We estimate a DSGE (dynamic stochastic general equilibrium) model where rare large shocks can occur, by replacing the commonly used Gaussian assumption with a Student's t-distribution. Results from the Smets and Wouters (American Economic Review 2007; 97: 586–606) model estimated on the usual set of macroeconomic time series over the 1964–2011 period indicate that (i) the Student's t specification is strongly favored by the data even when we allow for low-frequency variation in the volatility of the shocks, and (ii)) the estimated degrees of freedom are quite low for several shocks that drive US business cycles, implying an important role for rare large shocks. This result holds even if we exclude the Great Recession period from the sample. We also show that inference about low-frequency changes in volatility—and, in particular, inference about the magnitude of Great Moderation—is different once we allow for fat tails. Copyright © 2014 John Wiley & Sons, Ltd.