  Time-varying parameter (TVP) regression models can involve a huge number of
coefficients. Careful prior elicitation is required to yield sensible posterior
and predictive inferences. In addition, the computational demands of Markov
Chain Monte Carlo (MCMC) methods mean their use is limited to the case where
the number of predictors is not too large. In light of these two concerns, this
paper proposes a new dynamic shrinkage prior which reflects the empirical
regularity that TVPs are typically sparse (i.e. time variation may occur only
episodically and only for some of the coefficients). A scalable MCMC algorithm
is developed which is capable of handling very high dimensional TVP regressions
or TVP Vector Autoregressions. In an exercise using artificial data we
demonstrate the accuracy and computational efficiency of our methods. In an
application involving the term structure of interest rates in the eurozone, we
find our dynamic shrinkage prior to effectively pick out small amounts of
parameter change and our methods to forecast well.
