  We propose a two-stage least squares (2SLS) estimator whose first stage is
the equal-weighted average over a complete subset with $k$ instruments among
$K$ available, which we call the complete subset averaging (CSA) 2SLS. The
approximate mean squared error (MSE) is derived as a function of the subset
size $k$ by the Nagar (1959) expansion. The subset size is chosen by minimizing
the sample counterpart of the approximate MSE. We show that this method
achieves the asymptotic optimality among the class of estimators with different
subset sizes. To deal with averaging over a growing set of irrelevant
instruments, we generalize the approximate MSE to find that the optimal $k$ is
larger than otherwise. An extensive simulation experiment shows that the
CSA-2SLS estimator outperforms the alternative estimators when instruments are
correlated. As an empirical illustration, we estimate the logistic demand
function in Berry, Levinsohn, and Pakes (1995) and find the CSA-2SLS estimate
is better supported by economic theory than the alternative estimates.
