Seemingly unrelated regression

In econometrics, seemingly unrelated regression (SUR), model developed in Zellner (1962), is a technique for analyzing a system of multiple equations with cross-equation parameter restrictions 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 across the equations. SUR is an extension of the linear regression model which allows correlated errors between equations.

Suppose that the Gauss-Markov assumptions hold for all the equations. Then the OLS estimates are BLUE. However, by using the SUR method to estimate the equations jointly, efficiency is improved.

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 &times; M matrix representing the covariance of residuals between the equations. Even though each equation satisfies the OLS assumptions, the joint model exhibits serial correlation due to the correlation of the error terms. Standard OLS estimation, then, will be inefficient (unless all the equations have the identical explanatory variables). Thus, SUR uses generalized least squares to estimate $$\beta$$:



\hat{\beta}_{SUR} = \left(X^\prime V^{-1} X\right)^{-1} X^\prime V^{-1} Y $$

where



V(Y)=\Sigma\otimes I_N $$

where $$\otimes$$ is the Kronecker product and V(Y) is an M &times; N matrix.

Once SUR model estimates are obtained, inferences are mainly about testing the validity of cross-equation parameter restrictions.