The First-stage F Test with Many Weak Instruments

A widely adopted approach for detecting weak instruments is to use the first-stage $F$ statistic. While this method was developed with a fixed number of instruments, its performance with many instruments remains insufficiently explored. We show that the first-stage $F$ test exhibits distorted sizes for detecting many weak instruments, regardless of the choice of pretested estimators or Wald tests. These distortions occur due to the inadequate approximation using classical noncentral Chi-squared distributions. As a byproduct of our main result, we present an alternative approach to pre-test many weak instruments with the corrected first-stage $F$ statistic. An empirical illustration with Angrist and Keueger (1991)'s returns to education data confirms its usefulness.