Since OLS is applied to data with homoscedastic errors, the Gauss Markov theorem applies, and therefore the GLS estimate is the best linear unbiased estimator for " ? ".
2.
If the sample errors have equal variance ? 2 and are uncorrelated, then the least-squares estimate of ? is BLUE ( best linear unbiased estimator ), and its variance is easily estimated with
3.
The Gauss Markov theorem and Aitken demonstrate that the best linear unbiased estimator ( BLUE ), the unbiased estimator with minimum variance, has each weight equal to the reciprocal of the variance of the measurement.
4.
In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator.
5.
In other words it is the expectation of the square of the weighted sum ( across parameters ) of the differences between the estimators and the corresponding parameters to be estimated . ( Since we are considering the case in which all the parameter estimates are unbiased, this mean squared error is the same as the variance of the linear combination . ) The "'best linear unbiased estimator "'( BLUE ) of the vector \ beta of parameters \ beta _ j is one with the smallest mean squared error for every vector \ lambda of linear combination parameters.
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