Weak instruments and the first stage F-statistic in IV models with a nonscalar error covariance structure

We analyze the usefulness of the first stage F-statistic for detecting weak instruments in the IV model with a nonscalar error covariance structure. More in particular, we question the validity of the rule of thumb of a first stage F-statistic of 10 or higher for models with correlated errors arising from either a group structure or serial correlation. Using asymptotic expansion techniques we derive bias approximations for IV and OLS estimators in this generalized IV model. We relate these bias approximations to expected values of both standard and robust versions of the first stage F-statistic. Our theoretical and simulation results indicate that the standard first stage F-statistic overestimates the strength of instruments. In addition, there does not seem to be a close correspondence between the robust version of the F-statistic and weak instruments as measured by relative bias of IV with respect to OLS.