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This paper proposes methods for Bayesian inference in time-varying parameter(TVP) quantile regression (QR) models featuring conditional heteroskedasticity.I use data augmentation schemes to render the model conditionally Gaussian anddevelop an efficient Gibbs sampling algorithm. Regularization of thehigh-dimensional parameter space is achieved via flexible dynamic shrinkagepriors. A simple version of TVP-QR based on an unobserved component model isapplied to dynamically trace the quantiles of the distribution of inflation inthe United States, the United Kingdom and the euro area. In an out-of-sampleforecast exercise, I find the proposed model to be competitive and performparticularly well for higher-order and tail forecasts. A detailed analysis ofthe resulting predictive distributions reveals that they are sometimes skewedand occasionally feature heavy tails.

Modeling tail risks of inflation using unobserved component quantile regressions