Griffiths-Valenzuela (2002) A Gibbs Sampler for the Estimation of Seemingly Unrelated Regressions with Cross Set Restrictions on the Parameters

Summary

Griffiths and Valenzuela (2002) extend Bayesian Seemingly Unrelated Regressions (SUR) estimation to a multi-set structure where $H$ groups of SUR equations share a common coefficient vector $\eta$ while each group maintains its own set-specific parameters $\Theta_h$ and error covariance matrix $\Omega_h$. They derive an all-conjugate three-block Gibbs sampler — Inverted-Wishart (IW) for each $\Omega_h$, Normal for each $\Theta_h$ (within-set generalized least squares, GLS), and Normal for $\eta$ (pooled GLS across all $H$ sets) — and apply it to two Australian datasets: household equivalence scale estimation via the Extended Linear Expenditure System (ELES) with 5,532 households across 8 demographic types, and Merino wool translog cost function estimation across 23 years.

Key Claims

• Model: $Y_h = Z_h\Theta_h + X_h\eta + e_h$, $h = 1,\ldots,H$ sets; $e_h \sim N(0, \Omega_h \otimes I_{M_h})$; each set has $M_h$ observations, $N_h$ equations, set-specific $\Theta_h$, and common $\eta$ across all $H$ sets
• Each set has its own unrestricted $\Omega_h$ — unlike Baltagi (1995) who assumes the same covariance matrix across sets
• Block 1: $\Omega_h \mid \Theta_h, \eta, Y \sim \text{IW}(A_h, M_h)$ where $A_h$ is the residual outer product matrix within set $h$
• Block 2: $\Theta_h \mid \Omega_h, \eta, Y \sim N(\hat\Theta_h, [Z_h'(\Omega_h^{-1} \otimes I_{M_h})Z_h]^{-1})$ — within-set GLS estimator; the $\eta$ contribution is subtracted from the dependent variable before fitting
• Block 3: $\eta \mid \{\Omega_h\}, \{\Theta_h\}, Y \sim N(\hat\eta, W^{-1})$ where $W = \sum_h X_h'(\Omega_h^{-1} \otimes I_{M_h})X_h$ and $\hat\eta = W^{-1}Q$, $Q = \sum_h X_h'(\Omega_h^{-1} \otimes I_{M_h})(\tilde y_h - \tilde Z_h\Theta_h)$ — pooled GLS update across all $H$ sets; this block is the novel contribution relative to single-set SUR Gibbs
• Application 1 (ELES equivalence scales): 5,532 Australian households, $H=8$ demographic types, $N_h=11$ commodities; $y_{ih} = \theta_{ih} + \eta_i \cdot x_h$ where $\theta_{ih}$ are set-specific intercepts, $\eta_i$ are common marginal budget shares; equivalence scales $s_{ih} = a_{ih}/a_{ir}$; general scales from equating indirect utility functions; 18,000 draws (3,000 burn-in), SHAZAM; first child raises two-adult budget by ~23%; economies of scale greatest for the 2nd child
• Application 2 (Merino wool translog): 310 observations, $H=23$ years (sets), time-varying intercepts $\Theta_h$ with common $\alpha_i$, $\alpha_{ij}$ technology parameters; 23,000 draws (3,000 burn-in)
• Conference paper presented at the 2002 Australasian Meeting of the Econometric Society, Brisbane; not formally published in a journal

Concepts Introduced or Extended

• Gibbs Sampler
• Seemingly Unrelated Regression
• Equivalence Scale

Entities Mentioned

• William E. Griffiths
• Ma. Rebecca Valenzuela
• Percy (1992)
• Chib-Greenberg (1995b)

Quotes

"The key feature of our model is that $\eta$ is common to all sets, while $\Theta_h$ and $\Omega_h$ are set-specific."

My Take

The paper's main methodological contribution is Block 3 — the pooled-GLS update for $\eta$ that aggregates information across all $H$ sets. This is an elegant and computationally natural extension of single-set SUR Gibbs (Percy 1992, Chib-Greenberg 1995b) to the multi-set case. The three-block sampler is fully conjugate, requiring no Metropolis steps, and scales cleanly with $H$. The main limitation is that no convergence diagnostics are reported (unusual for a 2002 Bayesian paper), and the comparison with Baltagi (1995) is informal — there is no formal equivalence test or simulation study comparing the two $\Omega_h$ specifications. The ELES application provides plausible equivalence scale estimates, but the translog application adds little methodological insight beyond showing the sampler runs on a different dataset.