The Robust F-Statistic as a Test for Weak Instruments

TL;DR

This paper extends the effective F-statistic approach for testing weak instruments to a broader class of GMM estimators, introducing a novel estimator and simplifying the calculation of critical values, with implications for bias reduction.

Contribution

It generalizes the weak instrument test to GMM estimators, introduces the GMMf estimator based on first-stage residuals, and simplifies critical value calculations without simulation.

Findings

GMMf estimator outperforms 2SLS in bias in weak instrument scenarios

Effective F-statistic applies to a broader class of GMM estimators

Simplified critical value calculations eliminate the need for simulations

Abstract

Montiel Olea and Pflueger (2013) proposed the effective F-statistic as a test for weak instruments in terms of the Nagar bias of the two-stage least squares (2SLS) estimator relative to a benchmark worst-case bias. We show that their methodology applies to a class of linear generalized method of moments (GMM) estimators with an associated class of generalized effective F-statistics. The standard nonhomoskedasticity robust F-statistic is a member of this class. The associated GMMf estimator, with the extension f for first-stage, is a novel and unusual estimator as the weight matrix is based on the first-stage residuals. As the robust F-statistic can also be used as a test for underidentification, expressions for the calculation of the weak-instruments critical values in terms of the Nagar bias of the GMMf estimator relative to the benchmark simplify and no simulation methods or Patnaik (1949) distributional approximations are needed. In the grouped-data IV designs of Andrews (2018), where the robust F-statistic is large but the effective F-statistic is small, the GMMf estimator is shown to behave much better in terms of bias than the 2SLS estimator, as expected by the weak-instruments test results.