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Inference under First-Order Degeneracy

Xinyue Bei, Manu Navjeevan

TL;DR

The paper addresses inference when the gradient of the target transformation vanishes, causing the delta method to break down in local regions of degeneracy. It formalizes a local degeneracy framework via a Gaussian-shift limit and proves that regular and quantile-unbiased estimation are intrinsically impossible in these regions. To overcome this, it develops minimum-distance based inference with test inversion, deriving conditions under which standard chi-square critical values remain valid and offering bootstrap corrections when needed. Simulations and an empirical mediation study show the proposed approach achieves uniform size control and tighter confidence intervals than existing methods, with practical implications for causal mediation analyses and other first-order degeneracy scenarios.

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.

Inference under First-Order Degeneracy

TL;DR

The paper addresses inference when the gradient of the target transformation vanishes, causing the delta method to break down in local regions of degeneracy. It formalizes a local degeneracy framework via a Gaussian-shift limit and proves that regular and quantile-unbiased estimation are intrinsically impossible in these regions. To overcome this, it develops minimum-distance based inference with test inversion, deriving conditions under which standard chi-square critical values remain valid and offering bootstrap corrections when needed. Simulations and an empirical mediation study show the proposed approach achieves uniform size control and tighter confidence intervals than existing methods, with practical implications for causal mediation analyses and other first-order degeneracy scenarios.

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.
Paper Structure (19 sections, 26 theorems, 191 equations, 3 figures, 2 tables)

This paper contains 19 sections, 26 theorems, 191 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Overview and Examples
  4. Impossibility Results
  5. Preliminaries
  6. Analysis in the Limiting Experiment
  7. Hypothesis Testing and Infinite Dimensional Models
  8. Hypothesis Testing
  9. Infinite Dimensional Models
  10. Minimum Distance Based Inference
  11. Two-Dimensional $\theta$ and Indefinite $H$
  12. General Cases
  13. Simulation
  14. Empirical Application
  15. Conclusion
  16. ...and 4 more sections

Key Result

Proposition 3.1

Suppose assm:lan holds, and let $\Psi_n$ be a sequence of estimators satisfying eq:ncov. Then there exists a randomized statistic $\Psi(Z,U)$, where $Z$ is drawn from the Gaussian shift experiment and $U \sim \mathrm{Unif}(0,1)$ independent of $Z$, such that

Figures (3)

  • Figure 4.1: Acceptance Regions under Low (Left) and High (Right) Curvature.
  • Figure B.1: Illustration of Proposition \ref{['prop:tau=000020large']}.
  • Figure B.2: Lemma \ref{['lem:key=000020lemma']}: Acceptance Region of Linear and Curved Null.

Theorems & Definitions (72)

  • Example 2.1: Mediation Analysis
  • Example 2.2: Impulse Response Function
  • Example 2.3: Breakdown Point Analysis
  • Example 2.4: Weak IV Bias and Size Distortion
  • Example 2.5: Explained Variance in Linear Regression
  • Example 3.1: Smooth Parametric Models
  • Example 3.2: Plug-In Estimators
  • Definition 3.1: Regularity
  • Proposition 3.1: Limit Experiment
  • Remark 3.1
  • ...and 62 more