The time series model selection problem is strongly rooted in residual analysis. In a typical application many competing models are fitted, wherein the model features are so divergent that direct comparison statistics, such as the likelihood ratio or Akaike Information Criterion, are meaningless. This is because a time series model involves identification of transformation, unit roots, fixed regression effects, and serial correlation patterns. Given this difficulty, we instead advocate an indirect comparison through the entropy of the residual series, assessed through serial correlation and marginal Gaussianity. The ideal residual - namely, a residual series that behaves like Gaussian white noise - is our benchmark against which all model residuals are compared, and provides the basis by which to judge diverse models' superiority. We propose an entropy information criterion that is minimized at the ideal residual, and increases with deviations from either a Gaussian marginal or a non-white serial correlation pattern. Distribution theory is provided, with supporting empirical studies as well as illustrations on retail data. It is known that underfitting (the failure to specify enough model structure) is flagged through the presence of residual serial correlation; a key finding of our research is that overfitting (specifying superfluous model structure) is flagged through the presence of light-tailed residuals.