Heteroskedasticity — error variance that changes across observations — does not bias OLS. The coefficients stay unbiased and consistent; what breaks is the variance formula, so the conventional standard errors, t-statistics, and confidence intervals are wrong. Heteroskedasticity-robust (White / Eicker–Huber–White, or “sandwich”) standard errors fix exactly that, and only that: the point estimates are unchanged.
The logic is in the name. The true variance is a sandwich,
Var(β̂) = (X'X)⁻¹ (X'ΩX) (X'X)⁻¹, Ω = diag(σ_i²)We can’t observe each σ_i², but the squared residuals ε̂_i² are consistent building blocks for the “meat” in the middle:
Var̂(β̂) = (X'X)⁻¹ ( Σ_i w_i·ε̂_i²·x_i x_i' ) (X'X)⁻¹The HC variants differ only in the small-sample weight w_i:
- HC0 (White):
w_i = 1. - HC1:
w_i = n/(n−k)— a degrees-of-freedom scaling. - HC2:
w_i = 1/(1−h_ii), using the leverageh_ii. - HC3:
w_i = 1/(1−h_ii)²— approximates a jackknife; best in small samples, and a sensible default.
Where this sits relative to its neighbours:
- Versus weighted least squares: robust SE keeps the (inefficient) OLS estimates and only fixes inference; WLS instead changes the estimates to regain efficiency when you know the variance structure. Robust SE is what you use when heteroskedasticity is a nuisance and its form is unknown; WLS when the form is known and efficiency is the goal.
- It buys you nothing against bias. Exogeneity and a correct mean model are still required — omitted variables and endogeneity are untouched.
- The i.i.d.-robust sandwich also assumes errors are independent. Under clustering use cluster-robust SE; under serial correlation, HAC; under spatial correlation, a spatial-error model or spatially-robust inference.
Common slips: believing robust SE repairs bias (it doesn’t); using HC0/HC1 in small samples, where they under-cover (prefer HC3); and applying i.i.d.-robust errors when the real problem is clustered or spatially correlated dependence.