Inference with Many Weak Instruments and Heterogeneity

TL;DR

This paper develops a new valid inference method for linear instrumental variable models with many weak instruments and heterogeneous effects, addressing size distortions in existing tests.

Contribution

It introduces a novel score-based test with a leave-three-out variance estimator that is asymptotically the most powerful unbiased test under heterogeneity.

Findings

Existing tests can be arbitrarily oversized with many weak instruments.

The proposed test provides bounded confidence sets in applications.

The new method outperforms existing procedures in simulations and applications.

Abstract

This paper considers inference in a linear instrumental variable regression model with many potentially weak instruments, in the presence of heterogeneous treatment effects. I first show that existing test procedures, including those that are robust to either weak instruments or heterogeneous treatment effects, can be arbitrarily oversized. I propose a novel and valid test based on a score statistic and a ``leave-three-out" variance estimator. In the presence of heterogeneity and within the class of tests that are functions of the leave-one-out analog of a maximal invariant, this test is asymptotically the uniformly most powerful unbiased test. In two applications to judge and quarter-of-birth instruments, the proposed inference procedure also yields a bounded confidence set while some existing methods yield unbounded or empty confidence sets.