Empirical Bayes Estimation in Heterogeneous Coefficient Panel Models

TL;DR

This paper develops a nonparametric empirical Bayes G-modeling approach for short-panel linear models with multidimensional heterogeneity (HIVDX), allowing random intercepts, slopes, dynamics, and non-spherical error structures. By treating the individual parameters $ heta_i=(eta_i, ext{delta}_i)$ as i.i.d. draws from an unknown prior $G_*$ and estimating it via the nonparametric MLE, it achieves shrinkage and improved predictive performance relative to unit-level MLEs. The authors establish identification under rank and one-to-one conditions, prove NP MLE and EB-consistency, and provide regret-consistency results for functionals of $ heta_i$, with a Wasserstein–Fisher–Rao gradient flow algorithm to compute $ hat G$. Empirical analysis on PSID earnings demonstrates substantial cross-sectional heterogeneity and sizable predictive gains from EB shrinkage, while Monte Carlo experiments corroborate modest to large MSE reductions across parameters and improved predictive $R^2$. The framework thus offers a robust, flexible toolkit for inference and prediction in heterogeneous dynamic panels, with clear directions for extending inference and scalability.

Abstract

We develop an empirical Bayes (EB) G-modeling framework for short-panel linear models with multidimensional heterogeneity and nonparametric prior. Specifically, we allow heterogeneous intercepts, slopes, dynamics, and a non-spherical error covariance structure. We establish identification and consistency of the nonparametric maximum likelihood estimator (NPMLE) under general conditions, and provide low-level sufficient conditions for several models of empirical interest. Conditions for regret consistency of the resulting EB estimators are also established. The NPMLE is computed using a Wasserstein-Fisher-Rao gradient flow algorithm adapted to panel regressions. Using data from the Panel Study of Income Dynamics, we find that the slope coefficient for potential experience is substantially heterogeneous and negatively correlated with the random intercept, and that error variances and autoregressive coefficients vary significantly across individuals. The EB estimates reduce mean squared prediction errors relative to individual maximum likelihood estimates.

Figures (4)

• Figure 1: Shrinkage Effects: Comparison of $(\hat{\theta}^{\mathrm{MLE}}_i)_{i=1}^N$ and $(\hat{\theta}^{\mathrm{EB}}_i)_{i=1}^N$
• Figure 2: Pseudo-out-of-sample predictions and prediction errors
• Figure 3: EB Estimates of $(a_i, b_i)$ and Cross-Sectional Variance of Earnings
• Figure 4: Pairwise scatter plots of $\hat{\theta}_i^{\mathrm{EB}}$

Theorems & Definitions (25)

• Theorem 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Theorem 2
• Theorem 3
• Proposition 5
• proof : Proof of Theorem \ref{['thm:identification']}
• proof : Proof of Theorem \ref{['thm:consistency']}