Seemingly Unrelated Regression (SUR)

Seemingly unrelated regression (SUR) estimates several linear equations jointly by feasible generalized least squares, exploiting correlation between their error terms to gain efficiency.

SUR(겉보기 무관 회귀)은 여러 회귀식을 함께 추정하면서 식 사이 오차 상관을 이용해 FGLS로 효율을 높이는 방법입니다.

I have several related outcome equations, like demand for different products, whose errors are probably correlated; should I estimate them together?

오차가 서로 상관될 것 같은 여러 결과 회귀식을 따로 추정해야 하나요, 함께 추정해야 하나요?

Seemingly unrelated regression fits each outcome's own equation but estimates them jointly, using the estimated covariance of errors across equations to improve precision. The gain appears only when errors are correlated and the equations have different predictors; with identical predictors the estimates equal equation-by-equation OLS.

When to use Seemingly Unrelated Regression (SUR)

• Several outcomes are measured on the same units and their errors are likely correlated
• Equations contain different sets of predictors
• You want to test restrictions across equations

When to use something else

• All equations share exactly the same predictors and you only need coefficients; use multivariate multiple regression instead
• Some predictors are endogenous; use three-stage least squares or instrumental variable methods instead

Assumptions and requirements

• Each equation is linear and correctly specified
• Errors are correlated across equations but independent across units
• Predictors are exogenous
• Error covariance is constant across units

What the analysis reports

• Coefficients for each equation with standard errors and tests
• Equation and system R²
• Breusch-Pagan test that the error covariance is diagonal
• Residual correlations across equations

How to report it

A seemingly unrelated regression of [outcome 1] and [outcome 2] showed correlated errors, Breusch-Pagan χ²([df]) = [value], p = [value]; [predictor] predicted [outcome 1], b = [value], SE = [value], p = [value].

Related methods

• Multivariate Multiple Regression
• Multiple Linear Regression

Run Seemingly Unrelated Regression (SUR) on Quorant