This paper investigates the problem of model selection and testing in decision theory. We consider a consumption space equipped with an endogenous notion of ‘abstract numeraire,’ and characterize those preferences for which the quantity of numeraire needed to compensate an agent between a pair of alternatives provides a consistent, cardinal measure of the magnitude of preference. This framework includes all quasilinear or homothetic preferences on classical consumption spaces, stationary preferences over dated rewards, von Neumann-Morgenstern preferences on lottery spaces, and a wide range of preferences over monetary acts, including those represented by subjective expected utility, Choquet expected utility, maxmin expected utility, variational, and dual-self functionals. For data consisting of observed or experimentally elicited compensation differences, we show a simple least-squares methodology provides a systematic means of estimating the ‘best-fit’ preferences for an inconsistent data set from a given model, and allows for meaningful comparisons of goodness of fit across models. In the presence of cross-sectional data, our approach allows for nonparametric statistical testing of rationalizability at the population level.