Seemingly Unrelated Regressions

Imagine estimating a separate return-predicting regression for each of ten sector ETFs, each with its own sector-specific predictors. Run one at a time with ordinary least squares and each equation is estimated as if the others don't exist. But the error terms — the part of each sector's return the regression can't explain — are usually correlated across sectors, because broad market-wide shocks hit every sector's residual on the same day. Seemingly unrelated regressions (SUR) exploits exactly that correlation: it estimates all the equations jointly, using the cross-equation residual correlation to make every equation's coefficient estimates more precise than fitting each one alone.

The mechanism is a two-step, feasible generalized least squares procedure: first, run each equation separately by ordinary least squares to get initial residuals; second, use those residuals to estimate the cross-equation error covariance matrix, and then re-estimate all equations jointly using that covariance structure to weight the combined system. The gain from doing this shrinks as the equations' predictors become more similar (SUR reduces to plain equation-by-equation OLS when every equation uses identical predictors) and grows as the residual correlation across equations gets larger.

The name reflects the situation, not the method: equations that look "seemingly unrelated" because they involve different variables for different entities turn out to share information through their correlated errors — and SUR is precisely the tool for extracting that shared information. One practical footnote: if every equation happens to use exactly the same set of predictors, joint estimation offers no efficiency gain over equation-by-equation OLS at all, since there's no extra cross-equation information left for SUR to exploit.