The sensitivity of OLS when the variance matrix is (partially) unknown

We consider the standard linear regression model y=Xβ+u with all standard assumptions, except that the variance matrix is assumed to be σ2Ω(θ), where Ω depends on m unknown parameters θ1,…, θm. Our interest lies exclusively in the mean parameters β or Xβ. We introduce a new sensitivity statistic (B1) which is designed to decide whether ŷ (or β̂) is sensitive to covariance misspecification. We show that the Durbin–Watson test is inappropriate in this context, because it measures the sensitivity of σ̂2 to covariance misspecification. Our results demonstrate that the estimator β̂ and the predictor ŷ are not very sensitive to covariance misspecification. The statistic is easy to use and performs well even in cases where it is not strictly applicable.