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5.5 The Theoretical Foundations of Ordinary Least Squares 211

The Gauss–Markov Theorem for bn1 Key Concept

If the three least squares assumptions in Key Concept 4.3 hold and if errors are 5.5
homoskedastic, then the OLS estimator bn1 is the Best (most efficient) Linear
conditionally Unbiased Estimator (BLUE).

squares assumptions plus the assumption that the errors are homoskedastic. Con-
sequently, if the three least squares assumptions hold and the errors are homo-
skedastic, then OLS is BLUE. The Gauss–Markov theorem is stated in Key
Concept 5.5 and proven in Appendix 5.2.

Limitations of the Gauss–Markov theorem. The Gauss–Markov theorem provides
a theoretical justification for using OLS. However, the theorem has two important
limitations. First, its conditions might not hold in practice. In particular, if the error
term is heteroskedastic—as it often is in economic applications—then the OLS
estimator is no longer BLUE. As discussed in Section 5.4, the presence of hetero-
skedasticity does not pose a threat to inference based on heteroskedasticity-robust
standard errors, but it does mean that OLS is no longer the efficient linear condi-
tionally unbiased estimator. An alternative to OLS when there is heteroskedasticity
of a known form, called the weighted least squares estimator, is discussed below.

The second limitation of the Gauss–Markov theorem is that even if the condi-
tions of the theorem hold, there are other candidate estimators that are not linear
and conditionally unbiased; under some conditions, these other estimators are
more efficient than OLS.

Regression Estimators Other Than OLS

Under certain conditions, some regression estimators are more efficient than OLS.

The weighted least squares estimator. If the errors are heteroskedastic, then OLS
is no longer BLUE. If the nature of the heteroskedasticity is known—specifically,
if the conditional variance of ui given Xi is known up to a constant factor of
proportionality—then it is possible to construct an estimator that has a smaller
variance than the OLS estimator. This method, called weighted least squares
(WLS), weights the ith observation by the inverse of the square root of the condi-
tional variance of ui given Xi. Because of this weighting, the errors in this weighted
regression are homoskedastic, so OLS, when applied to the weighted data, is BLUE.