Back to Feedback: Dynamics and Heterogeneity in Panel Data
Abstract
Many popular estimation methods in panel data rely on the assumption that the covariates of interest are strictly exogenous. However, this assumption is empirically restrictive in a wide range of settings. In this paper I argue that credible empirical work requires meaningfully relaxing strict exogeneity assumptions. Econometricians have developed methods that allow for sequential exogeneity, which in contrast with strict exogeneity allows for the presence of feedback from past outcomes to future covariates or treatments. I review some of the classic work on linear models with constant coefficients, and then describe some approaches that allow for coefficient heterogeneity in models with feedback. Finally, in the last two parts of the paper I review recent work that allows for sequential exogeneity in nonlinear panel data models, and mention possible extensions to network settings.
Summary
This paper addresses the restrictive assumption of strict exogeneity in panel data models, which is often violated in empirical settings where past outcomes influence future covariates (feedback). The author, Stéphane Bonhomme, argues that credible empirical work requires relaxing this assumption and allowing for sequential exogeneity, where current covariates can depend on past outcomes. The paper reviews classic methods for linear models with constant coefficients, then explores approaches for handling coefficient heterogeneity in models with feedback. Finally, it examines recent work on sequential exogeneity in nonlinear panel data models and discusses potential extensions to network settings. The paper emphasizes the importance of addressing feedback bias, which can lead to inconsistent estimates in short panel data when strict exogeneity is incorrectly assumed, impacting popular methods like difference-in-differences. The paper highlights the pitfalls of relying on strict exogeneity and explores various solutions. It discusses sequential moment restrictions and associated GMM estimators, addressing the challenges of weak instruments and instrument proliferation. It also reviews quasi-likelihood approaches based on Gaussian specifications and bias correction techniques using large-T asymptotics to improve estimator performance in moderate-sized panels. The paper emphasizes that while pre-trend checks are commonly used to validate strict exogeneity, they are often insufficient and may not detect underlying feedback bias. By systematically reviewing these methods, the paper provides a comprehensive guide for researchers seeking to address dynamic feedback and heterogeneity in panel data analysis.
Key Insights
- •Strict exogeneity, a common assumption in panel data models, is often unrealistic because it prohibits feedback from past outcomes to future covariates.
- •Assuming strict exogeneity when covariates are only sequentially exogenous leads to biased OLS estimators in finite T and inconsistency in short panel data, even with fixed effects. The bias is approximately inversely proportional to T.
- •Two-way fixed effects estimators and event study regressions are also biased and inconsistent in short panels under sequential exogeneity.
- •Pre-trend checks are insufficient to validate strict exogeneity assumptions, as they may not detect underlying feedback bias, particularly in settings with limited data variation.
- •The popular Arellano-Bond (ABond) GMM estimator, while addressing endogeneity, can suffer from weak instruments and instrument proliferation issues, leading to biased estimates when the number of instruments is large relative to the sample size.
- •Quasi-likelihood approaches, based on Gaussian specifications, and bias correction methods using large-T asymptotics offer alternative estimation strategies with improved finite-sample properties compared to standard GMM estimators.
- •In models with coefficient heterogeneity, the unconditional and conditional versions of sequential exogeneity have different implications for the (partial) identification of average effects.
Practical Implications
- •Researchers should carefully consider the plausibility of strict exogeneity in their panel data applications, particularly when using difference-in-differences or event study designs, and explicitly address potential feedback mechanisms.
- •When strict exogeneity is questionable, researchers should explore alternative estimation methods that allow for sequential exogeneity, such as GMM estimators with careful instrument selection, quasi-likelihood approaches, or bias-corrected estimators.
- •Practitioners should be aware of the limitations of pre-trend checks and avoid relying solely on them to validate strict exogeneity assumptions.
- •The findings have direct implications for policy evaluation, particularly when assessing the impact of treatments (e.g., job training, policy reforms) that may be influenced by past outcomes.
- •Future research should focus on developing more robust estimation methods for nonlinear panel data models with feedback and heterogeneity, as well as extending these methods to dynamic network settings.