The assumption of rational expectations is potentially a serious source of misspecification in DSGE models. Many recent theories of expectations formation have relaxed rational expectations and improved the predictive properties of benchmark macroeconomic models. Problematically, the space of possible theoretical deviations from rational expectations is very large, especially since the aggregate consequences of deviations from rational expectations in equilibrium may not be directly measurable using existing surveys of expectations. This paper provides evidence on which small reduced-form state-contingent deviations from rational expectations yield the most improvement in replicating features of macroeconomic time series, and which aspects of model misspecification are and are not ameliorated by these small deviations from rational expectations. The findings: a) The data favor deviations from rational expectations among firms in which they over-estimate the persistence of inflation. b) Relaxing rational expectations in a New Keynesian model partially substitutes for the additional structural mechanisms in the larger Smets Wouters (2007) model. Relaxing rational expectations within the Smets Wouters (2007) model improves that model’s ability to reproduce some of the spectral coherencies between output growth, investment growth, and labor supply. The mechanism is a combination of shocks to beliefs themselves and the role that deviations from rational expectations have in changing the propagation of other shocks.