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17.5 Weighted Least Squares 739

estimate of the true unemployment rate (Y*i ). Thus Yi measures Y*i with error,
where the source of the error is random survey error, so Yi = Y*i + vi, where vi is
the measurement error arising from the survey. In this example, it is plausible that
the survey sample size, Wi, is not itself a determinant of the true state unemploy-
ment rate. Thus the population regression function does not depend on Wi; that
is, E(Y*i ͉ Xi,Wi) = b0 + b1Xi. We therefore have the two equations

Y*i = b0 + b1Xi + ui* and (17.28)
Yi = Y*i + vi, (17.29)

where Equation (17.28) models the relationship between the state economic pol-

icy variable and the true state unemployment rate and Equation (17.29) repre-

sents the relationship between the measured unemployment rate Yi and the true
unemployment rate Yi*.

The model in Equations (17.28) and (17.29) can lead to a population regres-

sion in which the conditional variance of the error depends on Wi but not on Xi.
The error term ui* in Equation (17.28) represents other factors omitted from this
regression, while the error term vi in Equation (17.29) represents measurement
error arising from the unemployment rate survey. If u*i is homoskedastic, then

var(u*i 0 Xi, Wi) = s2u* is constant. The survey error variance, however, depends
inversely on the survey sample size Wi; that is, var(vi 0 Xi, Wi) = a>Wi where a is a

constant. Because vi is random survey error, it is safely assumed to be uncorrelated

with ui*, so var(ui* + vi 0 Xi, Wi) = s2u* + a>Wi Thus, substituting Equation (17.28)

into Equation (17.29) leads to the regression model with heteroskedasticity

Yi = b0 + b1Xi + ui, (17.30)

var(ui ͉ Xi, Wi) = u0 + u1 a 1 b , (17.31)
Wi

where ui = u*i + vi, u0 = s2u*, u1 = a, and E(ui ͉ Xi, Wi) = 0.
If u0 and u1 were known, then the conditional variance function in Equation

(17.31) could be used to estimate b0 and b1 by WLS. In this example, u0 and u1 are
unknown, but they can be estimated by regressing the squared OLS residual [from

OLS estimation of Equation (17.30)] on 1>Wi. Then the estimated conditional
variance function can be used to construct the weights in feasible WLS.

It should be stressed that it is critical that E(ui 0 Xi, Wi) = 0; if not, the

weighted errors will have nonzero conditional mean and WLS will be inconsistent.

Said differently, if Wi is in fact a determinant of Yi, then Equation (17.30) should
be a multiple regression equation that includes both Xi and Wi.