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A Neyman-Orthogonalization Approach to the Incidental Parameter Problem

Stéphane Bonhomme, Koen Jochmans, Martin Weidner

Abstract

A popular approach to perform inference on a target parameter in the presence of nuisance parameters is to construct estimating equations that are orthogonal to the nuisance parameters, in the sense that their expected first derivative is zero. Such first-order orthogonalization allows the estimator of the nuisance parameters to converge at a slower-than-parametric rate. It may, however, not suffice when the nuisance parameters are very imprecisely estimated. Leading examples are models for panel and network data that feature fixed effects. In this paper, we show how, in the conditional-likelihood setting, estimating equations can be constructed that are orthogonal to any chosen order $q$, in that their leading $q$ expected derivatives are zero. This yields estimators of target parameters that are unaffected by the presence of nuisance parameters to order $q$. In an empirical illustration, we apply our method to a fixed-effect model of team production.

A Neyman-Orthogonalization Approach to the Incidental Parameter Problem

Abstract

A popular approach to perform inference on a target parameter in the presence of nuisance parameters is to construct estimating equations that are orthogonal to the nuisance parameters, in the sense that their expected first derivative is zero. Such first-order orthogonalization allows the estimator of the nuisance parameters to converge at a slower-than-parametric rate. It may, however, not suffice when the nuisance parameters are very imprecisely estimated. Leading examples are models for panel and network data that feature fixed effects. In this paper, we show how, in the conditional-likelihood setting, estimating equations can be constructed that are orthogonal to any chosen order , in that their leading expected derivatives are zero. This yields estimators of target parameters that are unaffected by the presence of nuisance parameters to order . In an empirical illustration, we apply our method to a fixed-effect model of team production.

Paper Structure

This paper contains 57 sections, 12 theorems, 217 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem statement and motivation
  3. Setup
  4. Example: Neyman-Scott model.
  5. Example: CES production function.
  6. The role of first-order orthogonality and its limitations
  7. Example: Neyman-Scott model (continued).
  8. Higher-order orthogonality
  9. Example: Neyman-Scott model (continued)
  10. Estimation based on orthogonalized functions
  11. Definition of higher-order orthogonality
  12. Estimation
  13. Achieving higher-order Neyman-orthogonality
  14. Main result
  15. Intuition and discussion
  16. ...and 42 more sections

Key Result

Theorem 1

Suppose that $\varSigma_{w_qw_q}(x;\theta,\eta)$ is invertible and let Then the function satisfies $\mathbb{E}_{\theta,\eta} ( \nabla^q_\eta \, u_q^*(Z;\theta,\eta,\mu) \, \vert \, X=x ) = 0$. This implies that $u_q^*$ is Neyman-orthogonal to order $q$, as defined above.

Figures (2)

  • Figure 1: Comparing two estimates of $\beta_0$
  • Figure 2: Production function estimate

Theorems & Definitions (16)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • proof : Proof of Theorem \ref{['theo_neyman']}
  • Remark 2
  • Lemma 3
  • Lemma 4
  • ...and 6 more