What heteroskedasticity does to OLS

Question

You estimate y = X\beta + \varepsilon by OLS, but the error variance is not constant: residuals fan out as firm size grows, so \operatorname{Var}(\varepsiloni \mid X) depends on the regressors. What is still valid, what breaks, and how do you fix it? Be precise about which property of OLS is lost.

Accepted Answer

What still holds. Heteroskedasticity does not touch exogeneity (\mathbb{E}[\varepsilon \mid X] = 0), so OLS remains unbiased and consistent: \hat{\beta} still converges to the true \beta. The point estimates are fine. What breaks. Two things: 1. Efficiency. OLS is no longer BLUE. The Gauss-Markov theorem needs constant variance; without it there exist linear unbiased estimators with smaller variance, so OLS is inefficient. 2. Inference, the practical killer. The usual standard-error formula \hat{\sigma}^2 (X^	op X)^{-1} assumes a single \sigma^2 and is now wrong. Your t-statistics, p-values,…

What heteroskedasticity does to OLS

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