Non-constant error variance and the loss of efficiency

Question

You regress hourly wages on years of experience for 3,000 workers. The residual plot fans out: experienced, high-wage workers scatter far more widely than entry-level workers. A colleague says, "The slope is still unbiased, so who cares?"

Name the problem and identify what property of OLS is lost, beyond the standard errors being wrong.

Accepted Answer

The problem is heteroskedasticity. The error spread widens with fitted wages, so the error variance is not constant. Your colleague is right that the slope stays unbiased, but that is not the whole story. What is lost: efficiency. With constant-variance errors, ordinary least squares is the best linear unbiased estimator, meaning no other linear unbiased estimator has a smaller sampling variance. Under heteroskedasticity that guarantee is gone: OLS is still unbiased, but it is no longer the minimum-variance choice. A method that down-weights the noisy high-wage observations (weighted least sq…

Non-constant error variance and the loss of efficiency

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