  We address the problem of dynamic variable selection in time series
regression with unknown residual variances, where the set of active predictors
is allowed to evolve over time. To capture time-varying variable selection
uncertainty, we introduce new dynamic shrinkage priors for the time series of
regression coefficients. These priors are characterized by two main
ingredients: smooth parameter evolutions and intermittent zeroes for modeling
predictive breaks. More formally, our proposed Dynamic Spike-and-Slab (DSS)
priors are constructed as mixtures of two processes: a spike process for the
irrelevant coefficients and a slab autoregressive process for the active
coefficients. The mixing weights are themselves time-varying and depend on
lagged values of the series. Our DSS priors are probabilistically coherent in
the sense that their stationary distribution is fully known and characterized
by spike-and-slab marginals. For posterior sampling over dynamic regression
coefficients, model selection indicators as well as unknown dynamic residual
variances, we propose a Dynamic SSVS algorithm based on forward-filtering and
backward-sampling. To scale our method to large data sets, we develop a Dynamic
EMVS algorithm for MAP smoothing. We demonstrate, through simulation and a
topical macroeconomic dataset, that DSS priors are very effective at separating
active and noisy coefficients. Our fast implementation significantly extends
the reach of spike-and-slab methods to large time series data.
