[Paper Review] Panel Data Quantile Regression with Grouped Fixed Effects

Jiaying Gu, Stanislav Volgushev|arXiv (Cornell University)|Jan 15, 2018

Statistical Methods and Inference22 references3 citations

TL;DR

This paper proposes a novel method for panel data quantile regression with grouped fixed effects, where individual heterogeneity is modeled as latent groups rather than individual-specific effects. Using convex optimization to estimate both group membership and model parameters, the approach achieves consistent group estimation and asymptotically normal estimators, offering a balance between flexibility and interpretability in heterogeneous panel data models.

ABSTRACT

This paper introduces estimation methods for grouped latent heterogeneity in panel data quantile regression. We assume that the observed individuals come from a heterogeneous population with a finite number of types. The number of types and group membership is not assumed to be known in advance and is estimated by means of a convex optimization problem. We provide conditions under which group membership is estimated consistently and establish asymptotic normality of the resulting estimators. Simulations show that the method works well in finite samples when T is reasonably large. To illustrate the proposed methodology we study the effects of the adoption of Right-to-Carry concealed weapon laws on violent crime rates using panel data of 51 U.S. states from 1977 - 2010.

Motivation & Objective

• To address the challenge of high-dimensional individual fixed effects in panel data quantile regression by modeling latent heterogeneity through grouped fixed effects.
• To estimate both the number of groups and group membership simultaneously without assuming prior knowledge of group structure.
• To establish theoretical consistency and asymptotic normality of the resulting estimators under regularity conditions.
• To provide a method that balances flexibility (unrestricted correlation between individual effects and covariates) with interpretability (fewer distinct effect values).
• To demonstrate the method's finite-sample performance and empirical relevance through simulations and an application to U.S. state-level crime data.

Proposed method

• Employs a convex optimization framework to estimate group structure and model parameters jointly, minimizing a penalized quantile regression objective with a group-specific penalty.
• Uses an information criterion (IC) with a data-dependent penalty function $ p_{n,T} $ to select the optimal number of groups.
• Applies an $ \ell_1 $-type penalty to shrink individual effects toward group means, promoting group structure.
• Estimates group membership via convex clustering, leveraging the structure of the optimization problem to identify homogeneous subpopulations.
• Derives asymptotic normality of the common parameter estimators under regularity conditions, including moment and rate conditions on $ n $ and $ T $.
• Uses Hendricks-Koenker sandwich standard errors with Hall-Sheather bandwidth for inference on coefficient estimates.

Experimental results

Research questions

• RQ1 Can grouped fixed effects be consistently estimated in panel data quantile regression when the number of groups and group membership are unknown?
• RQ2 How does the proposed convex optimization method perform in finite samples, especially when $ T $ is moderately large?
• RQ3 What is the impact of the Right-to-Carry concealed weapon law on violent crime rates across different quantiles of the crime rate distribution?
• RQ4 How does the grouped fixed effect estimator compare to standard fixed effect and mean regression estimators in terms of bias, efficiency, and coverage?
• RQ5 What is the role of the penalty parameter $ p_{n,T} $ in determining the number of groups and estimator performance?

Key findings

• The method consistently estimates the number of groups and group membership under regularity conditions, with consistency established for both the group structure and the common parameters.
• Asymptotic normality of the common parameter estimators is established, enabling valid inference via standard errors and confidence intervals.
• Simulations show good finite-sample performance, with high correct group estimation rates and low RMSE for $ \hat{\beta}^{IC}(\tau) $ when $ T $ is reasonably large.
• The coverage rates for 95% confidence intervals are close to nominal levels, indicating accurate inference.
• In the empirical application, the grouped fixed effect estimator reveals heterogeneous effects of RTC laws across states, with varying impacts across quantiles of the crime rate distribution.
• The estimated group structure shows that states with similar crime patterns are clustered together, improving interpretability over individual fixed effects.