Weak Instruments, First-Stage Heteroskedasticity and the Robust F-test

This paper is concerned with the findings related to the robust first-stage F-statistic in the Monte Carlo analysis of Andrews (2018), who found in a heteroskedastic design that even for very large values of the robust F-statistic, the standard 2SLS con dence intervals had large coverage distortions. This finding appears to discredit the robust F-statistic as a test for underidenti fication. However, it is shown here that large values of the robust F-statistic do imply that there is first-stage information, but this may not be utilised well by the 2SLS estimator, or the standard GMM estimator. An estimator that corrects for this is a robust GMM estimator, with the robust weight matrix not based on the structural residuals, but on the first-stage residuals. For the grouped data setting of Andrews (2018), this estimator gives the weights to the group speci fic estimators according to the group specific concentration parameters in the same way as 2SLS does under homoskedasticity, which is formally shown using weak instrument asymptotics. This estimator is much better behaved than the 2SLS estimator in this design, behaving well in terms of relative bias and Wald test size distortion at more ‘standard’ values of the robust F-statistic. We further derive the conditions under which the Stock and Yogo (2005) weak instruments critical values apply to the robust F-statistic in relation to the behaviour of this GMM estimator.