  In finite samples, the use of a slightly endogenous but highly relevant
instrument can reduce mean-squared error (MSE). Building on this observation, I
propose a novel moment selection procedure for GMM -- the Focused Moment
Selection Criterion (FMSC) -- in which moment conditions are chosen not based
on their validity but on the MSE of their associated estimator of a
user-specified target parameter. The FMSC mimics the situation faced by an
applied researcher who begins with a set of relatively mild "baseline"
assumptions and must decide whether to impose any of a collection of stronger
but more controversial "suspect" assumptions. When the (correctly specified)
baseline moment conditions identify the model, the FMSC provides an
asymptotically unbiased estimator of asymptotic MSE, allowing us to select over
the suspect moment conditions. I go on to show how the framework used to derive
the FMSC can address the problem of inference post-moment selection. Treating
post-selection estimators as a special case of moment-averaging, in which
estimators based on different moment sets are given data-dependent weights, I
propose simulation-based procedures for inference that can be applied to a
variety of formal and informal moment-selection and averaging procedures. Both
the FMSC and confidence interval procedures perform well in simulations. I
conclude with an empirical example examining the effect of instrument selection
on the estimated relationship between malaria and income per capita.
