14.4 Seemingly unrelated regressions
Zellner (1962) considered \(G\) equations that are related only through their disturbances: \[ y_i = X_i\beta _i + u_i, \qquad i = 1, \dots , G, \qquad E(u_{it}u_{jt}) = \sigma _{ij}, \qquad E(u_{it}u_{js}) = 0 \ (t \neq s) . \] Each equation satisfies the classical assumptions on its own, and least squares is unbiased equation by equation. But demand equations for several goods are hit by common shocks; so are investment equations for firms in the same industry, or death rates for different groups of road users in the same month. Stacking the equations as \(y = X\beta + u\), with \(X = \operatorname {diag}(X_1, \dots , X_G)\) block diagonal, gives \(\Var (u) = \Sigma \otimes I_n\), where \(\otimes \) is the Kronecker product. Aitken’s theorem then gives the GLS estimator \[ b_{\text {SUR}} = \bigl [X'(\Sigma ^{-1} \otimes I_n)X\bigr ]^{-1}X'(\Sigma ^{-1} \otimes I_n)y , \] using \((\Sigma \otimes I)^{-1} = \Sigma ^{-1} \otimes I\). In practice \(\Sigma \) is estimated from the least squares residuals, \(\hat \sigma _{ij} = e_i'e_j/n\), giving feasible SUR.
Theorem 14.4.1. SUR is identical to least squares equation by equation if either
- (i)
- \(\Sigma \) is diagonal; or
- (ii)
- every equation has the same regressors, \(X_i = X_0\) for all \(i\).
More generally, if the regressors of one equation are a subset of those of another, SUR gives least squares for the equation with fewer regressors.
Proof. (i) If \(\Sigma \) is diagonal, \(X'(\Sigma ^{-1} \otimes I)X\) is block diagonal with blocks \(X_i'X_i/\sigma _{ii}\), and \(X'(\Sigma ^{-1} \otimes I)y\) has blocks \(X_i'y_i/\sigma _{ii}\); the \(\sigma _{ii}\) cancel.
(ii) Now \(X = I_G \otimes X_0\). Using \((A \otimes B)(C \otimes D) = AC \otimes BD\), \[ X'(\Sigma ^{-1} \otimes I)X = \Sigma ^{-1} \otimes X_0'X_0, \qquad X'(\Sigma ^{-1} \otimes I)y = (\Sigma ^{-1} \otimes X_0')y, \] so \(b_{\text {SUR}} = (\Sigma \otimes (X_0'X_0)^{-1})(\Sigma ^{-1} \otimes X_0')y = (I_G \otimes (X_0'X_0)^{-1}X_0')y\), which is least squares on each equation. The subset result is proved in the same way, partitioning the larger regressor matrix; we verify it numerically in Software 14.7.2. □
The efficiency gain comes from information in the other equations’ residuals about this equation’s disturbance, and it requires both correlated disturbances and regressors that differ across equations. It is largest when the correlation is high and the regressors of the different equations are weakly correlated with each other. Whether SUR is worth the trouble can be tested: under \(H_0\colon \Sigma \) diagonal, the Breusch–Pagan LM statistic \(n\sum _{i<j}r_{ij}^2\), with \(r_{ij}\) the correlation between the least squares residuals of equations \(i\) and \(j\), is asymptotically \(\chi ^2_{G(G-1)/2}\).
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