Dynamic Function-on-Scalars Regression

Abstract: 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.