Bayesian SVAR Identification

Definition

Bayesian SVAR identification treats the identification of a structural VAR as a problem of prior information rather than of exact restrictions. Writing the structural form $A\,y_t = c + B_1 y_{t-1}+\cdots+u_t$ with contemporaneous impact matrix $A$ and structural shocks $u_t$, the mapping from the reduced form to $A$ is not pinned down by the data alone. Instead of fixing elements of $A$ (dogmatic restrictions), the analyst places an informative, non-dogmatic prior on the economically interpretable structural parameters and reports the posterior for $A$ and the impulse responses (Baumeister-Hamilton 2019).

Key Ideas

• Traditional restrictions are extreme priors. A recursive (Cholesky) ordering is a prior that puts point mass on certain elements of $A$ being zero; agnostic sign restrictions correspond to a completely uninformative prior over the admissible region. Both are endpoints of a continuum — the honest choice lives in the "vast middle ground" of proper informative priors.
• Priors on interpretable parameters. Put the prior on the objects economists actually hold beliefs about — price elasticities of supply and demand, pass-through coefficients, the equilibrium impacts of shocks — not on an abstract rotation matrix. This draws on all available (if imperfect) structural knowledge.
• Identification uncertainty widens the bands. Posterior credible intervals for impulse responses combine sampling uncertainty with uncertainty about the identifying structure, unlike frequentist SVAR bands that condition on the restrictions being exactly correct.
• "Agnostic" is not assumption-free. A flat prior over rotations still induces an implicit — and often influential — prior on the structural parameters, so purportedly assumption-free sign-restriction results can be prior-driven.
• Set vs. point identification. When the prior is weak the model is only set-identified (a range of $A$ consistent with the data and prior); as the prior tightens toward a dogmatic restriction it approaches point identification. Bayesian inference handles both cases uniformly.
• Extensions. The framework accommodates measurement error in the observed variables and downweighting of earlier data when structural stability is in doubt.
• Early precursor (Koop 1992b). A Bayesian analysis of aggregate shocks that computes the full posterior density of impulse responses and variance decompositions (by Monte Carlo / Gibbs) while imposing the identifying restrictions as uncertain rather than exact. Working within the Blanchard-Quah long-run-restriction tradition, it already found posterior standard deviations substantially larger — and more skewed — than classical bootstrap intervals, prefiguring the modern point that identification uncertainty must widen the bands.

How It Works

1. Specify the structural form and the parameters of $A$ (e.g., contemporaneous elasticities) to be identified.
2. Elicit an informative prior on those structural parameters (and on lag/covariance parameters), normalized so the admissible set of $A$ integrates to a finite positive number (a proper prior).
3. Combine with the (Gaussian) likelihood of the reduced form; draw from the posterior of $A$ by importance sampling / Metropolis over the structural parameters.
4. Map each posterior draw of $A$ to impulse responses, historical decompositions, and variance decompositions; report posterior medians and credible bands that reflect both data and prior uncertainty.
5. Probe robustness by varying the priors — the sensitivity of conclusions to identification is now explicit in the results.

Why It Matters

• Reconciles the identification wars. Recursive orderings, long-run restrictions, and sign restrictions become special cases of one Bayesian object, clarifying exactly what each assumes and what it costs.
• Honest error bands. By letting doubt about the structure enter the posterior, it stops SVARs from reporting spuriously tight intervals that hide the fragility of the identifying assumptions.
• Substantive payoff. In the oil market it overturns conclusions built on very tight supply-elasticity priors — raising the estimated short-run supply elasticity (~0.15) and the importance of supply shocks — showing that identification assumptions, not just data, drove earlier findings.

Open Questions

• Prior elicitation and defense. The method is only as credible as the priors on elasticities; eliciting, justifying, and stress-testing them becomes the central task.
• Prior sensitivity vs. data information. How much the posterior is driven by prior versus likelihood depends on sample information about the contemporaneous relations, which is often weak.
• Computation at scale. Sampling the structural parameters in larger systems with rich priors is demanding, and efficient posterior exploration remains an active area.

Related

• Structural Identification
• Sign Restriction Identification
• Narrative Identification
• Bayesian VAR
• Impulse Response Function
• Monetary Policy Shocks
• Christiane Baumeister
• James D. Hamilton
• Gary Koop
• Baumeister-Hamilton (2019) Structural Interpretation of Vector Autoregressions with Incomplete Identification
• Koop (1992b) Aggregate Shocks and Macroeconomic Fluctuations: A Bayesian Approach
• Fiscal Policy Shocks
• Caldara-Kamps (2017) The Analytics of SVARs
• Oil Price Shocks
• Lutz Kilian