SUR — Seemingly Unrelated Regressions

SUR (Seemingly Unrelated Regressions) estimates multiple equations jointly when they appear independent but their errors are correlated across equations. By exploiting that correlation (via GLS), SUR yields more efficient estimates than running OLS on each equation separately.

When to use

Use SUR when multiple equations share common shocks (e.g. expenditure systems across goods, demand functions across sectors) — no endogeneity but correlated errors. If there is endogeneity ⇒ 3SLS.

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Model specification

A system of $m$ equations $Y_g = X_g \beta_g + \varepsilon_g$ with $\mathrm{Cov}(\varepsilon_g, \varepsilon_h) \ne 0$. SUR is estimated by FGLS using the cross-equation error covariance $\Sigma$:

$$\hat{\beta}_{SUR} = \big(X'(\Sigma^{-1} \otimes I)X\big)^{-1} X'(\Sigma^{-1}\otimes I)Y$$

When equations share the same regressors or errors are uncorrelated, SUR reduces to equation-by-equation OLS.

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Running in EcoLab

1. Modeling module → IV & simultaneous equations family → SUR.
2. Declare the equations (each with its own $Y_g$ and $X_g$).
3. Run; read system-wide coefficients + the error-correlation matrix; export the replication code.

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Replication code

Stata

* ── SUR estimation ────────────────────────────────
* Equation 1: y1 depends on x1, x2
* Equation 2: y2 depends on x3, x4
sureg (eq1: y1 x1 x2) (eq2: y2 x3 x4)

* Test cross-equation restrictions
test [eq1]x1 = [eq2]x3

R

# ── SUR estimation ────────────────────────────────
library(systemfit)

# Define the system of equations
eq1 <- y1 ~ x1 + x2
eq2 <- y2 ~ x3 + x4
sys <- list(eq1 = eq1, eq2 = eq2)

# Estimate with SUR (Seemingly Unrelated Regressions)
model_sur <- systemfit(sys, method = "SUR", data = df)
summary(model_sur)

# Cross-equation error correlation matrix
model_sur$residCov

Python

# ── SUR estimation ────────────────────────────────
from linearmodels.system import SUR

# Define equations as a dictionary of formulas
equations = {
    "eq1": {"dependent": df["y1"],
            "exog": df[["x1", "x2"]]},
    "eq2": {"dependent": df["y2"],
            "exog": df[["x3", "x4"]]},
}

model = SUR(equations)
results = model.fit(cov_type="robust")
print(results)

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Limitations

• Efficiency gains are small when errors are weakly correlated or equations share regressors.
• Sensitive to misspecification in any equation.

Video tutorial

Video Tutorial: Guide to running SUR in EcoLab

See also

• 3SLS · IV/2SLS · Catalog