Contract structures typically embedded in a recent generation of models with nominal inertia have been criticized for a failure to generate persistent responses of output and inflation to nominal shocks. In this paper, we argue that this failure does not reflect an inherent limitation of the contract structure, but rather, very strong assumptions about central bank credibility. We show that an optimization-based DGE model with four-quarter price and wage contracts can generate highly persistent inflation and output responses when the private sector must use signal extraction to make inferences about the central bank's inflation target based on only observing the policy instrument. In particular, we consider the disinflation experiences of the United States and several other industrial countries in the early 1980s, and we demonstrate that the model matches the dynamics of both expected and actual inflation and generates sacrifice ratios which are roughly in line with empirical estimates.