  We propose a novel class of dynamic shrinkage processes for Bayesian time
series and regression analysis. Building upon a global-local framework of prior
construction, in which continuous scale mixtures of Gaussian distributions are
employed for both desirable shrinkage properties and computational
tractability, we model dependence among the local scale parameters. The
resulting processes inherit the desirable shrinkage behavior of popular
global-local priors, such as the horseshoe prior, but provide additional
localized adaptivity, which is important for modeling time series data or
regression functions with local features. We construct a computationally
efficient Gibbs sampling algorithm based on a P\'olya-Gamma scale mixture
representation of the proposed process. Using dynamic shrinkage processes, we
develop a Bayesian trend filtering model that produces more accurate estimates
and tighter posterior credible intervals than competing methods, and apply the
model for irregular curve-fitting of minute-by-minute Twitter CPU usage data.
In addition, we develop an adaptive time-varying parameter regression model to
assess the efficacy of the Fama-French five-factor asset pricing model with
momentum added as a sixth factor. Our dynamic analysis of manufacturing and
healthcare industry data shows that with the exception of the market risk, no
other risk factors are significant except for brief periods.
