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Double Machine Learning for Static Panel Models with Fixed Effects

Paul S. Clarke, Annalivia Polselli

TL;DR

This paper extends double machine learning to static panel models with fixed effects by developing CRE, exact first-difference, and approximate within grouping estimators that leverage Neyman-orthogonal scores to robustly estimate treatment effects when nuisance functions are high-dimensional or nonlinear. The authors formalize learning procedures for nuisance components, justify approximations, and derive an exact orthogonal score under panel assumptions, enabling valid inference with cross-fitting. An empirical UK study on the National Minimum Wage and voting behavior demonstrates practical benefits of the FD (Exact) approach and ensemble learning for nuisance estimation, while simulations highlight when linear, nonlinear-smooth, or discontinuous nuisance structures favor different learners. Collectively, the work provides concrete guidance for practitioners: use first-difference with ensemble nuisance learners and rely on cross-fitting to obtain reliable causal estimates in panel settings with fixed effects. Extensions to heterogeneous treatment effects and broader policy targets are discussed as avenues for future research.

Abstract

Recent advances in causal inference have seen the development of methods which make use of the predictive power of machine learning algorithms. In this paper, we develop novel double machine learning (DML) procedures for panel data in which these algorithms are used to approximate high-dimensional and nonlinear nuisance functions of the covariates. Our new procedures are extensions of the well-known correlated random effects, within-group and first-difference estimators from linear to nonlinear panel models, specifically, Robinson (1988)'s partially linear regression model with fixed effects and unspecified nonlinear confounding. Our simulation study assesses the performance of these procedures using different machine learning algorithms. We use our procedures to re-estimate the impact of minimum wage on voting behaviour in the UK. From our results, we recommend the use of first-differencing because it imposes the fewest constraints on the distribution of the fixed effects, and an ensemble learning strategy to ensure optimum estimator accuracy.

Double Machine Learning for Static Panel Models with Fixed Effects

TL;DR

This paper extends double machine learning to static panel models with fixed effects by developing CRE, exact first-difference, and approximate within grouping estimators that leverage Neyman-orthogonal scores to robustly estimate treatment effects when nuisance functions are high-dimensional or nonlinear. The authors formalize learning procedures for nuisance components, justify approximations, and derive an exact orthogonal score under panel assumptions, enabling valid inference with cross-fitting. An empirical UK study on the National Minimum Wage and voting behavior demonstrates practical benefits of the FD (Exact) approach and ensemble learning for nuisance estimation, while simulations highlight when linear, nonlinear-smooth, or discontinuous nuisance structures favor different learners. Collectively, the work provides concrete guidance for practitioners: use first-difference with ensemble nuisance learners and rely on cross-fitting to obtain reliable causal estimates in panel settings with fixed effects. Extensions to heterogeneous treatment effects and broader policy targets are discussed as avenues for future research.

Abstract

Recent advances in causal inference have seen the development of methods which make use of the predictive power of machine learning algorithms. In this paper, we develop novel double machine learning (DML) procedures for panel data in which these algorithms are used to approximate high-dimensional and nonlinear nuisance functions of the covariates. Our new procedures are extensions of the well-known correlated random effects, within-group and first-difference estimators from linear to nonlinear panel models, specifically, Robinson (1988)'s partially linear regression model with fixed effects and unspecified nonlinear confounding. Our simulation study assesses the performance of these procedures using different machine learning algorithms. We use our procedures to re-estimate the impact of minimum wage on voting behaviour in the UK. From our results, we recommend the use of first-differencing because it imposes the fewest constraints on the distribution of the fixed effects, and an ensemble learning strategy to ensure optimum estimator accuracy.
Paper Structure (15 sections, 26 equations, 1 figure, 6 tables)

This paper contains 15 sections, 26 equations, 1 figure, 6 tables.

Table of Contents

  1. Empirical Application
  2. Conclusion
  3. Derivations of Results
  4. Derivation of learning procedure for CRE in Section 5.1.1
  5. Justification of Approximation Approach in Section 5.1.2.a
  6. Derivation of Exact Approach in Section 5.1.2.b
  7. Derivation of Neyman Orthogonal Score Function
  8. Monte Carlo Simulations
  9. Monte Carlo Simulation Design
  10. Monte Carlo Simulation Results
  11. Hyperparameter Tuning
  12. The Grid Search Algorithm
  13. Hyperparameter Tuning in Monte Carlo Simulations
  14. Hyperparameter Tuning in the Empirical Application
  15. Variable Description

Figures (1)

  • Figure 1: Comparison of distributions of $\widehat{\theta}$ across learners (Panel A) and with different tuned hyperparameters (Panel B) with the CRE approach.