What if your weights are wrong? WLS versus robust SEs

Question

You suspect heteroskedasticity but are not confident about its exact form. A colleague proposes weighted least squares using a guessed variance model; another proposes plain OLS with robust standard errors.

Discuss what happens to WLS if the guessed weights are wrong, and give a rule for choosing between the two approaches.

Accepted Answer

WLS with wrong weights is still unbiased. Weighting changes which observations the fit leans on, not the exogeneity that guarantees unbiasedness. So even with mis-specified weights, WLS gives an unbiased, consistent estimate. What you lose is the efficiency gain: wrong weights do not achieve minimum variance, and can even be worse than OLS if they up-weight the noisiest points. But the WLS standard errors become unreliable. The clean, efficient-looking standard errors from a weighted regression are only valid if the weights correctly equalize the variances. If they do not, the transformed err…

What if your weights are wrong? WLS versus robust SEs

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