Estimating the rank of the coefficient matrix is a major challenge in multivariateregression, including vector autoregression (VAR). In this paper, we develop a novelfully Bayesian approach that allows for rank estimation. The key to our approachis reparameterizing the coefficient matrix using its singular value decomposition andconducting Bayesian inference on the decomposed parameters. By implementing astochastic search variable selection on the singular values of the coefficient matrix,the ultimate selected rank can be identified as the number of nonzero singular values.Our approach is appropriate for small multivariate regressions as well as for higherdimensional models with up to about 40 predictors. In macroeconomic forecastingusing VARs, the advantages of shrinkage through proper Bayesian priors are well doc-umented. Consequently, the shrinkage approach proposed here that selects or averagesover low rank coefficient matrices is evaluated in a forecasting environment. We showin both simulations and empirical studies that our Bayesian approach provides fore-casts that are better than those with two of the most promising benchmark methods,dynamic factor models and factor augmented VARs.