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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: either when the error terms are in fact uncorrelated between the equations (so that they are truly unrelated), or 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 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 : Here ''i'' represents the equation number, is the observation index and we are taking the transpose of the column vector. The number of observations ''R'' is assumed to be large, so that in the analysis we take , whereas the number of equations ''m'' remains fixed. Each equation ''i'' has a single response variable ''y''''ir'', and a ''k''''i''-dimensional vector of regressors ''x''''ir''. 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 ''y''''i'' and ''ε''''i'' are ''R''×1 vectors, ''X''''i'' is a ''R''×''k''''i'' matrix, and ''β''''i'' is a ''k''''i''×1 vector. Finally, if we stack these ''m'' vector equations on top of each other, the system will take form 〔 : The assumption of the model is that error terms ''ε''''ir'' are independent across time, but may have cross-equation contemporaneous correlations. Thus we assume that whenever , whereas . Denoting the ''m×m'' skedasticity matrix of each observation, the covariance matrix of the stacked error terms ''ε'' will be equal to 〔〔 : where ''I''''R'' is the ''R''-dimensional identity matrix and ⊗ denotes the matrix Kronecker product. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Seemingly unrelated regressions」の詳細全文を読む スポンサード リンク
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