We compare sparse and dense representations of predictive models in macroeconomics, microeconomics, and finance. To deal with a large number of possible predictors, we specify a prior that allows for both variable selection and shrinkage. The posterior distribution does not typically concentrate on a single sparse model, but on a wide set of models that often include many predictors.

Vector autoregressions (VARs) are flexible time series models that can capture complex dynamic interrelationships among macroeconomic variables. However, their dense parameterization leads to unstable inference and inaccurate out-of-sample forecasts, particularly for models with many variables. A solution to this problem is to use informative priors in order to shrink the richly parameterized unrestricted model toward a parsimonious naıve benchmark, and thus reduce estimation uncertainty. This paper studies the optimal choice of the informativeness of these priors, which we treat as additional parameters, in the spirit of hierarchical modeling. This approach, theoretically grounded and easy to implement, greatly reduces the number and importance of subjective choices in the setting of the prior. Moreover, it performs very well in terms of both out-of-sample forecasting—as well as factor models—and accuracy in the estimation of impulse response functions.