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Event Studies with Feedback

Irene Botosaru, Laura Liu

TL;DR

The paper tackles the challenge that event studies with persistent outcomes can conflate direct treatment effects with indirect effects via endogenous covariate adjustments. It develops a dynamic panel event-study framework that separately models the direct structural response and the covariate-feedback mechanism, under sequential exogeneity and a homogeneous feedback restriction, achieving point identification of the common parameters $\theta$, the distribution of latent heterogeneity $H(\lambda_i \mid \mathcal{I}_i^0)$, and the covariate-feedback densities $\{f_t(\cdot)\}$. A dynamic decomposition algorithm enables counterfactual analysis and quantifies the relative importance of direct versus indirect channels for treatment-effect dynamics. The framework supports nonlinear extensions and group-specific feedback, broadening applicability to policy contexts where covariates adjust endogenously, and provides a practical tool for policy analysis and counterfactual planning with explicit decomposition into the two channels.

Abstract

Event studies often conflate direct treatment effects with indirect effects operating through endogenous covariate adjustment. We develop a dynamic panel event study framework that separates these effects. The framework allows for persistent outcomes and treatment effects and for covariates that respond to past outcomes and treatment exposure. Under sequential exogeneity and homogeneous feedback, we establish point identification of common parameters governing outcome and treatment effect dynamics, the distribution of heterogeneous treatment effects, and the covariate feedback process. We propose an algorithm for dynamic decomposition that enables researchers to assess the relative importance of each effect in driving treatment effect dynamics.

Event Studies with Feedback

TL;DR

The paper tackles the challenge that event studies with persistent outcomes can conflate direct treatment effects with indirect effects via endogenous covariate adjustments. It develops a dynamic panel event-study framework that separately models the direct structural response and the covariate-feedback mechanism, under sequential exogeneity and a homogeneous feedback restriction, achieving point identification of the common parameters , the distribution of latent heterogeneity , and the covariate-feedback densities . A dynamic decomposition algorithm enables counterfactual analysis and quantifies the relative importance of direct versus indirect channels for treatment-effect dynamics. The framework supports nonlinear extensions and group-specific feedback, broadening applicability to policy contexts where covariates adjust endogenously, and provides a practical tool for policy analysis and counterfactual planning with explicit decomposition into the two channels.

Abstract

Event studies often conflate direct treatment effects with indirect effects operating through endogenous covariate adjustment. We develop a dynamic panel event study framework that separates these effects. The framework allows for persistent outcomes and treatment effects and for covariates that respond to past outcomes and treatment exposure. Under sequential exogeneity and homogeneous feedback, we establish point identification of common parameters governing outcome and treatment effect dynamics, the distribution of heterogeneous treatment effects, and the covariate feedback process. We propose an algorithm for dynamic decomposition that enables researchers to assess the relative importance of each effect in driving treatment effect dynamics.
Paper Structure (7 sections, 2 theorems, 18 equations, 1 algorithm)

This paper contains 7 sections, 2 theorems, 18 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and Assumptions
  3. Identification
  4. Counterfactual Analysis and Dynamic Decomposition
  5. Extensions
  6. Proof of Theorem \ref{['th:theorem']}
  7. Supplemental Appendix

Key Result

Theorem 1

Suppose $\left\{Y_{it},X_{it},\{D_{it}^j\}_{j\in\mathcal{J}}\right\}_{t=1}^T$ follow eq:outcome and eq:delta_ar. Let Assumptions ass:seq_exog--ass:homog_feedback hold, and suppose the regularity conditions of botosaru2025time for identification are satisfied (i.i.d. sampling over $i$, conditional in

Theorems & Definitions (5)

  • Theorem 1: Identification
  • Remark 1: Time Effects
  • proof
  • Lemma 1: Elimination of Time Effects
  • proof