  This paper proposes methods for Bayesian inference in time-varying parameter
(TVP) quantile regression (QR) models. We use data augmentation schemes to
facilitate the conditional likelihood, and render the model conditionally
Gaussian to develop an efficient Gibbs sampling algorithm. Regularization of
the high-dimensional parameter space is achieved via flexible dynamic shrinkage
priors. A simple version of the TVP-QR based on an unobserved component (UC)
model is applied to dynamically trace the quantiles of the distribution of
inflation in the United States (US), the United Kingdom (UK) and the euro area
(EA). We conduct an out-of-sample inflation forecasting exercise to assess
predictive accuracy of the proposed framework versus several benchmarks using
metrics to capture performance in different parts of the distribution. The
proposed model is competitive and performs particularly well for higher-order
and tail forecasts. We analyze the resulting predictive distributions and find
that they are often skewed and feature heavier than normal tails.
