Griffiths (2001) Bayesian Inference in the Seemingly Unrelated Regressions Model

Summary

A self-contained chapter on Bayesian inference for the $M$-equation Seemingly Unrelated Regressions (SUR) model, covering three Markov chain Monte Carlo (MCMC) algorithms (Gibbs over $\beta$ and $\Sigma$ jointly, Gibbs equation-by-equation, and Metropolis-Hastings random walk), three extensions (inequality restrictions, nonlinear SUR, missing data/Tobit), and predictive density computation. The noninformative prior $f(\beta,\Sigma) \propto |\Sigma|^{-(M+1)/2}$ yields an analytically intractable marginal posterior for $\beta$, motivating the MCMC approach. Three empirical applications illustrate the methods: wheat yield forecasting under mild inequality constraints, translog cost estimation under severe monotonicity constraints, and nonlinear expenditure functions for Thai households.

Key Claims

• Under the noninformative prior, $\beta|\Sigma \sim \text{MVN}(\hat\beta_{GLS}, [X'(\Sigma^{-1}\otimes I_T)X]^{-1})$ (multivariate normal) and $\Sigma|\beta \sim \text{IW}(T, A)$ (inverted Wishart), where $[A]_{ij} = (y_i - X_i\beta_i)'(y_j - X_j\beta_j)$ — both full conditionals are standard, enabling Gibbs sampling.
• The equation-by-equation Gibbs sampler draws $\beta_i|\beta_{-i}$ from a multivariate $t$ with $v_i = T - K_i$ degrees of freedom (eq. 19), avoiding the need to update $\Sigma$ explicitly.
• The marginal posterior predictive $f(y^*|\beta,y)$ is multivariate $t$ with $v^* = T - M + 1$ df, mean $X^*\beta$, and scale $A/(v^* - 2)$; in practice it is approximated by averaging over MCMC draws of $\beta$.
• Inequality restrictions are handled by including the indicator $I_S(\beta)$ in the prior; $\beta|\Sigma$ then follows a truncated MVN, sampled via rejection or by substituting $z = Z\gamma + e$ for linear equality restrictions.
• Nonlinear SUR requires Metropolis-Hastings because $\beta|\Sigma$ is no longer Gaussian; the Gibbs algorithms are inapplicable.
• Percy (1992) generates $(y^*, \beta, \Sigma)$ jointly in Gibbs; Griffiths derives the predictive pdf conditional on $\beta$ analytically and averages over MCMC draws — more efficient but restricted to linear models.

Concepts Introduced or Extended

• Seemingly Unrelated Regression
• Gibbs Sampler
• Metropolis-Hastings
• Inverted Wishart Distribution
• Noninformative Prior for VAR
• Predictive Density
• Inequality Restriction in Bayesian Inference

Entities Mentioned

• William E. Griffiths
• Zellner (1962)
• Percy (1992)
• Zellner (1971)

Quotes

"The general Gibbs sampler cycles through all elements of the parameter vector, drawing each one from its conditional posterior distribution."

"In many situations restrictions of the form $g(\beta) \geq 0$ that are likely to hold will not hold for all values of $\beta$. These can be treated as prior information, with a prior density that is nonzero only for those values of $\beta$ that satisfy the restrictions."

My Take

A comprehensive pedagogical reference for Bayesian SUR. The equation-by-equation Gibbs sampler (Section 3.2) is the most practically useful result — it eliminates the need to invert the full $\Sigma$ at each iteration and reduces to univariate $t$ conditionals. The nonlinear and inequality-restriction extensions are well-motivated by the three applications but the treatment is somewhat brief on convergence diagnostics. The comparison with Percy (1992) clarifies an important design choice in MCMC-based predictive inference.