This paper introduces a form of boundedly-rational expectations into an otherwise standard New Keynesian Phillips curve. The representative agent's perceived law of motion allows for both temporary and permanent shocks to in ‡ation, the latter intended to capture the possibility of evolving shifts in the central bank's in ‡ation target. The agent's perceived optimal forecast rule de…ned by the Kalman …lter is parameterized to be consistent with the observed moments of the in ‡ation time series. From the agent's perspective, the use of a variable Kalman gain parameter is justi…ed by movements in the perceived "signalto-noise ratio," which measures the relative variances of the permanent and temporary shocks to in ‡ation. I show that this simple model of in ‡ation expectations can generate time-varying in ‡ation dynamics similar to those observed in long-run U.S. data. The U.S. signal-to-noise ratio identi…ed using the model's methodology exhibits an upward drift during the 1970s, followed by downward drift from the mid-1990s onwards. This pattern suggests that the perceived signal-to-noise ratio might be viewed as a inverse measure of the central bank's credibility for maintaining a constant in ‡ation target. Model-based values for expected in ‡ation track quite well with movements in survey-based measures of U.S. expected in ‡ation.