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This paper is concerned with the findings related to the robust first-stageF-statistic in the Monte Carlo analysis of Andrews (2018), who found in aheteroskedastic grouped-data design that even for very large values of therobust F-statistic, the standard 2SLS confidence intervals had large coveragedistortions. This finding appears to discredit the robust F-statistic as a testfor underidentification. However, it is shown here that large values of therobust F-statistic do imply that there is first-stage information, but this maynot be utilized well by the 2SLS estimator, or the standard GMM estimator. Anestimator that corrects for this is a robust GMM estimator, denoted GMMf, withthe robust weight matrix not based on the structural residuals, but on thefirst-stage residuals. For the grouped-data setting of Andrews (2018), thisGMMf estimator gives the weights to the group specific estimators according tothe group specific concentration parameters in the same way as 2SLS does underhomoskedasticity, which is formally shown using weak instrument asymptotics.The GMMf estimator is much better behaved than the 2SLS estimator in theAndrews (2018) design, behaving well in terms of relative bias and Wald-testsize distortion at more standard values of the robust F-statistic. We show thatthe same patterns can occur in a dynamic panel data model when the errorvariance is heteroskedastic over time. We further derive the conditions underwhich the Stock and Yogo (2005) weak instruments critical values apply to therobust F-statistic in relation to the behaviour of the GMMf estimator.

Weak Instruments, First-Stage Heteroskedasticity, the Robust F-Test and a GMM Estimator with the Weight Matrix Based on First-Stage Residuals