Sims-Zha (1998) Bayesian Methods for Dynamic Multivariate Models

Summary

Sims and Zha (1998) provide a unified derivation of the Bayesian posterior for identified structural Vector AutoRegressions (VARs), framing everything in terms of a prior on the structural-form parameters $A_+|A_0$ rather than the reduced-form $B|A_0$ . The key computational insight is a Kronecker-product symmetry condition on the lag-coefficient prior that reduces a single $(mk) \times (mk)$ factorization to $m$ separate $k \times k$ problems — a roughly 400 $\times$ speedup for a 20-variable, 6-lag system. Three types of dummy observations implement unit-root, cointegration, and initial-condition beliefs, each with an explicit limiting interpretation as $\mu \to \infty$ .

Key Claims

The structural VAR likelihood factors into $|A_0|^T \exp[-\tfrac{1}{2} \text{trace}(ZA)'(ZA)]$ where $Z = [Y\; -X]$ and $A = [A_0'; A_+'']'$ , enabling joint prior specification on $(A_0, A_+)$ .
The prior $\pi(a) = \pi_0(a_0)\,\phi(a_+ - \mu(a_0); H(a_0))$ yields a conditional posterior $q(a_+|a_0)$ that is Gaussian in closed form, while the marginal $q(a_0)$ integrates out $a_+$ analytically.
Preserving Kronecker structure requires $H(a_0) = B \otimes G$ with $B = c\,I_m$ (eq. 12). With equation-specific prior covariances $G_i$ this reduces $m$ of the $(mk)$ -dimensional computations to $m$ independent $k \times k$ problems.
Three dummy-observation designs implement: (i) equation-specific random-walk tightness (Table 1, hyperparameters $\mu_0$ – $\mu_4$ ); (ii) sums-of-coefficients prior (Table 2, $\mu_5$ ); (iii) dummy initial observation (Table 3, $\mu_6$ ). As $\mu_5 \to \infty$ , all first differences are forced to zero (unit roots, no cointegration). As $\mu_6 \to \infty$ , the system is centered at the pre-sample mean, allowing unit roots and cointegration simultaneously.
Reduced-form $B|A_0$ prior (Section 5.2, Litterman-style) requires a 2.8-hour mode search plus 16 min/1,000 draws; and produces empirically problematic posteriors — unemployment bands miss the actual data, price inflation badly overpredicted. The structural $A_+|A_0$ prior (Section 5.1) requires only 31 sec/1,000 draws and produces well-calibrated 68% bands.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The key computational issue … is how to exploit the Kronecker product structure in the likelihood when we do not wish to impose the cross-equation restrictions that arise from a reduced form prior." (p. 955)

My Take

The paper's main contribution is clarifying the computational bottleneck that prevented large Bayesian VARs from being practical: the Kronecker symmetry condition (eq. 12) is not merely a convenience but the central organizing principle. The comparison between $A_+|A_0$ and $B|A_0$ priors in Sections 5.1–5.2 makes a compelling empirical case — the reduced-form prior imposes implicit cross-equation restrictions that show up as posteriors that badly miss the data. The dummy-observation taxonomy is rigorous and flexible, covering unit-root, cointegration, and intercept beliefs within a single augmented data matrix. The paper is also notable for its honest empirical comparisons: Fig. 3 shows the $B|A_0$ model's failure on unemployment and inflation explicitly rather than burying it.