Page 739 -

738 Chapter 17 The Theory of Linear Regression with One Regressor

Example #1: The variance of u is quadratic in X. Suppose that the conditional vari-
ance is known to be the quadratic function

var(ui 0 Xi) = u0 + u1X2i , (17.27)

where u0 and u1 are unknown parameters, u0 7 0, and u1 Ú 0.
Because u0 and u1 are unknown, it is not possible to construct the weighted

variables Yi, X0i, and X1i. It is, however, possible to estimate u0 and u1, and to use
those estimates to compute estimates of var(ui ͉ Xi). Let un0 and un1 be estimators of

muYXn0ia1>aito2(nwrdvihsauer1tr(,heuaeibn0OX0dXLni)l0e,SitXtneav0skiateir=m(suta1iht0>Xoe2rip)ovlaf=acrte(hunueo0if0cX+tohie)une,f1faXiinnci2tdie.eXnrDnct1eseipfii=ntnbetXh0)te1h.i>re2ewgvreeaisrgs(hiuotien0 Xdoir)fe.YTgnriheeosnWsoXLrns0SiYneasint=id-

Implementation of this estimator requires estimating the conditional variance
function, that is, estimating u0 and u1 in Equation (17.27). One way to estimate u0
and u1 consistently is to regress uni2 on X2i using OLS, where un2i is the square of the
ith OLS residual.

Suppose that the conditional variance has the form in Equation (17.27) and
that un0 and un1 are consistent estimators of u0 and u1. Under Assumptions #1
through #3 of Key Concept 17.1, plus additional moment conditions that arise
because u0 and u1 are estimated, the asymptotic distribution of the WLS estimator
is the same as if u0 and u1 were known. Thus the WLS estimator with u0 and u1
estimated has the same asymptotic distribution as the infeasible WLS estimator
and is in this sense asymptotically BLUE.

Because this method of WLS can be implemented by estimating unknown
parameters of the conditional variance function, this method is sometimes called
feasible WLS or estimated WLS.

Example #2: The variance depends on a third variable. WLS also can be used
when the conditional variance depends on a third variable, Wi, which does not
appear in the regression function. Specifically, suppose that data are collected on
three variables, Yi, Xi, and Wi, i = 1, c, n; the population regression function
depends on Xi but not Wi; and the conditional variance depends on Wi but not Xi.
That is, the population regression function is E(Yi ͉ Xi, Wi) = b0 + b1Xi and the
conditional variance is var(ui ͉ Xi, Wi) = lh(Wi), where l is a constant and h is a
function that must be estimated.

For example, suppose that a researcher is interested in modeling the relation-
ship between the unemployment rate in a state and a state economic policy vari-
able (Xi). The measured unemployment rate (Yi), however, is a survey-based