GMM with Many Weak Moment Conditions and Nuisance Parameters: General Theory and Applications to Causal Inference

TL;DR

The paper develops a general theory for estimating a finite-dimensional parameter under many weak moment conditions and growing nuisance parameters, introducing a two-step continuously updated estimator (CUE) with global Neyman orthogonality and cross-fitting. It shows consistency and asymptotic normality in a many-weak-moments regime and provides a consistent variance estimator that accounts for the nonstandard U-statistic term. The framework is illustrated through three causal-inference examples—additive and multiplicative structural mean models with many instruments and proximal causal inference with weak proxies—demonstrating reduced bias and robust inference relative to traditional GMM or 2SLS. An over-identification test extends the J-test to assess model specification under growing moment conditions. The work lays groundwork for future efficiency theory, longitudinal extensions, and inference for complex, weakly-identified objects such as dose–response curves.

Abstract

Weak identification arises in many statistical problems when key variables exhibit weak correlations-for example, when instrumental variables correlate weakly with treatment, or when proxy variables correlate weakly with unmeasured confounders. Under weak identification, standard estimation methods such as the generalized method of moments (GMM) can produce substantial bias, both in finite samples and asymptotically. This challenge is compounded in modern applications that require estimating many nuisance parameters. This paper develops a framework for estimation and inference of a finite-dimensional target parameter in general moment models with the number of weak moment conditions and nuisance parameters growing with sample size. We analyze a general two-step debiasing estimator that accommodates flexible, possibly nonparametric first-step estimation of nuisance parameters, in which Neyman orthogonality plays a more critical role in obtaining debiased inference than in conventional settings with strong identification. Under a many-weak-moment asymptotic regime, we establish the estimator's consistency and asymptotic normality. We provide high-level conditions for the general setting and demonstrate their application to two important special cases: inference with weak instruments and inference with weak proxies.

Figures (3)

• Figure 1: Neyman orthogonality plays a critical role under many weak moments. Data are simulated from an additive structural mean model, with the red dashed line indicating the true parameter value. Panels show sampling distributions across 1,000 simulations for: (a) CUE with orthogonal weak moments, (b) GMM with orthogonal weak moments, (c) CUE with non-orthogonal weak moments, and (d) CUE with non-orthogonal strong moments. Under weak identification, the orthogonalized CUE (panel a) remains unbiased, while the non-orthogonal CUE (panel c) exhibits substantial bias—despite being consistent under strong identification (panel d). Even with orthogonal moments, GMM (panel b) remains biased. Simulation details appear in Supplemental Material S13.2.
• Figure S.1: Sampling distribution of the CUE and GMM estimator under ASMM and MSMM settings. The dashed red lines represents the ground truth in each setting.
• Figure S.2: Sampling distribution of the CUE and 2SLS estimator under proximal causal inference settings. The dashed red lines represents the ground truth in each setting.

Theorems & Definitions (117)

• Definition 1: Neyman Orthogonality
• Lemma 1: Permanence properties of Neyman orthogonality
• Theorem 1: Consistency
• Theorem 2
• Theorem 3: Over-identification test
• Theorem S1: Consistency of variance estimator
• Theorem S2
• Theorem S3
• Theorem S4
• Lemma S1