Uniform Priors for Impulse Responses

Structural vector autoregressions (SVARs) identified with sign restrictions are a widely used tool for estimating dynamic causal effects in macroeconomics. Critics—notably Baumeister and Hamilton (2015) and Watson (2020)—have called for caution because the standard practice of using a uniform prior over the set of orthogonal matrices (with respect to the Haar measure) induces non-uniform marginal prior distributions over the identified sets of individual impulse responses. This paper formally challenges that caution: through an if-and-only-if theorem the authors show that the uniform prior over orthogonal matrices is not only sufficient but also necessary to induce a uniform joint prior distribution over the identified set for the vector of impulse responses—a result that holds for any prior distribution over the reduced-form parameters. The paper additionally shows how to conduct posterior inference based on a uniform joint prior for the vector of impulse responses, which requires modifying the prior for the reduced-form parameters away from the standard Minnesota prior while retaining the uniform prior over orthogonal matrices. An application to Watson’s (2020) empirical example finds that joint credible sets under this new prior are similar to, but wider than, those obtained under the standard approach, and that imposing tighter identifying restrictions sharpens inference under both priors.

Daniel F. Waggoner | Macro Paper Warehouse

Uniform Priors for Impulse Responses