Robust Two-Step Confidence Sets, and the Trouble with the First Stage F-Statistic

When weak identification is a concern researchers frequently calculate confidence sets in two steps, first assessing the strength of identification and then, on the basis of this initial assessment, deciding whether to use an identification-robust confidence set. Unfortunately, two-step procedures of this sort can generate highly misleading confidence sets, and we demonstrate that two-step confidence sets based on the first stage F-statistic can have extremely poor coverage in linear instrumental variables models with heteroskedastic errors. To remedy this issue, we introduce a simple approach to detecting weak identification and constructing two-step confidence sets which we show controls coverage distortions under weak identification in general nonlinear GMM models, while also indicating strong identification with probability tending to one if the model is well-identified. Applying our approach to linear IV we show that it is competitive with approaches based on the first-stage F-statistic under homoscedasticity but performs far better under heteroskedasticity