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740 Chapter 17 The Theory of Linear Regression with One Regressor

General method of feasible WLS. In general, feasible WLS proceeds in five steps:

1. Regress Yi on Xi by OLS and obtain the OLS residuals uni, i = 1, c, n.

2. Estimate a model of the conditional variance function var(ui 0 Xi). For example,

if the conditional variance function has the form in Equation (17.27), this entails
regressing un2i on X2i . In general, this step entails estimating a function for the
conditional variance, var(ui ͉ Xi).
3. Use the estimated function to compute predicted values of the conditional

variance function, var(ui 0 Xi).

4. Weight the dependent variable and regressor (including the intercept) by the
inverse of the square root of the estimated conditional variance function.

5. Estimate the coefficients of the weighted regression by OLS; the resulting
estimators are the WLS estimators.

Regression software packages typically include optional weighted least
squares commands that automate the fourth and fifth of these steps.

Heteroskedasticity-Robust Standard Errors or WLS?

There are two ways to handle heteroskedasticity: estimating b0 and b1 by WLS or
estimating b0 and b1 by OLS and using heteroskedasticity-robust standard errors.
Deciding which approach to use in practice requires weighing the advantages and
disadvantages of each.

The advantage of WLS is that it is more efficient than the OLS estimator of
the coefficients in the original regressors, at least asymptotically. The disadvan-
tage of WLS is that it requires knowing the conditional variance function and
estimating its parameters. If the conditional variance function has the quadratic
form in Equation (17.27), this is easily done. In practice, however, the functional
form of the conditional variance function is rarely known. Moreover, if the func-
tional form is incorrect, then the standard errors computed by WLS regression
routines are invalid in the sense that they lead to incorrect statistical inferences
(tests have the wrong size).

The advantage of using heteroskedasticity-robust standard errors is that they
produce asymptotically valid inferences even if you do not know the form of the
conditional variance function. An additional advantage is that heteroskedasticity-
robust standard errors are readily computed as an option in modern regression
packages, so no additional effort is needed to safeguard against this threat. The
disadvantage of heteroskedasticity-robust standard errors is that the OLS estima-
tor will have a larger variance than the WLS estimator (based on the true condi-
tional variance function).