When you know the variance structure, weight instead of just correcting

Question

Your errors are heteroskedastic, but this time you know the form: $\operatorname{Var}(\varepsilon_i \mid X) = \sigma^2 w_i$ for a known positive $w_i$ (say variance grows with firm size).

Robust standard errors would fix your inference. Why might you prefer weighted least squares instead, and what does it buy you?

Accepted Answer

Robust SEs fix inference but not efficiency. Under heteroskedasticity, OLS is still unbiased, and White standard errors give you valid t-statistics. But OLS is no longer BLUE: it wastes information by treating high-variance and low-variance observations as equally trustworthy. Robust SEs correct your uncertainty estimate, they do not make the estimator any sharper. Weighted least squares recovers efficiency. If you know each observation's relative variance wi, weight each by 1/wi and minimize \sumi \frac{(yi - xi'\beta)^2}{wi}. Equivalently, divide every row (both yi and xi) by \sqrt{wi}; the…

When you know the variance structure, weight instead of just correcting

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