George-Sun-Ni (2008) Bayesian Stochastic Search for VAR Model Restrictions

Summary

George, Sun, and Ni extend the stochastic search variable selection (SSVS) framework of George-McCulloch (1993, 1997) from univariate regression to vector autoregression (VAR) models, simultaneously searching restrictions on the VAR coefficient matrix $\Phi$ and the off-diagonal elements of the upper-triangular Cholesky factor $\Psi$ (where $\Sigma^{-1} = \Psi\Psi'$). The upper-triangular parameterisation is essential for global identification of $\Sigma$: arbitrary $\Psi$ satisfies only local rank conditions and may be non-unique (Bekker-Pollock 1986; Amisano-Giannini 1997). A five-component conjugate prior yields a five-step Gibbs sampler whose every conditional is of standard form (Gamma, Normal, or Bernoulli) with no Metropolis-Hastings steps. Rao-Blackwell model-averaged forecasts substantially outperform unrestricted maximum likelihood estimation (MLE) across simulated and empirical settings.

Key Claims

• Searching over $2^{(h+Lp)p + p(p-1)/2}$ competing VAR submodels is made tractable by a mixture-prior Gibbs sampler that visits high-posterior-probability models without enumeration.
• The $\gamma$ indicators for $\Phi$ do not depend on $Y$ given $\psi$ — the same key property inherited from George-McCulloch (1993) that permits efficient Bernoulli simulation.
• Restricting $\Phi$ improves $\Psi$ selection: the two searches are interdependent (demonstrated in simulation Examples 1 and 2).
• Rao-Blackwell (RB) averaging over visited models reduces forecast mean-squared error by 20–67% relative to MLE in simulation; Kullback-Leibler (KL) distance from true model substantially smaller for RB than for unrestricted VAR.
• Diagonal entries $\psi_{ii}$ are always included (Gamma prior), guaranteeing that every sampled $\Sigma^{-1} = \Psi\Psi'$ is positive definite by construction.
• Semiautomatic tuning sets $\tau_{0i} = c_0 \hat\sigma_{\phi_{s_i}}$ and $\tau_{1i} = c_1 \hat\sigma_{\phi_{s_i}}$ ($c_0 = 0.1$, $c_1 = 10$) from the ordinary least squares (OLS) standard error, analogous to George-McCulloch (1997).
• Empirically (7-variable producer price index (PPI)→consumer price index (CPI) VAR, L=12): contemporaneous structure confirms supply-chain transmission from import prices (IM)→food-and-energy CPI (FC)→CPI in the 1969–80 period; post-1981 the finished-goods PPI (FF)–crude-materials PPI (CM) link breaks and the Fed Funds Rate becomes endogenously responsive to commodity prices, import prices, and unemployment contemporaneously.

How It Works

Model Setup

VAR: $y_t' = z_t' C + \sum_{j=1}^L y_{t-j}' A_j + \varepsilon_t'$, $\varepsilon_t \sim \mathcal{N}_p(0,\Sigma)$; stack as $Y = X\Phi + \mathcal{E}$ (eq. 4). MLE: $\hat\Phi_M = (X'X)^{-1}X'Y$ (eq. 8).

Precision decomposition: $\Sigma^{-1} = \Psi\Psi'$ where $\Psi$ is upper triangular with positive diagonal (eq. 9). Row $j$ of $\Psi$: diagonal element $\psi_{jj} > 0$; off-diagonal row vector $\eta_j = (\psi_{1j},\ldots,\psi_{j-1,j})'$; restriction indicator vector $\omega_j$.

Five-Component Prior (Section 2.2)

1. Always-included coefficients $\phi_{non}$: $\mathcal{N}_{n-m}(\tilde\phi_{non}, M_{non})$ (eq. 11).
2. Restricted $\phi_m$: mixture $(\phi_{s_i}|\gamma_i) \sim (1-\gamma_i)\mathcal{N}(0,\tau_{0i}^2) + \gamma_i\mathcal{N}(0,\tau_{1i}^2)$ (eq. 13); $\gamma_i \sim \text{Bernoulli}(p_i)$ (eq. 14).
3. Off-diagonal $\psi_{ij}$: mixture $(\psi_{ij}|\omega_{ij}) \sim (1-\omega_{ij})\mathcal{N}(0,\kappa_{0ij}^2) + \omega_{ij}\mathcal{N}(0,\kappa_{1ij}^2)$ (eq. 16); $\omega_{ij} \sim \text{Bernoulli}(q_{ij})$ (eq. 17).
4. Diagonal $\psi_{ii}^2 \sim \text{Gamma}(a_i, b_i)$, default $a_i = b_i = 0.01$ (eq. 18).
5. Semiautomatic tuning: $\tau_{ki} = c_k\hat\sigma_{\phi_{s_i}}$ from OLS.

