Minimum Distance Approach to Inference with Many Instruments

TL;DR

This paper introduces a new minimum distance inference method for linear models with many instruments, improving efficiency and providing valid confidence intervals even when certain restrictions are not met.

Contribution

It develops a novel minimum distance estimator that is equivalent to existing estimators under specific conditions and introduces a more efficient estimator using an optimal weight matrix.

Findings

The estimator matches the random effects estimator with a specific weight matrix.

A more efficient estimator is constructed based on the optimal weight matrix.

Confidence intervals are valid even when the proportionality restriction is violated.

Abstract

I analyze a linear instrumental variables model with a single endogenous regressor and many instruments. I use invariance arguments to construct a new minimum distance objective function. With respect to a particular weight matrix, the minimum distance estimator is equivalent to the random effects estimator of Chamberlain and Imbens (2004), and the estimator of the coefficient on the endogenous regressor coincides with the limited information maximum likelihood estimator. This weight matrix is inefficient unless the errors are normal, and I construct a new, more efficient estimator based on the optimal weight matrix. Finally, I show that when the minimum distance objective function does not impose a proportionality restriction on the reduced-form coefficients, the resulting estimator corresponds to a version of the bias-corrected two-stage least squares estimator. I use the objective function to construct confidence intervals that remain valid when the proportionality restriction is violated.