This paper studies the joint dynamics of U.S. inflation and a term structure of average inflation predictions taken from the Survey of Professional Forecasters (SPF). We estimate these joint dynamics by combining an unobserved components (UC) model of inflation and a sticky‐information forecast mechanism. The UC model decomposes inflation into trend and gap components, and innovations to trend and gap inflation are affected by stochastic volatility. A novelty of our model is to allow for time‐variation in inflation‐gap persistence as well as in the frequency of forecast updating under sticky information. The model is estimated with sequential Monte Carlo methods that include a particle learning filter and a Rao–Blackwellized particle smoother. Based on data from 1968Q4 to 2018Q3, estimates show that (i) longer horizon average SPF inflation predictions inform estimates of trend inflation; (ii) inflation gap persistence is countercyclical before the Volcker disinflation and acyclical afterwards; (iii) by 1990 sticky‐information inflation forecast updating is less frequent than it was earlier in the sample; and (iv) the drop in the frequency of the sticky‐information forecast updating occurs at the same time persistent shocks become less important for explaining movements in inflation. Our findings support the view that stickiness in survey forecasts is not invariant to the inflation process.