Inference under First-Order Degeneracy

TL;DR

This paper addresses the challenge of inference in models with first-order degeneracy, such as causal mediation analysis, proposing methods for valid confidence intervals despite intrinsic limitations.

Contribution

It introduces minimum-distance methods and bootstrap procedures to construct valid confidence intervals in degenerate models, with theoretical guarantees and practical demonstrations.

Findings

Minimum-distance methods provide valid confidence intervals.

Bootstrap procedures work when chi-square critical values are invalid.

Simulations and empirical application show improved inference performance.

Abstract

We study inference in models where a transformation of parameters exhibits first-order degeneracy -- that is, its gradient is zero or close to zero, making the standard delta method invalid. A leading example is causal mediation analysis, where the indirect effect is a product of coefficients and the gradient degenerates near the origin. In these local regions of degeneracy the limiting behaviors of plug-in estimators depend on nuisance parameters that are not consistently estimable. We show that this failure is intrinsic -- around points of degeneracy, both regular and quantile-unbiased estimation are impossible. Despite these restrictions, we develop minimum-distance methods that deliver uniformly valid confidence intervals. We establish sufficient conditions under which standard chi-square critical values remain valid, and propose a simple bootstrap procedure when they are not. We demonstrate favorable power in simulations and in an empirical application linking teacher gender attitudes to student outcomes.