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. 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 B {\displaystyle \mathrm {B} } are restricted to be equal to zero, or as the generalization of the general linear model where the regressors on the right-hand-side are allowed to be different in each equation. The SUR model can be further generalized into the simultaneous equations model, where the right-hand side regressors are allowed to be the endogenous variables as well.
The model Suppose there are m regression equations
y i r = x i r T β i + ε i r , i = 1 , … , m . {\displaystyle y_{ir}=x_{ir}^{\mathsf {T}}\;\!\beta _{i}+\varepsilon _{ir},\quad i=1,\ldots ,m.}
Here i represents the equation number, r = 1, …, R is the individual observation, and we are taking the transpose of the x i r {\displaystyle x_{ir}} column vector. The number of observations R is assumed to be large, so that in the analysis we take R → ∞ {\displaystyle \infty } , whereas the number of equations m remains fixed. 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
y i = X i β i + ε i , i = 1 , … , m , {\displaystyle y_{i}=X_{i}\beta _{i}+\varepsilon _{i},\quad i=1,\ldots ,m,}
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 observations, but may have cross-equation correlations within observations. 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
Ω ≡ E [ ε ε T | X ] = Σ ⊗ I R , {\displaystyle \Omega \equiv \operatorname {E} [\,\varepsilon \varepsilon ^{\mathsf {T}}\,|X\,]=\Sigma \otimes I_{R},}
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 Σ {\displaystyle \Sigma } :
σ ^ i j = 1 R ε ^ i T ε ^ j . {\displaystyle {\hat {\sigma }}_{ij}={\frac {1}{R}}\,{\hat {\varepsilon }}_{i}^{\mathsf {T}}{\hat {\varepsilon }}_{j}.}
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