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The dynamic stochastic general equilibrium (DSGE) models used to study business cycles typically assume that exogenous disturbances are independent first-order autoregressions. This paper relaxes this tight and arbitrary restriction by allowing for disturbances that have a rich contemporaneous and dynamic correlation structure. Our first contribution is a new Bayesian econometric method that uses conjugate conditionals to allow for feasible and quick estimation of DSGE models with correlated disturbances. Our second contribution is a reexamination of U.S. business cycles. We find that allowing for correlated disturbances resolves some conflicts between estimates from DSGE models and those from vector autoregressions and that a key missing ingredient in the models is countercyclical fiscal policy. According to our estimates, government spending and technology disturbances play a larger role in the business cycle than previously ascribed, while changes in markups are less important.