In 90 seconds, Vera explains why inverse-variance weighting can improve efficiency and inference under heteroskedasticity but cannot repair bias from omitted variables, reverse causality, or measurement error. This article remains the reference for the full definitions, formulas, and limits.
Production note: Vera's presenter footage and English voice were generated with HeyGen; HyperFrames produced the teaching graphics, animation, Chinese subtitles, and final composite.
Ordinary least squares treats every observation as equally informative. When some observations are noisier than others — heteroskedasticity, Var(ε_i) = σ_i² varying with i — that even-handedness is wasteful. OLS stays unbiased and consistent, but it is no longer efficient (not BLUE) and its textbook standard errors are wrong.
Weighted least squares re-weights the fit:
β̂_WLS = argmin_β Σ_i w_i (y_i − x_i'β)² = (X'WX)⁻¹ X'Wy, W = diag(w_1, …, w_n)With w_i = 1/σ_i², the observations you can trust more (smaller variance) pull harder on the fit. Mechanically it is just OLS on the transformed data √w_i·(y_i, x_i), and it is the diagonal special case of generalized least squares (GLS). Common weight choices:
| situation | weight |
|---|---|
a mean of n_i underlying units | w_i = n_i (variance ∝ 1/n_i) |
| a known standard error per point (e.g. a second stage run on estimates) | w_i = 1/SE_i² |
variance a known function of x | w_i = 1/σ²(x_i) |
| variance unknown | estimate it first (FGLS), then weight |
Done right, WLS restores efficiency and gives correct inference under heteroskedasticity. But it is routinely asked to do things it cannot:
- It does not fix bias. WLS still requires the mean model to be correct. Omitted variables, simultaneity, endogeneity — reweighting touches none of them. A weighted regression of a sorted outcome on a self-selected regressor is exactly as non-causal as the unweighted one. Efficiency is not identification.
- It does not rescue a generated dependent variable. If
y_iis itself an estimate from a prior step, inverse-variance weighting correctly handles the differing precision — but it does not repair the downward bias in standard errors that comes from ignoring first-stage estimation uncertainty. That needs a separate correction (a bootstrap, or an explicit errors-in-variables treatment). - Wrong weights can make things worse. Badly misspecified weights can leave WLS less efficient than plain OLS. When the weights are estimated rather than known, pair WLS with heteroskedasticity-robust (HC) standard errors as a guard.
The clean mental model: heteroskedasticity is a variance problem, and WLS is a variance tool. Endogeneity is a mean / identification problem, and no choice of weights will solve it. Reach for WLS to make honest estimates more precise — never to make a biased estimate causal.