This paper considers a prototypical monetary business cycle model for the U.S. economy, in which the equilibrium is undetermined if monetary policy is 'inactive'. In previous multivariate studies it has been common practice to restrict parameter estimates to values for which the equilibrium is unique. We show how the likelihood-based estimation of dynamic stochastic general equilibrium models can be extended to allow for indeterminacies and sunspot fluctuations. We propose a posterior odds test for the hypothesis that the data are best explained by parameters that imply determinacy. Our empirical results show that the Volcker-Greenspan policy regime is consistent with determinacy, whereas the pre-Volcker regime is not. We find that before 1979 non-fundamental sunspot shocks may have contributed significantly to inflation and interest rate volatility, but essentially did not affect output fluctuations.