Griffiths-Skeels-Chotikapanich (2002) Sample Size Requirements for Estimation in SUR Models

Summary

Demonstrates that the standard textbook requirement for estimating Seemingly Unrelated Regressions (SUR) models — $T \geq \max(M, k_{\max} + 1)$ — is both incomplete and potentially seriously misleading. Derives necessary conditions separately for two-stage feasible generalized least squares (FGLS) (Theorem 1: $T \geq M + \rho - \eta$ ) and for maximum likelihood (ML)/Bayesian estimation (Theorem 2: $T \geq M + \rho$ ), where $\rho = \text{rank}([X_1,\ldots,X_M])$ and $\eta$ captures how many equations have exclusive regressors. The ML/Bayesian requirement is always at least as stringent as two-stage, and strictly more stringent when equations have partially distinct regressors — because ML/Bayes minimizes a degree- $2M$ polynomial in $\beta$ rather than a quadratic. Empirically confirmed via experiments where ML/Gibbs sampler broke down at $M \geq 7$ while two-stage worked up to $M=18$ ( $T=19$ , $K_j=3$ per equation).

Key Claims

Theorem 1 (Two-stage FGLS): $T \geq M + \rho - \eta$ is necessary for $\hat\Sigma$ to be nonsingular with probability one, where $\rho = \text{rank}([X_1,\ldots,X_M])$ and $\eta = \text{rank}(D) \leq \min(d, \rho - \omega)$ with $d$ = number of equations whose regressors don't span $X$ and $\omega$ = dimension of regressors common to all equations. With probability one, $\eta = \min(d, \rho - \omega)$ , giving $T \geq M + \omega$ (high overlap) or $T \geq M + \rho - d$ (high distinctness).
Theorem 2 (ML and Bayesian): $T \geq M + \rho$ is necessary for the likelihood to be bounded and the posterior to be proper. Proof: $\text{rank}[Y, -X] = \min(T, M + \rho)$ with probability one, so rank deficiency of $E$ requires $T < M + \rho$ .
The gap between two-stage and ML/Bayes requirements: two-stage OLS residuals $\hat{e}_j = M_{X_j}y_j$ are orthogonal to their own regressors, enabling the rank decomposition $\hat{E} = M_V Y + D$ ; ML/Bayes residuals have no such constraint and solve a higher-dimensional optimization.
For the multivariate regression model ( $X_1 = \ldots = X_M$ ): both estimators require $T \geq M + k^*$ , so the gap vanishes.
For two-equation SUR with fully distinct regressors (model 17): two-stage requires only $T \geq 2$ ; ML/Bayes require $T \geq M + \rho = 4$ .
Standard software does not reliably detect the failure mode: SHAZAM (ML) sometimes found local maxima instead of detecting unboundedness; the Bayesian Gibbs sampler got stuck or broke from singularities.
Existence of moments of the two-stage estimator (Srivastava-Raj 1979) requires $T > M + K^* + 1$ , which is more demanding than mere feasibility.

Concepts Introduced or Extended

Seemingly Unrelated Regression
Maximum Likelihood Estimation
Feasible GLS
Noninformative Prior for VAR

Entities Mentioned

Quotes

"It is somewhat surprising, therefore, that the sample size requirements for joint estimation of the parameters in this model do not appear to have been correctly stated in the literature."

"For a 10-equation model with 3 distinct explanatory variables and a constant in each equation, two-stage estimation requires $T \geq 11$ … whereas ML and Bayesian estimation need $T > 40$ ."

My Take

A short, rigorous correctness paper with immediate practical implications. Theorem 2 is the critical result for Bayesian practitioners: any SUR system with $T < M + \rho$ produces an improper posterior under the noninformative prior, but standard Markov chain Monte Carlo (MCMC) may not detect this (the Gibbs sampler may simply get stuck or produce garbage). The two-stage bound (Theorem 1) is looser but depends on the overlap structure $\omega$ in a way that rewards heterogeneous regressor design. The paper motivates careful pre-flight checking of rank conditions before any SUR estimation.