The selection problem among models for the seasonal behavior in time series is considered. The central decision of interest is between models with seasonal unit roots and with deterministic cycles. In multivariate models, also the number of stochastic seasonal factors is a discrete parameter of interest. To enable restricting attention to data-admissible models, a new attempt is made at defining data admissibility. Among data-admissible model classes, statistical decision rules are constructed on the basis of weighting priors and decision-bounds analysis. The procedure is applied to some exemplary economics series. Many univariate series select models without seasonal unit roots but the bivariate experiments enhance the importance of seasonal unit roots with restricted influence of seasonal constants. The framework of decision-bounds analysis offers a convenient alternative to sequences of classical hypothesis tests.