Seemingly Unrelated Regressions - Estimation

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:

\hat\sigma_{ij} = \frac1T\, \hat\varepsilon_i' \hat\varepsilon_j .

In the second step we run generalized least squares regression for (1) using the variance matrix :

\hat\beta = \Big( X'(\hat\Sigma^{-1}\otimes I_T) X \Big)^{\!-1} X'(\hat\Sigma^{-1}\otimes I_T)\,y .

This estimator is unbiased in small samples assuming the error terms εit have symmetric distribution; in large samples it is consistent and asymptotically normal with limiting distribution

\sqrt{T}(\hat\beta - \beta) \ \xrightarrow{d}\ \mathcal{N}\Big(\,0,\; \Big(\tfrac1T X'(\Sigma^{-1}\otimes I_T) X \Big)^{\!-1}\,\Big) .

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, then estimate again using GLS, and so on, until convergence is achieved; the iterative ordinary least squates (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 & Gilbert (1968) ran a Monte-Carlo study and established that all three methods — IGLS, IOLS and ML — yield the 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.

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