Five-Step Gibbs Sampler (Appendix A.3, eqs. 51–56)

1. $(\psi_{ii}^2|\phi,\omega;Y) \sim \text{Gamma}(a_i + T/2,\, B_i)$ where $B_i$ involves Schur complements of $S(\Phi) = (Y-X\Phi)'(Y-X\Phi)$ (eq. 51).
2. $(\eta_j|\phi,\omega,\psi;Y) \sim \mathcal{N}_{j-1}(\mu_j, \Delta_j)$ via normal-normal conjugacy (eq. 53).
3. $\omega_{ij} \sim \text{Bernoulli}$ via ratio of normal densities evaluated at current $\psi_{ij}$ (eq. 30).
4. $(\phi|\Psi,Y) \sim \mathcal{N}$ from normal-normal conjugate (eq. 56).
5. $(\gamma_i|\phi,\gamma_{(-i)},\eta,\psi;Y) \sim \text{Bernoulli}$ — does not depend on $Y$ (eq. 37; cf. George-McCulloch (1993)).

Identification Result

Upper-triangular $\Psi$ is globally identified: unique mapping $\Sigma \to \Psi$ via Cholesky. Arbitrary $\Psi$ satisfies only local rank conditions (Bekker-Pollock 1986); the decomposition may be non-unique with multiple valid $\Psi$ matrices.

Simulation Results (Section 4)

Example 1 — 6-variable VAR, $L=1$, $T=50$, 100 replications, 50K burn-in + 10K kept:

• $\Phi$ match: 30.9%, $\Psi$ match: 1.7%, joint match: 0.6% (out of $2^{51}$ models).
• RB forecast improvement over MLE: 47–67%; KL distance UB vs. RB: mean 44.36.

Example 2 — 4-variable VAR, $L=2$, 36 $\Phi$ elements + 6 $\Psi$ elements:

• $\Phi$ match: 0.02%, $\Psi$ match: 14.1%; RB improvement: 20–48%; KL mean 16.35.
• Key finding: restricting $\Phi$ aids $\Psi$ selection; joint search outperforms either alone.

Empirical Application (Section 5)

7-variable PPI-to-CPI VAR, $L=12$, two periods (1969:1–1980:12 and 1981:1–2001:12):

• Variables: Finished-goods PPI (FF), crude-materials PPI (CM), import-price index (IM), food-and-energy CPI (FC), CPI, unemployment rate (UNEMP), Fed Funds Rate (FFR).
• $|\Phi| = 595$ candidate $\Phi$ restrictions, $|\Psi| = 28$ candidate $\Psi$ restrictions.

1969–80: Supply-chain contemporaneous structure confirmed; links FF↔CM, CM↔IM, IM↔FC, IM&FC↔CPI, CPI↔UNEMP, UNEMP↔FFR. PPI lagged coefficients NOT significant predictors of CPI. Key CPI equation: $6.579\,\varepsilon_{CPI} = -0.060\,\varepsilon_{FF} + 0.022\,\varepsilon_{CM} + 1.054\,\varepsilon_{IM} + 0.533\,\varepsilon_{FC} + u_{CPI}$ (standard deviation, SD ≈ 0.152%).

Post-1981: FF-CM link breaks; FFR becomes endogenously responsive to IM, CPI, and UNEMP shocks contemporaneously — consistent with Volcker-era inflation targeting.

Concepts Introduced or Extended

• Variable Selection
• Vector Autoregression
• Markov Chain Monte Carlo
• Gibbs Sampler

Entities Mentioned

• Edward I. George
• Dongchu Sun
• Shawn Ni
• Robert E. McCulloch

Quotes

"The key advantage of SSVS over other model selection approaches is that, rather than selecting a single model, it uses all the information in the data by averaging over a large class of potential models." (paraphrase of paper's motivation)

"The Φ-search Gibbs sampler step 5 shows that γ does not depend on Y — the same property exploited in George and McCulloch (1993)." (eq. 37 and surrounding discussion)

My Take

The paper's main technical contribution is clean: it shows that the upper-triangular Cholesky decomposition of the precision matrix is both necessary for global identification and sufficient to keep all Gibbs conditionals in standard form. The result that restricting Φ helps Ψ selection (and vice versa) is intuitive but the simulation quantification is useful. The empirical application is informative but the two-period comparison is qualitative — no formal test is offered for whether the contemporaneous structure genuinely changed. The semiautomatic tuning rule inherited from George-McCulloch (1997) is convenient but the sensitivity of results to $c_0$ and $c_1$ deserves more attention in high-p settings.