Using Invalid Instruments on Purpose: Focused Moment Selection and Averaging for GMM

TL;DR

This paper introduces the Focused Moment Selection Criterion (FMSC), a novel GMM approach that optimizes mean-squared error by selecting moment conditions based on their impact on the target estimator, with applications to instrument choice.

Contribution

The paper develops the FMSC, a new moment selection method for GMM that considers MSE of the target parameter, and proposes inference procedures for post-moment selection estimation.

Findings

FMSC performs well in simulations for moment selection.

Simulation-based inference procedures are effective post-moment selection.

Empirical example demonstrates the impact of instrument choice on estimated effects.

Abstract

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.