This paper provides an alternative explanation for the recent empirical evidence (Stock and Watson, 2007 and Cogley et al., 2010) showing that a random walk dynamic component accounts for much of the persistence in inflation. I use a time-dependent sticky price model and study the mapping from disturbances to the price reset hazard function to inflation dynamics. I show that shocks to the hazard function propagate the dynamics of the price distribution, which contains a unit root. Furthermore, I derive a principle for simple interest rate reaction rules to eliminate the unit root from the equilibrium dynamics of inflation.