Bilodeau-Duchesne (2000) Robust Estimation of the SUR Model

Summary

Bilodeau and Duchesne adapt S-estimators — high breakdown-point regression estimators introduced by Rousseeuw and Yohai (1984) — to the seemingly unrelated regression (SUR) model. The resulting estimator minimises $|\Sigma|$ subject to a constraint on a ρ-function applied to the Mahalanobis residuals $d_i = (e_i'\Sigma^{-1}e_i)^{1/2}$ , bridging single-equation robust regression and multivariate robust location/scatter estimation. Unlike Zellner's generalized least squares (GLS) (zero breakdown point) and Koenker-Portnoy's M-estimators (not affine equivariant), the S-estimator can detect multivariate outliers across all equations simultaneously. A modified Ruppert (1992) algorithm provides fast computation; $\sqrt{n}$ -consistency and asymptotic normality justify bootstrap standard errors (SE).

Key Claims

Zero breakdown point of GLS: Zellner's two-stage Aitken estimator is a least-squares estimator in the stacked system and has breakdown point (BP) = 0 — one outlier can move estimates arbitrarily far.
S-estimator definition for SUR: $\min_{(\beta,\Sigma)} |\Sigma|$ subject to $\frac{1}{n}\sum_{i=1}^n \rho[(e_i'\Sigma^{-1}e_i)^{1/2}] = b$ , with $b = E_{F_0}\rho(|r|)$ ensuring consistency at the elliptical target $E_q(0,I)$ . Breakdown point $\lambda = b/\rho(c) \leq 0.5$ .
Estimating equations (reduce to the maximum likelihood estimator, MLE, when $\rho(d)=d^2$ ): $\beta = \{X'(\Sigma^{-1}\otimes D_u)X\}^{-1}X'(\Sigma^{-1}\otimes D_u)y$ and $\Sigma = q(Y-\tilde{X}B)'D_u(Y-\tilde{X}B)/\sum_i v(d_i)$ , where $u(d)=\rho'(d)/d$ , $v(d)=\rho'(d)d-\rho(d)+b$ .
Affine equivariance: The S-estimator satisfies $\hat\beta(\mathbf{D}y+\mathbf{X}\mathbf{C}v,\mathbf{X}\mathbf{C})=\mathbf{D}\mathbf{C}^{-1}\hat\beta(y,\mathbf{X})+v$ , for block-diagonal $\mathbf{D}$ , $\mathbf{C}$ — unlike Koenker-Portnoy M-estimators.
$\sqrt{n}$ -consistency: Contrary to least median of squares / minimum volume ellipsoid (LMS/MVE) (convergence rate $n^{1/3}$ , non-normal limit), S-estimators converge at rate $n^{1/2}$ with asymptotic normal distribution, so bootstrap SE is valid by resampling centered multivariate residuals.
Efficiency-BP tradeoff: BP=50% → efficiency≈29%; BP=40% → $\lambda^{-1}\approx74\%$ for $\hat\beta$ , $\sigma_1^{-1}\approx58\%$ for $\hat\Sigma$ . Table 2 reports $(\lambda, \sigma_1, \sigma_2)$ for $q=2,\ldots,10$ and BP $r=0.1,\ldots,0.5$ .
Modified Ruppert algorithm: Adapted to the SUR context — step 1 uses ordinary least squares (OLS) rather than MLE on subsamples; local improvement via line search between the current iterate and the subsample OLS solution; Ruppert's condition on the scale enforces descent in $|\Sigma|$ .
Grunfeld application: GE + Westinghouse 1935–1954 ( $n=20$ , BP=40%). Robust correlation $\hat\rho_S=0.85$ vs. MLE's $\hat\rho=0.77$ . Multivariate diagnostic plot (bivariate Mahalanobis distance of residuals) reveals 1946 and 1952 as joint outliers — partly hidden in univariate residual plots. Postwar explanation: transitional year (1946) and GE's jet-engine expansion (1952).

Concepts Introduced or Extended

Seemingly Unrelated Regression

Entities Mentioned

Quotes

"Unlike the MLE and Koenker & Portnoy's estimator (1990), the S-estimator gives weight to the n observations in the multivariate regression $Y = \tilde{X}B + E$ according to multivariate residuals $e_i$ . It is thus suited to detect not only univariate (in each of the q models) outliers but multivariate outliers as well."

My Take

A technically clean paper that fills a clear gap: SUR robustness via a principled high-breakdown estimator. The asymptotic normality result (which requires only elliptical errors, not normality) is the main theoretical contribution; it makes bootstrap inference credible in small samples. The paper is peripheral to this wiki's Bayesian time-series focus — there is no Bayesian content, no time-series dynamics, and the application (20 annual observations, two equations) is a classic cross-sectional SUR illustration. The connection to the wiki is through the SUR model and its presence in Bayesian VAR and hierarchical modelling contexts (Percy 1992, Chib-Greenberg 1995b).