Weak-instrument-robust subvector inference in instrumental variables regression: A subvector Lagrange multiplier test and properties of subvector Anderson-Rubin confidence sets

TL;DR

This paper introduces a new weak-instrument-robust subvector Lagrange multiplier test for instrumental variables regression, providing properties of subvector confidence sets and their relation to classical tests, enhancing inference reliability.

Contribution

It develops the first weak-instrument-robust subvector test that recovers degrees of freedom and characterizes subvector confidence sets, connecting them to Wald and Anderson-Rubin tests.

Findings

The proposed test is asymptotically size-correct under certain conditions.

Subvector confidence sets are centered around a k-class estimator.

Bounded confidence sets correspond to Wald-based sets with data-dependent levels.

Abstract

We propose a weak-instrument-robust subvector Lagrange multiplier test for instrumental variables regression. We show that it is asymptotically size-correct under a technical condition or as the number of instruments grows to infinity. This is the first weak-instrument-robust subvector test for instrumental variables regression to recover the degrees of freedom of the commonly used non-weak-instrument-robust Wald test. Additionally, we provide a closed-form solution for subvector confidence sets obtained by inverting the subvector Anderson-Rubin test. We show that they are centered around a k-class estimator. We show that the subvector confidence sets for single coefficients of the causal parameter are jointly bounded if and only if Anderson's likelihood-ratio test rejects the null hypothesis that the first-stage regression parameter is of reduced rank, that is, that the causal parameter is not identified. Finally, we show that if a confidence set obtained by inverting the Anderson-Rubin test is bounded and nonempty, it is equal to a Wald-based confidence set with a data-dependent confidence level. We explicitly compute this Wald-based confidence set and its confidence level.