Page 738 -

17.5 Weighted Least Squares 737

by the square root of h and then regressing this modified dependent variable on
the modified regressor using OLS. Specifically, divide both sides of the single-
variable regressor model by 2h(Xi) to obtain

Yi = b0X0i + b1X1i + ui, (17.25)

where Yi = Yi> 2h(Xi), X0i = 1> 2h(Xi), X1i = Xi>2h(Xi), and ui = ui> 2h(Xi).
The WLS estimator is the OLS estimator of b1 in Equation (17.25); that

is, it is the estimator obtained by the OLS regression of Yi on X0i and X1i,
where the coefficient on X0i takes the place of the intercept in the unweighted
regression.

Under the first three least squares assumptions in Key Concept 17.1 plus the

known heteroskedasticity assumption in Equation (17.24), WLS is BLUE. The

reason that the WLS estimator is BLUE is that weighting the variables has made

the error term ui in the weighted regression homoskedastic. That is,

var(ui 0 Xi) = var c ui 0 Xi d = var(ui ͉ Xi) = lh(Xi) = l, (17.26)
1h(Xi) h(Xi) h(Xi)

so the conditional variance of ui, var(ui ͉ Xi), is constant. Thus the first four least
squares assumptions apply to Equation (17.25). Strictly speaking, the Gauss–Markov

theorem was proven in Appendix 5.2 for Equation (17.1), which includes the

intercept b0, so it does not apply to Equation (17.25), in which the intercept is
replaced by b0X0i. However, the extension of the Gauss–Markov theorem for
multiple regression (Section 18.5) does apply to estimation of b1 in the weighted
population regression, Equation (17.25). Accordingly, the OLS estimator of b1 in
Equation (17.25)—that is, the WLS estimators of b1:is BLUE.

In practice, the function h typically is unknown, so neither the weighted vari-

ables in Equation (17.25) nor the WLS estimator can be computed. For this rea-

son, the WLS estimator described here is sometimes called the infeasible WLS

estimator. To implement WLS in practice, the function h must be estimated, the

topic to which we now turn.

WLS with Heteroskedasticity
of Known Functional Form

If the heteroskedasticity has a known functional form, then the heteroskedasticity
function h can be estimated and the WLS estimator can be calculated using this
estimated function.