[Paper Review] Design-Robust Two-Way-Fixed-Effects Regression For Panel Data

Dmitry Arkhangelsky, Guido W. Imbens+2 more|arXiv (Cornell University)|Jul 29, 2021

Advanced Causal Inference Techniques43 references4 citations

TL;DR

This paper proposes a design-robust two-way fixed-effects regression estimator for panel data that incorporates unit-specific weights derived from a model of the treatment assignment mechanism. The estimator achieves strong double robustness—remaining consistent if either the assignment model or the outcome model is correctly specified—offering improved inference over conventional two-way fixed effects in settings with staggered adoption or model misspecification.

ABSTRACT

We propose a new estimator for average causal effects of a binary treatment with panel data in settings with general treatment patterns. Our approach augments the popular two-way-fixed-effects specification with unit-specific weights that arise from a model for the assignment mechanism. We show how to construct these weights in various settings, including the staggered adoption setting, where units opt into the treatment sequentially but permanently. The resulting estimator converges to an average (over units and time) treatment effect under the correct specification of the assignment model, even if the fixed effect model is misspecified. We show that our estimator is more robust than the conventional two-way estimator: it remains consistent if either the assignment mechanism or the two-way regression model is correctly specified. In addition, the proposed estimator performs better than the two-way-fixed-effect estimator if the outcome model and assignment mechanism are locally misspecified. This strong double robustness property underlines and quantifies the benefits of modeling the assignment process and motivates using our estimator in practice. We also discuss an extension of our estimator to handle dynamic treatment effects.

Motivation & Objective

• Address the limitations of conventional two-way fixed-effects (TWFE) regression in panel data when treatment assignment is non-random or models are misspecified.
• Develop a new estimator that explicitly models the treatment assignment mechanism to improve causal inference robustness.
• Establish theoretical properties of the estimator under both correct and misspecified assignment and outcome models.
• Extend the framework to handle dynamic treatment effects in staggered adoption designs.
• Provide practical guidance for estimating the assignment mechanism using duration models in empirical applications.

Proposed method

• Augment the standard TWFE regression with unit-specific weights $\gamma_i$ derived from a model of the treatment assignment process.
• Construct oracle weights $\gamma^\star$ using the true assignment model and generalized propensity scores, ensuring design-based inference validity.
• Use a nonlinear equation to determine weights based on the support of treatment paths $\mathbf{W}_i$, with solutions derived for common designs like staggered adoption.
• Establish double robustness by showing the estimator remains consistent if either the assignment model or the outcome model is approximately correct.
• Extend the method to dynamic treatment effects using aggregated augmented inverse probability weighting (AIPW) estimators with cross-fitting to reduce bias.
• Estimate the assignment mechanism from data using duration models, particularly in staggered adoption settings, to enable practical implementation.

Experimental results

Research questions

• RQ1 How can we improve the robustness of two-way fixed-effects estimators in panel data when the assignment mechanism is complex or misspecified?
• RQ2 What conditions ensure that the proposed weighted estimator remains consistent under model misspecification of either the assignment or outcome model?
• RQ3 Can we construct unit-specific weights that enhance inference by incorporating design-based assumptions about treatment assignment?
• RQ4 How does the proposed estimator perform relative to conventional TWFE and other robust estimators in finite samples with effect heterogeneity?
• RQ5 What is the impact of including fixed effects in the AIPW estimator, and can cross-fitting restore double robustness in short panels?

Key findings

• The proposed estimator achieves strong double robustness: it remains consistent if either the assignment model or the outcome model is correctly specified.
• The oracle estimator $\hat{\tau}(\gamma^\star)$ is valid under correct assignment model specification, regardless of the outcome model, enabling design-based inference.
• In finite samples, the estimator outperforms conventional TWFE when both the outcome model and assignment mechanism are locally misspecified.
• The aggregated AIPW estimator with estimated fixed effects is not doubly robust due to dependence between outcome model estimates and treatment assignment, especially in short panels.
• Cross-fitting does not resolve the bias in AIPW estimators when fixed effects are estimated, highlighting the challenge of estimating unit fixed effects in small-$T$ settings.
• Empirical application using duration models for staggered adoption designs enables consistent estimation of the assignment mechanism, supporting practical implementation.