Seemingly Unrelated Regressions (SUR)
Also known as: SUR, Zellner's SUR, seemingly unrelated regression equations, Görünürde İlişkisiz Regresyon (SUR)
Seemingly Unrelated Regressions, introduced by Arnold Zellner in 1962, is a system regression method that estimates several linear equations jointly when their error terms are correlated across equations. By exploiting that cross-equation correlation through generalized least squares, it is more efficient than estimating each equation separately by OLS.
Key highlights
- More efficient than equation-by-equation OLS when the equation errors are correlated across equations.
- Allows each equation to carry its own, possibly different, set of predictors.
- Well suited to jointly estimated economic systems such as demand and cost-function systems.
Intuition
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How it works
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When to use it
Use SUR when you have a system of several continuous-outcome linear equations whose error terms are plausibly correlated across equations but not autocorrelated, on cross-sectional or panel data with a reasonably large sample (about 100 observations or more). It shines in demand systems, cost-function systems, and other settings where related equations share unobserved influences. If the equations have identical regressors, or the cross-equation error correlation is negligible, SUR reduces to ordinary OLS and offers no gain.
Strengths & limitations
- More efficient than equation-by-equation OLS when the equation errors are correlated across equations.
- Allows each equation to carry its own, possibly different, set of predictors.
- Well suited to jointly estimated economic systems such as demand and cost-function systems.
- Offers no efficiency gain over OLS when the cross-equation error correlation is zero or when all equations share the same regressors.
- Relies on a correctly specified linear system; misspecification in one equation can contaminate the others.
- Assumes errors are correlated across equations but free of autocorrelation, and typically needs a sizeable sample (around 100+).
Common pitfalls
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Applications
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Frequently asked
How is SUR different from running separate OLS regressions?
Separate OLS treats each equation in isolation. SUR estimates them jointly and weights the system by the cross-equation error covariance, so information about how the errors move together makes every coefficient more efficient — provided that correlation is actually present.
When does SUR give the same answer as OLS?
Two cases: when the error terms are uncorrelated across equations, or when all equations contain exactly the same regressors. In both, SUR collapses to equation-by-equation OLS and there is nothing to gain.
What does the Σ⁻¹ ⊗ I term mean?
Σ is the covariance matrix of the errors across the M equations. The Kronecker product Σ⁻¹ ⊗ I spreads that cross-equation weighting across all observations, so the generalized least squares estimator down-weights noisy equations and exploits the correlation structure.
How does SUR relate to 3SLS?
Three-Stage Least Squares extends the SUR idea to systems with endogenous regressors: it combines instrumental-variables (2SLS) estimation with the same cross-equation error-correlation weighting that SUR uses. SUR itself assumes the regressors are exogenous.
Sources
- 1.Zellner, A. (1962). An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias. Journal of the American Statistical Association, 57(298), 348-368.
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ScholarGate. (2026, June 1). Seemingly Unrelated Regression. ScholarGate. https://scholargate.app/econometrics/seemingly-unrelated-regression