Here we present a novel method for modeling stationary time series. Our approach is to construct the model with a specified marginal family and build the dependence structure around it. We show that the resulting time series is linear with a simple autocorrelation structure. We construct models that parallel existing structures, namely state-space models, autoregressive conditional heteroscedasticity (ARCH) models, and generalized ARCH models. We use Bayesian techniques to estimate the resulting models. We also demonstrate that the models perform well compared with competing methods for the applications considered, count models and volatility models.