Orthogonal Moments in Likelihood Models

2026-09-25

AI Summary

This study addresses the incidental parameter problem arising in panel and network data, where numerous imprecisely estimated nuisance parameters induce severe estimation bias. To mitigate this issue, the proposed method integrates likelihood modeling, Neyman orthogonality theory, and subspace projection techniques. By comparing three distinct notions of orthogonality, this work develops a nested subspace projection approach for constructing orthogonal moments, yielding estimating equations that are robust to nuisance parameter perturbations. The primary contribution lies in providing explicit construction schemes for orthogonal moments across binary choice, count, and nonlinear regression models. Collectively, these advances establish a systematic theoretical framework and practical toolkit for robust statistical inference under high-dimensional, complex data structures.

Abstract

Many models, such as fixed-effect models for panel or network data, are hard to estimate because they feature nuisance parameters that are both numerous and estimated imprecisely. This, in general, causes an incidental-parameter problem in the estimator of the parameters of interest. The problem can be alleviated by working with an estimating equation whose expectation is insensitive to the value of the nuisance parameters. We discuss and contrast three notions of insensitivity, also called orthogonality, in the context of likelihood models: Neyman orthogonality, Neyman orthogonality to order q, and full orthogonality. Orthogonal moments are obtained by projecting the estimating equation on nested subspaces, which are spanned by, respectively, the scores of the nuisance parameters, the first q derivatives of the likelihood ratio with respect to the nuisance parameters, and all likelihood ratios of the model. We give explicit constructions in binary-choice, count-data, and nonlinear regression models.

Problem

• incidental-parameter problem
• nuisance parameters
• likelihood models
• orthogonal moments
• fixed-effect models

Innovation

• Orthogonal Moments
• Incidental-Parameter Problem
• Neyman Orthogonality
• Likelihood Models
• Estimating Equations