In econometrics, the seemingly unrelated regressions (SUR) or seemingly unrelated regression equations (SURE) model, proposed by Arnold Zellner in (1962), is a generalization of a linear regression model that consists of several regression equations, each having its own dependent variable and potentially different sets of exogenous explanatory variables. Each equation is a valid linear regression on its own and can be estimated separately, which is why the system is called seemingly unrelated, although some authors suggest that the term seemingly related would be more appropriate, since the error terms are assumed to be correlated across the equations.
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The model can be estimated equation-by-equation using standard ordinary least squares (OLS). Such estimates are consistent, however generally not as efficient as the SUR method, which amounts to feasible generalized least squares with a specific form of the variance-covariance matrix. Two important cases when SUR is in fact equivalent to OLS are when the error terms are in fact uncorrelated between the equations (so that they are truly unrelated) and when each equation contains exactly the same set of regressors on the right-hand-side.
The SUR model can be viewed as either the simplification of the general linear model where certain coefficients in matrix
The model
Suppose there are m regression equations
Here i represents the equation number, r = 1, …, R is the time period and we are taking the transpose of the
Each equation i has a single response variable yir, and a ki-dimensional vector of regressors xir. If we stack observations corresponding to the i-th equation into R-dimensional vectors and matrices, then the model can be written in vector form as
where yi and εi are R×1 vectors, Xi is a R×ki matrix, and βi is a ki×1 vector.
Finally, if we stack these m vector equations on top of each other, the system will take the form
The assumption of the model is that error terms εir are independent across time, but may have cross-equation contemporaneous correlations. Thus we assume that E[ εir εis | X ] = 0 whenever r ≠ s, whereas E[ εir εjr | X ] = σij. Denoting Σ = [[σij]] the m×m skedasticity matrix of each observation, the covariance matrix of the stacked error terms ε will be equal to
where IR is the R-dimensional identity matrix and ⊗ denotes the matrix Kronecker product.
Estimation
The SUR model is usually estimated using the feasible generalized least squares (FGLS) method. This is a two-step method where in the first step we run ordinary least squares regression for (1). The residuals from this regression are used to estimate the elements of matrix
In the second step we run generalized least squares regression for (1) using the variance matrix
This estimator is unbiased in small samples assuming the error terms εir have symmetric distribution; in large samples it is consistent and asymptotically normal with limiting distribution
Other estimation techniques besides FGLS were suggested for SUR model: the maximum likelihood (ML) method under the assumption that the errors are normally distributed; the iterative generalized least squares (IGLS), were the residuals from the second step of FGLS are used to recalculate the matrix
Equivalence to OLS
There are two important cases when the SUR estimates turn out to be equivalent to the equation-by-equation OLS, so that there is no gain in estimating the system jointly. These cases are:
- When the matrix Σ is known to be diagonal, that is, there are no cross-equation correlations between the error terms. In this case the system becomes not seemingly but truly unrelated.
- When each equation contains exactly the same set of regressors, that is X1 = X2 = … = Xm. That the estimators turn out to be numerically identical to OLS estimates follows from Kruskal's theorem, or can be shown via the direct calculation.
Statistical packages
syslin
procedure.sureg
command.sure
command