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210 Chapter 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

with a discussion of alternative estimators that are more efficient than OLS when
the conditions of the Gauss–Markov theorem do not hold.

Linear Conditionally Unbiased Estimators and
the Gauss–Markov Theorem

If the three least squares assumptions (Key Concept 4.3) hold and if the error is
homoskedastic, then the OLS estimator has the smallest variance, conditional on
X1, c, Xn, among all estimators in the class of linear conditionally unbiased esti-
mators. In other words, the OLS estimator is the Best Linear conditionally Unbi-
ased Estimator—that is, it is BLUE. This result is an extension of the result,
summarized in Key Concept 3.3, that the sample average Y is the most efficient
estimator of the population mean among the class of all estimators that are unbi-
ased and are linear functions (weighted averages) of Y1, c, Yn.

Linear conditionally unbiased estimators. The class of linear conditionally unbi-

ased estimators consists of all estimators of b1 that are linear functions of
Y1, c, Yn and that are unbiased, conditional on X1, c, Xn. That is, if b1 is a
linear estimator, then it can be written as

n

b1 = ia= 1aiYi ( b1 is linear), (5.24)

where the weights a1, c, an can depend on X1, c, Xn but not on Y1, c, Yn.
The estimator ∼b1 is conditionally unbiased if the mean of its conditional sampling
distribution, given X1, c, Xn, is b1. That is, the estimator ∼b1 is conditionally

unbiased if

E(b1 ͉ X1, c, Xn) = b1 (b1 is conditionally unbiased). (5.25)

The estimator b1 is a linear conditionally unbiased estimator if it can be written
in the form of Equation (5.24) (it is linear) and if Equation (5.25) holds (it is con-
ditionally unbiased). It is shown in Appendix 5.2 that the OLS estimator is linear
and conditionally unbiased.

The Gauss–Markov theorem. The Gauss–Markov theorem states that, under a set
of conditions known as the Gauss–Markov conditions, the OLS estimator bn1 has
the smallest conditional variance, given X1, c, Xn, of all linear conditionally
unbiased estimators of b1; that is, the OLS estimator is BLUE. The Gauss–Markov
conditions, which are stated in Appendix 5.2, are implied by the three least