A Comparison of Bayesian Multivariate Versus Univariate Normal Regression Models for Prediction

In many moderate dimensional applications we have multiple response variables that are associated with a common set of predictors. When the main objective is prediction of the response variables, a natural question is: do multivariate regression models that accommodate dependency among the response variables improve prediction compared to their univariate counterparts? Note that in this article, by univariate versus multivariate regression models we refer to regression models with a single versus multiple response variables, respectively. We assume that under both scenarios, there are multiple covariates. Our question is motivated by an application in climate science, which involves the prediction of multiple metrics that measure the activity, intensity, severity etc. of a hurricane season. Average sea surface temperatures (SSTs) during the hurricane season have been used as predictors for each of these metrics, in separate univariate regression models, in the literature. Since the true SSTs are yet to be observed during prediction, typically their forecasts from multiple climate models are used as predictors. Some climate models have a few missing values so we develop Bayesian univariate/multivariate normal regression models, that can handle missing covariates and variable selection uncertainty. Whether Bayesian multivariate normal regression models improve prediction compared to their univariate counterparts is not clear from the existing literature, and in this work we try to fill this gap.