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

statement “the variance of earnings is the same for men as it is for women.” In
other words, in this example, the error term is homoskedastic if the variance of
the population distribution of earnings is the same for men and women; if these
variances differ, the error term is heteroskedastic.

Mathematical Implications of Homoskedasticity

The OLS estimators remain unbiased and asymptotically normal. Because the
least squares assumptions in Key Concept 4.3 place no restrictions on the condi-
tional variance, they apply to both the general case of heteroskedasticity and the
special case of homoskedasticity. Therefore, the OLS estimators remain unbiased
and consistent even if the errors are homoskedastic. In addition, the OLS estima-
tors have sampling distributions that are normal in large samples even if the errors
are homoskedastic. Whether the errors are homoskedastic or heteroskedastic, the
OLS estimator is unbiased, consistent, and asymptotically normal.

Efficiency of the OLS estimator when the errors are homoskedastic. If the least

squares assumptions in Key Concept 4.3 hold and the errors are homoskedastic,
then the OLS estimators bn0 and bn1 are efficient among all estimators that are
linear in Y1, c, Yn and are unbiased, conditional on X1, c, Xn. This result,
which is called the Gauss–Markov theorem, is discussed in Section 5.5.

Homoskedasticity-only variance formula. If the error term is homoskedastic,

then the formulas for the variances of bn0 and bn1 in Key Concept 4.4 simplify. Con-

sequently, if the errors are homoskedastic, then there is a specialized formula that

can be used for the standard errors of bn0 and bn1. The homoskedasticity-only stan-
dard error of bn1, derived in Appendix (5.1), is SE(bn1) = 2sb2n1 where sbn21
homoskedasticity-only estimator of the variance of bn1: is the

sb2n1 = n su2n (homoskedasticity@only), (5.22)

a (Xi - X )2
i=1

where su2n is given in Equation (4.19). The homoskedasticity-only formula for the
standard error of bn0 is given in Appendix (5.1). In the special case that X is a
binary variable, the estimator of the variance of bn1 under homoskedasticity (that
is, the square of the standard error of bn1 under homoskedasticity) is the so-called
pooled variance formula for the difference in means, given in Equation (3.23).

Because these alternative formulas are derived for the special case that the

errors are homoskedastic and do not apply if the errors are heteroskedastic, they