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Robust Priors in Nonlinear Panel Models with Individual and Time Effects

Zizhong Yan, Zhengyu Zhang, Mingli Chen, Jingrong Li, Iván Fernández-Val

Abstract

We develop likelihood-based bias reduction for nonlinear panel models with additive individual and time effects. In two-way panels, integrated-likelihood corrections are attractive but challenging because the required integration is high dimensional and standard Laplace approximations may fail when the parameter dimension grows with the sample size. We propose a target-centered full-exponential Laplace--cumulant expansion that exploits the sparse higher-order derivative structure implied by additive effects, delivering a tractable approximation with a negligible remainder under large-$N,T$ asymptotics. The expansion motivates robust priors that yield bias reduction for both common parameters and fixed effects. We provide implementations for binary, ordered, and multinomial response models with two-way effects. For average partial effects, we show that the remaining first-order bias has a simple variance form and can be removed by a closed-form adjustment. Monte Carlo experiments and an empirical illustration show substantial bias reduction with accurate inference.

Robust Priors in Nonlinear Panel Models with Individual and Time Effects

Abstract

We develop likelihood-based bias reduction for nonlinear panel models with additive individual and time effects. In two-way panels, integrated-likelihood corrections are attractive but challenging because the required integration is high dimensional and standard Laplace approximations may fail when the parameter dimension grows with the sample size. We propose a target-centered full-exponential Laplace--cumulant expansion that exploits the sparse higher-order derivative structure implied by additive effects, delivering a tractable approximation with a negligible remainder under large- asymptotics. The expansion motivates robust priors that yield bias reduction for both common parameters and fixed effects. We provide implementations for binary, ordered, and multinomial response models with two-way effects. For average partial effects, we show that the remaining first-order bias has a simple variance form and can be removed by a closed-form adjustment. Monte Carlo experiments and an empirical illustration show substantial bias reduction with accurate inference.

Paper Structure

This paper contains 28 sections, 11 theorems, 80 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and Estimators
  3. Model
  4. Estimators and likelihood-based corrections
  5. Notation and Assumptions
  6. Characterization of the Robust Priors
  7. Target-Centered Full-Exponential Laplace Approximations
  8. Robust Priors for Generic Models
  9. Asymptotic Distribution and Inference
  10. Bias-reducing Penalties for Joint Likelihood
  11. Model-Specific Bias-reducing Priors
  12. One-Parameter Exponential Family Panels
  13. Exponential family: strict exogeneity.
  14. Exponential family: predetermined regressors.
  15. Poisson panels: strict exogeneity.
  16. ...and 13 more sections

Key Result

Lemma 3.1

Suppose Assumption assumption:main holds. Then, as $N,T\rightarrow{\infty}$, for any $\beta$ satisfying $\| \beta-\beta_0\|=\mathcal{O}_P((NT)^{-1/4})$, where $C^{st}$ is a constant term$C^{st}$ or $c^{st}$ is an $\mathcal{O}(1)$ constant whose value may change across contexts., and $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Female labor force participation --- MCMC diagnostics for selected coefficients in the dynamic logit model

Theorems & Definitions (23)

  • Remark 1: Assumptions \ref{['assumption:main']} and \ref{['assumption:ape']}
  • Lemma 3.1
  • Theorem 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Theorem 3: Bias-reducing priors for generic models
  • Remark 4
  • Corollary 3.1
  • Remark 5
  • ...and 13 more