Seemingly unrelated regression

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In econometrics, seemingly unrelated regression (SUR) is a technique for analyzing a model with multiple equations and correlated error terms.

An economic model may contain multiple equations which are independent of each other on the surface: they are not estimating the same dependent variable, they have different independent variables, etc. However, if the equations are using the same data, the errors may be correlated between the two equations. SUR is an extension of the linear regression model which allows correlated errors between equations.

The mathematics is very similar to computing Huber-White standard errors. Suppose we have a series of equations

[y_i = x_i \beta_i + \varepsilon_i]

where x, [\beta], and [\varepsilon] are vectors and i = 1, ..., M where M is the number of equations. Assume each equation has N observations. Let [\Sigma] be an M × M matrix representing the covariance of residuals between the equations. Then SUR is merely computing the GLS estimation for [\beta]:

[\hat_ = \left(X^\prime V^ X\right)^ X^\prime V^ Y]

where

[V(Y)=\Sigma\otimes I_N]

where [\otimes] is the Kronecker product and V(Y) is an M × N matrix.

References

[Nawar, Abdel-Hameed (2005) "Seemingly Unrelated Regressions (SUR): An Introduction by Examples," A Statistics Lecture Note, New York University, July, mimeo]

Chapter 17.

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