expected entropy of the predictive distribution interpretation of Akaike's minimum AIC procedure is exploited for the modeling and prediction of time series with trend and seasonal mean value functions and stationary covariance’s. The AIC criterion best one-step-ahead and best twelve-step-ahead prediction models are different. They exhibit the relative optimality properties for which they were designed. The results are related to open questions on optimal trend estimation and optimal seasonal adjustment of time series.