  We develop a modeling framework for dynamic function-on-scalars regression,
in which a time series of functional data is regressed on a time series of
scalar predictors. The regression coefficient function for each predictor is
allowed to be dynamic, which is essential for applications where the
association between predictors and a (functional) response is time-varying. For
greater modeling flexibility, we design a nonparametric reduced-rank functional
data model with an unknown functional basis expansion, which is data-adaptive
and, unlike most existing methods, modeled as unknown for appropriate
uncertainty quantification. Within a Bayesian framework, we introduce shrinkage
priors that simultaneously (i) regularize time-varying regression coefficient
functions to be locally static, (ii) effectively remove unimportant predictor
variables from the model, and (iii) reduce sensitivity to the dimension of the
functional basis. A simulation analysis confirms the importance of these
shrinkage priors, with notable improvements over existing alternatives. We
develop a novel projection-based Gibbs sampling algorithm, which offers
unrivaled computational scalability for fully Bayesian functional regression.
We apply the proposed methodology (i) to analyze the time-varying impact of
macroeconomic variables on the U.S. yield curve and (ii) to characterize the
effects of socioeconomic and demographic predictors on age-specific fertility
rates in South and Southeast Asia.
