Page 737 -

736 Chapter 17 The Theory of Linear Regression with One Regressor

Equation (17.23) matches the definition of a random variable with a Student t
distribution given in Appendix 17.1. That is, by using the degrees of freedom
adjustment to calculate the standard error, the t-statistic has the Student t distribu-
tion when the errors are normally distributed.

17.5 Weighted Least Squares

Under the first four extended least squares assumptions, the OLS estimator is
efficient among the class of linear (in Y1, c, Yn), conditionally (on X1, c, Xn)
unbiased estimators; that is, the OLS estimator is BLUE. This result is the Gauss–
Markov theorem, which was discussed in Section 5.5 and proven in Appendix 5.2.
The Gauss–Markov theorem provides a theoretical justification for using the OLS
estimator. A major limitation of the Gauss–Markov theorem is that it requires
homoskedastic errors. If, as is often encountered in practice, the errors are
heteroskedastic, the Gauss–Markov theorem does not hold and the OLS estimator
is not BLUE.

This section presents a modification of the OLS estimator, called weighted
least squares (WLS), which is more efficient than OLS when the errors are
heteroskedastic.

WLS requires knowing quite a bit about the conditional variance function,
var(ui ͉ Xi). We consider two cases. In the first case, var(ui ͉ Xi) is known up to a
factor of proportionality, and WLS is BLUE. In the second case, the functional
form of var(ui ͉ Xi) is known, but this functional form has some unknown param-
eters that can be estimated. Under some additional conditions, the asymptotic
distribution of WLS in the second case is the same as if the parameters of the
conditional variance function were in fact known, and in this sense the WLS esti-
mator is asymptotically BLUE. The section concludes with a discussion of the
practical advantages and disadvantages of handling heteroskedasticity using WLS
or, alternatively, heteroskedasticity-robust standard errors.

WLS with Known Heteroskedasticity

Suppose that the conditional variance var(ui 0 Xi) is known up to a factor of pro-

portionality; that is,

var(ui 0 Xi) = lh(Xi), (17.24)

where l is a constant and h is a known function. In this case, the WLS estimator
is the estimator obtained by first dividing the dependent variable and regressor