Waggoner-Zha (2000) A Gibbs Simulator for Restricted VAR Models

Summary

Waggoner and Zha (2000) solve the computational bottleneck in Bayesian estimation of identified structural Vector Autoregression (VAR) models under linear zero restrictions. They show that the non-Gaussian shape of the posterior for contemporaneous coefficients causes importance sampling to collapse to a single effective draw — the posterior lies on a curved, non-elliptic ridge that a Gaussian proposal cannot approximate. They then derive a Gibbs sampler in which each equation's conditional decomposes into a Univariate Wishart (UW) draw and standard Normal draws, computed via Lower-Upper (LU) decomposition of the other equations' current draws. The paper also characterizes when Gibbs draws across equations are exactly independent (block-recursive exclusion restrictions) and validates convergence via Gelman-Rubin diagnostics on Sims's (1986) 6-variable VAR.

Key Claims

• Under the Sims-Zha (1998) reference prior, the joint posterior $p(b_1,\ldots,b_n)$ is non-Gaussian due to the determinant factor $|\det[U_1 b_1 \mid \cdots \mid U_n b_n]|^T$; importance sampling with a Gaussian proposal collapses to a single effective draw (Figure 3b: single spike in importance weight distribution).
• Theorem 2 (Gibbs step): conditional on all other equations, $b_i = \sum_j \beta_j w_j$ where $\{w_j\}$ is an orthonormal basis from LU decomposition of current draws; $\beta_1 \sim \text{UW}[T^{-1}, k_i+1]$ and $\beta_j \sim \mathcal{N}(0, T^{-1})$ for $j \geq 2$.
• UW distribution: $f(x) \propto x^{(k-1)/2} e^{-Tx/2}$; sampled as $x = z^2$ with $z \sim \mathcal{N}(0, T^{-1})$.
• Theorem 3 / Corollary 1 (Independence): Gibbs draws are exactly independent across equations iff there exist permutation $\Gamma$ and orthogonal $\Pi$ such that $\Gamma A \Pi$ is upper triangular for all admissible $A$; block-recursive exclusion restrictions are sufficient.
• Gelman-Rubin $\hat{R} \to 1$ within ~500 iterations for all 20 free structural parameters; monetary policy and interest-rate-defense equations in the Sims (1986) system yield exactly independent draws.

Concepts Introduced or Extended

• Gibbs Sampler
• Structural Identification
• Sims-Zha Prior
• Vector Autoregression

Entities Mentioned

• Daniel Waggoner
• Tao Zha
• Christopher Sims

Quotes

"The joint posterior density for the structural parameters is not normally distributed even though the prior is normal... the method of importance sampling may produce a very poor approximation to the posterior distribution." (§1, paraphrase)

"When $A$ satisfies block-recursive exclusion restrictions... the draws from Theorem 2 are exactly independent across equations." (Corollary 1, paraphrase)

My Take

The UW + Normal decomposition is elegant: by rotating into an equation-specific orthonormal basis via LU decomposition, the non-Gaussian determinant term concentrates into a single scalar coefficient, and the remaining components factorize into standard Normals. The independence result (Corollary 1) is practically important — researchers with recursive or block-recursive identification know their draws are not merely approximately independent but exactly so, enabling much shorter chains. The limitation is scope: the algorithm handles only linear zero restrictions on a fixed-coefficient VAR. Sign restrictions (Uhlig 2005), regime-switching, and time-varying-parameter models require different approaches. The 2003b follow-up (Waggoner-Zha 2003b) extended the framework to general normalization and likelihood preservation.