On the sensitivity of the usual t- and F-tests to covariance misspecification

We consider the standard linear regression model with all standard assumptions, except that the disturbances are not white noise, but distributed N(0, σ2Ω(θ)) where Ω(0)=In. Our interest lies in testing linear restrictions using the usual F-statistic based on OLS residuals. We are not interested in finding out whether θ=0 or not. Instead we want to find out what the effect is of possibly nonzero θ on the F-statistic itself. We propose a sensitivity statistic φ for this purpose, discuss its distribution, and obtain a practical and easy-to-use decision rule to decide whether the F-test is sensitive or not to covariance misspecification when θ is close to zero. Some finite and asymptotic properties of ϕ are studied, as well as its behaviour in the special case of an AR(1) process near the unit root.