Эконометрикадағы байланыссыз регрессиялар моделі

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, the maximum likelihood (ML) method under the assumption that the errors are normally distributed; the iterative generalized least squares (IGLS), where the residuals from the second step of FGLS are used to recalculate the matrix , then estimate again using GLS, and so on, until convergence is achieved; the iterative ordinary least squares (IOLS) scheme, where estimation is performed on equation by equation basis, but every equation includes as additional regressors the residuals from the previously estimated equations in order to account for the cross equation correlations, the estimation is run iteratively until convergence is achieved. Kmenta and Gilbert (1968) ran a Monte Carlo study and established that all three methods—IGLS, IOLS and ML—yield numerically equivalent results, they also found that the asymptotic distribution of these estimators is the same as the distribution of the FGLS estimator, whereas in small samples neither of the estimators was more superior than the others. Zellner and Ando (2010) developed a direct Monte Carlo method for the Bayesian analysis of SUR model.

There are two important cases when the SUR estimates turn out to be equivalent to the equation by equation OLS. 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 1=X1 = X2 = = Xm. That the estimates turn out to be numerically identical to OLS estimates follows from Kruskal's tree theorem,