  This paper investigates the behavior of Stock and Yogo (2005)'s first-stage F
statistic and the Cragg-Donald statistic (Cragg and Donald, 1993) when the
number of instruments and the sample size go to infinity in a comparable
magnitude. Our theory shows that the first-stage F test is oversized for
detecting many weak instruments. We next propose an asymptotically valid
correction of the F statistic for testing weakness of instruments. The theory
is also used to construct confidence intervals for the strength of instruments.
As for the Cragg-Donald statistic, we obtain an asymptotically valid correction
in the case of two endogenous variables. Monte Carlo experiments demonstrate
the satisfactory performance of the proposed methods in both situations of a
single and multiple endogenous variables. The usefulness of the proposed tests
is illustrated by an analysis of the returns to education data in Angrist and
Keueger (1991).
