Uniform Priors for Impulse Responses
What this paper finds — and why it matters
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.
Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.
Q1. What is the core result and what does it imply for applied work?
The central result is an if-and-only-if theorem: the uniform prior over the set of orthogonal matrices is both sufficient and necessary for the conventional Bayesian approach to induce a uniform joint prior distribution over the identified set for the vector of impulse responses, for any prior over the reduced-form parameters. The critics’ concern about non-uniform individual marginal priors does not extend to the joint object: when inference targets the full vector of impulse responses, the standard Haar prior is exactly appropriate. Practitioners interested in joint inference on the shape and comovement of the impulse response function need not heed the call for caution.
Q2. Why does non-uniformity of individual marginal priors not imply non-uniformity of the joint distribution?
The marginal distribution extracted from a uniform joint distribution over a compact manifold need not be uniform; marginal uniformity and joint uniformity are different properties, and only the latter is required for observationally equivalent vectors to be distinguished solely by the identifying restrictions. Baumeister and Hamilton (2015) and Watson (2020) correctly note that individual impulse responses have non-uniform marginal priors under the Haar measure, but this is not the relevant criterion when the object of interest is the entire impulse response vector. The paper’s theorem shows the joint distribution is uniform, which is the property that ensures the identification restrictions—not the prior—drive the posterior shape.
Q3. How does one implement a uniform joint prior for the vector of impulse responses?
The authors show that a uniform joint prior for the vector of impulse responses requires a modified prior for the reduced-form parameters: one that is independent between (B, Σ) and Q, takes a model-dependent non-standard form for (B, Σ), and retains a uniform prior over orthogonal matrices. The induced reduced-form prior resembles but differs from both the standard Minnesota prior and Uhlig’s (2005) “weak prior.” Because the induced prior for (B, Σ, Q) is still a uniform-normal-inverse-Wishart (UNIW) distribution, the conventional sampling algorithm applies without modification; analysts supply the modified reduced-form prior while continuing to draw Q uniformly from the Haar measure.
Q4. What does the empirical illustration show?
In Watson’s (2020) empirical example, joint credible sets under the uniform-joint-prior approach are similar to but wider than those under the standard Minnesota-prior approach. The widening is consistent with theory: the uniform joint prior spreads probability mass more evenly over the identified set rather than concentrating it toward regions favored by the Minnesota prior. The finding that tighter identifying restrictions sharpen inference under both approaches reinforces the conclusion of Inoue and Kilian (2022b) that many sign restrictions help when the focus is on joint distributions.
Q5. How is the analysis generalized?
The paper extends the results to a broader class of objects of interest—any smooth function of impulse responses, such as combinations of structural elasticities and standard deviations—with an importance-sampling correction when the induced prior over orthogonal matrices is not uniform in the extended case. The generalization exploits the diffeomorphism between IR parameters and orthogonal reduced-form parameters, which allows the change-of-variables formula to apply to any smooth object of interest.
Key concepts
vector of impulse responses : the collection of impulse responses across all variables, shocks, and horizons, treated as a single vector object for joint inference; contrasted with individual impulse responses (the response of one variable to one shock at one horizon).
uniform prior over orthogonal matrices (Haar measure) : the unique probability measure on the set of n×n orthogonal matrices invariant under left and right multiplication; the standard prior used in Bayesian sign-restricted SVARs.
identified set : the set of vectors of impulse responses that are observationally equivalent given the data and the sign restrictions; the conventional approach draws uniformly from this set under the Haar prior.
uniform-normal-inverse-Wishart (UNIW) prior : the joint prior over orthogonal reduced-form parameters consisting of the Haar prior over Q and a normal-inverse-Wishart prior over (B, Σ); conjugate and computationally tractable.