  I analyze a linear instrumental variables model with a single endogenous
regressor and many instruments. I use invariance arguments to construct a new
minimum distance objective function. With respect to a particular weight
matrix, the minimum distance estimator is equivalent to the random effects
estimator of Chamberlain and Imbens (2004), and the estimator of the
coefficient on the endogenous regressor coincides with the limited information
maximum likelihood estimator. This weight matrix is inefficient unless the
errors are normal, and I construct a new, more efficient estimator based on the
optimal weight matrix. Finally, I show that when the minimum distance objective
function does not impose a proportionality restriction on the reduced-form
coefficients, the resulting estimator corresponds to a version of the
bias-corrected two-stage least squares estimator. I use the objective function
to construct confidence intervals that remain valid when the proportionality
restriction is violated.
