Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fixed Effects as Generated Regressors

Jiaqi Huang

TL;DR

This paper develops a debiased, Neyman-orthogonal framework for inference on parameters defined by cross-sectional moments that involve latent fixed effects from an auxiliary panel data regression. By constructing orthogonal moments that adjust for the first-order bias from estimating fixed effects, the approach yields a central limit theorem for the target parameter under general panel asymptotics and relaxed exogeneity. It leverages cross-fitting adapted to panel data, high-dimensional representations for the adjustment term, and optional Empirical Bayes or SURE shrinkage to improve nuisance parameter estimation. The methodology is demonstrated via Monte Carlo simulations showing reduced bias and credible size control when endogeneity is present, and an empirical application on experimental site selection in China showing that orthogonal moments yield more reliable inference about latent county specialization effects. Overall, the framework enables valid inference for nonlinear moments and generated regressors in settings with latent fixed effects, broadening applicability in economics and related fields.

Abstract

Many economic models feature moment conditions that involve latent variables. When the latent variables are individual fixed effects in an auxiliary panel data regression, we construct orthogonal moments that eliminate first-order bias induced by estimating the fixed effects. Machine Learning methods and Empirical Bayes methods can be used to improve the estimate of the nuisance parameters in the orthogonal moments. We establish a central limit theorem based on the orthogonal moments without relying on exogeneity assumptions between panel data residuals and the cross-sectional moment functions. In a simulation study where the exogeneity assumption is violated, the estimator based on orthogonal moments has smaller bias compared with other estimators relying on that assumption. An empirical application on experimental site selection demonstrates how the method can be used for nonlinear moment conditions.

Fixed Effects as Generated Regressors

TL;DR

This paper develops a debiased, Neyman-orthogonal framework for inference on parameters defined by cross-sectional moments that involve latent fixed effects from an auxiliary panel data regression. By constructing orthogonal moments that adjust for the first-order bias from estimating fixed effects, the approach yields a central limit theorem for the target parameter under general panel asymptotics and relaxed exogeneity. It leverages cross-fitting adapted to panel data, high-dimensional representations for the adjustment term, and optional Empirical Bayes or SURE shrinkage to improve nuisance parameter estimation. The methodology is demonstrated via Monte Carlo simulations showing reduced bias and credible size control when endogeneity is present, and an empirical application on experimental site selection in China showing that orthogonal moments yield more reliable inference about latent county specialization effects. Overall, the framework enables valid inference for nonlinear moments and generated regressors in settings with latent fixed effects, broadening applicability in economics and related fields.

Abstract

Many economic models feature moment conditions that involve latent variables. When the latent variables are individual fixed effects in an auxiliary panel data regression, we construct orthogonal moments that eliminate first-order bias induced by estimating the fixed effects. Machine Learning methods and Empirical Bayes methods can be used to improve the estimate of the nuisance parameters in the orthogonal moments. We establish a central limit theorem based on the orthogonal moments without relying on exogeneity assumptions between panel data residuals and the cross-sectional moment functions. In a simulation study where the exogeneity assumption is violated, the estimator based on orthogonal moments has smaller bias compared with other estimators relying on that assumption. An empirical application on experimental site selection demonstrates how the method can be used for nonlinear moment conditions.
Paper Structure (18 sections, 16 theorems, 188 equations, 1 figure, 5 tables, 1 algorithm)

This paper contains 18 sections, 16 theorems, 188 equations, 1 figure, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Individual fixed effects in cross-sectional moments
  3. Inference procedure
  4. Constructing Orthogonal Moment Conditions
  5. Implementation
  6. Asymptotic Theory
  7. Monte-Carlo Simulation
  8. Empirical Application
  9. Data and measurements
  10. Econometric Specification
  11. Results
  12. Conclusion
  13. Proof of results in section \ref{['as theory']}
  14. Consistency of Estimators
  15. Convergence rates for Elastic Net
  16. ...and 3 more sections

Key Result

Proposition 1

Under Assumption ass:cross-as:mom eb, and Assumption as:mom eb is needed only if EB or SURE variants of estimators $\tilde{\alpha}_{il}^*$ for $\alpha_i$ are used, for $\zeta$ and $g$ satisfying $\zeta+g\geq 1$, we have

Figures (1)

  • Figure 1: Power function for Panel A

Theorems & Definitions (36)

  • Example 1
  • Definition 1
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Lemma A.1
  • proof
  • Lemma A.2
  • ...and 26 more