  Identification-robust hypothesis tests are commonly based on the continuous
updating objective function or its score. When the number of moment conditions
grows proportionally with the sample size, the large-dimensional weighting
matrix prohibits the use of conventional asymptotic approximations and the
behavior of these tests remains unknown. We show that the structure of the
weighting matrix opens up an alternative route to asymptotic results when,
under the null hypothesis, the distribution of the moment conditions is
reflection invariant. In a heteroskedastic linear instrumental variables model,
we then establish asymptotic normality of conventional tests statistics under
many instrument sequences. A key result is that the additional terms that
appear in the variance are negative. Revisiting a study on the elasticity of
substitution between immigrant and native workers where the number of
instruments is over a quarter of the sample size, the many instrument-robust
approximation indeed leads to substantially narrower confidence intervals.
