Weak Identification with Many Instruments

TL;DR

This paper reviews recent methods for estimation and inference in linear instrumental variable regressions with many instruments, addressing weak identification issues and proposing new robust testing procedures.

Contribution

It introduces new weak-identification-robust tests that handle many exogenous regressors and instruments, with improved size and power properties.

Findings

Established new results for a jack-knifed LM test statistic.

Extended weak-identification tests to models with many regressors and instruments.

Proposed a test that maintains proper size and power in complex settings.

Abstract

Linear instrumental variable regressions are widely used to estimate causal effects. Many instruments arise from the use of ``technical'' instruments and more recently from the empirical strategy of ``judge design''. This paper surveys and summarizes ideas from recent literature on estimation and statistical inferences with many instruments for a single endogenous regressor. We discuss how to assess the strength of the instruments and how to conduct weak identification-robust inference under heteroskedasticity. We establish new results for a jack-knifed version of the Lagrange Multiplier (LM) test statistic. Furthermore, we extend the weak-identification-robust tests to settings with both many exogenous regressors and many instruments. We propose a test that properly partials out many exogenous regressors while preserving the re-centering property of the jack-knife. The proposed tests have correct size and good power properties.