Book excerpt: This article investigates the evidence of time-variation and asymmetry in the persistence of US inflation. We compare the out-of-sample performance of different forecasting models and find that quantile forecasts from an Auto-Regressive (AR) model with level-dependent volatility are at least as accurate as the forecasts of the Quantile Auto-Regressive model, in particular for the core inflation measures. Our results indicate that the persistence of core inflation has been relatively constant and high, but it declined for the headline inflation measures. We also find that the asymmetric persistence of inflation shocks can be mostly attributed to the positive relation between inflation level and its volatility.