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17.5 Weighted Least Squares 741
In practice, the functional form of var(ui 0 Xi) is rarely if ever known, which
poses a problem for using WLS in real-world applications. This problem is difficult
enough with a single regressor, but in applications with multiple regressors it is even
more difficult to know the functional form of the conditional variance. For this rea-
son, practical use of WLS confronts imposing challenges. In contrast, in modern
statistical packages it is simple to use heteroskedasticity-robust standard errors, and
the resulting inferences are reliable under very general conditions; in particular,
heteroskedasticity-robust standard errors can be used without needing to specify a
functional form for the conditional variance. For these reasons, it is our opinion that,
despite the theoretical appeal of WLS, heteroskedasticity-robust standard errors
provide a better way to handle potential heteroskedasticity in most applications.
Summary
1. The asymptotic normality of the OLS estimator, combined with the consistency
of heteroskedasticity-robust standard errors, implies that, if the first three least
squares assumptions in Key Concept 17.1 hold, then the heteroskedasticity-
robust t-statistic has an asymptotic standard normal distribution under the null
hypothesis.
2. If the regression errors are i.i.d. and normally distributed, conditional on the
regressors, then bn1 has an exact normal sampling distribution, conditional on
the regressors. In addition, the homoskedasticity-only t-statistic has an exact
Student tn–2 sampling distribution under the null hypothesis.
3. The weighted least squares (WLS) estimator is OLS applied to a weighted regres-
sion, where all variables are weighted by the square root of the inverse of the
conditional variance, var(ui 0 Xi), or its estimate. Although the WLS estimator is
asymptotically more efficient than OLS, to implement WLS you must know the
functional form of the conditional variance function, which usually is a tall order.
Key Terms weighted least squares (WLS) (736)
WLS estimator (736)
convergence in probability (726) infeasible WLS (737)
consistent estimator (726) feasible WLS (738)
convergence in distribution (728) normal p.d.f. (747)
asymptotic distribution (728) bivariate normal p.d.f. (748)
Slutsky’s theorem (729)
continuous mapping theorem (729) 
