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Difference-in-Differences with Time-varying Continuous Treatments using Double/Debiased Machine Learning

Michel F. C. Haddad, Martin Huber, José Eduardo Medina-Reyes, Lucas Z. Zhang

TL;DR

This paper develops a generalized difference-in-differences framework for time-varying continuous treatments, focusing on the average treatment effect on the treated (ATET) across non-zero dose comparisons. It integrates kernel-based ATET estimators with double/debiased machine learning (DML) and cross-fitting to flexibly adjust for high-dimensional covariates and treatment histories, applicable to both repeated cross-section and panel data. Identification rests on a conditional parallel trends assumption for non-zero doses and is implemented via doubly robust score functions, including instantaneous and lagged dose effects, with undersmoothed kernel bandwidths to ensure Neyman orthogonality and valid inference. The authors establish asymptotic normality and consistent variance estimates, verify finite-sample performance through simulations, and illustrate the approach with a Brazilian municipality study showing that higher second-dose vaccination rates reduce COVID-19 mortality after several weeks, in line with the expected lag from infection to death.

Abstract

We propose a difference-in-differences (DiD) framework designed for time-varying continuous treatments across multiple periods. Specifically, we estimate the average treatment effect on the treated (ATET) by comparing distinct non-zero treatment intensities. Identification rests on a conditional parallel trends assumption that accounts for observed covariates and past treatment histories. Our approach allows for lagged treatment effects and, in repeated cross-sectional settings, accommodates compositional changes in covariates. We develop kernel-based ATET estimators for both repeated cross-sections and panel data, leveraging the double/debiased machine learning framework to handle potentially high-dimensional covariates and histories. We establish the asymptotic properties of our estimators under mild regularity conditions and demonstrate via simulations that their undersmoothed versions perform well in finite samples. As an empirical illustration, we apply our estimator to assess the effect of the second-dose COVID-19 vaccination rate in Brazil and find that higher vaccination rates reduce COVID-19-related mortality after a lag of several weeks.

Difference-in-Differences with Time-varying Continuous Treatments using Double/Debiased Machine Learning

TL;DR

This paper develops a generalized difference-in-differences framework for time-varying continuous treatments, focusing on the average treatment effect on the treated (ATET) across non-zero dose comparisons. It integrates kernel-based ATET estimators with double/debiased machine learning (DML) and cross-fitting to flexibly adjust for high-dimensional covariates and treatment histories, applicable to both repeated cross-section and panel data. Identification rests on a conditional parallel trends assumption for non-zero doses and is implemented via doubly robust score functions, including instantaneous and lagged dose effects, with undersmoothed kernel bandwidths to ensure Neyman orthogonality and valid inference. The authors establish asymptotic normality and consistent variance estimates, verify finite-sample performance through simulations, and illustrate the approach with a Brazilian municipality study showing that higher second-dose vaccination rates reduce COVID-19 mortality after several weeks, in line with the expected lag from infection to death.

Abstract

We propose a difference-in-differences (DiD) framework designed for time-varying continuous treatments across multiple periods. Specifically, we estimate the average treatment effect on the treated (ATET) by comparing distinct non-zero treatment intensities. Identification rests on a conditional parallel trends assumption that accounts for observed covariates and past treatment histories. Our approach allows for lagged treatment effects and, in repeated cross-sectional settings, accommodates compositional changes in covariates. We develop kernel-based ATET estimators for both repeated cross-sections and panel data, leveraging the double/debiased machine learning framework to handle potentially high-dimensional covariates and histories. We establish the asymptotic properties of our estimators under mild regularity conditions and demonstrate via simulations that their undersmoothed versions perform well in finite samples. As an empirical illustration, we apply our estimator to assess the effect of the second-dose COVID-19 vaccination rate in Brazil and find that higher vaccination rates reduce COVID-19-related mortality after a lag of several weeks.

Paper Structure

This paper contains 13 sections, 3 theorems, 86 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Instantaneous effects in repeated cross-sections
  3. Time-varying treatment doses in repeated cross-sections
  4. Identification in panel data
  5. Estimation
  6. Asymptotic Theory
  7. Simulation study
  8. Empirical application
  9. Conclusion
  10. Appendix
  11. Neyman orthogonality of the DR expressions
  12. Proofs of asymptotic theories
  13. Descriptive statistics

Key Result

Lemma 1

Suppose Assumptions ass9, ass10, ass11 hold for the repeated cross-sections case, and Assumptions ass9, ass12, ass13 hold for the panel case. Then, the bias satisfies $B(h) = \Delta_{d_t,d_t',t} - \Delta_{d_t,d_t',t}^h = O(h^2)$ for $d_t, d_t'\in \text{supp}(D_t)$.

Figures (1)

  • Figure 1: Dose-2 Vaccination Rate Over Time by Municipality

Theorems & Definitions (3)

  • Lemma 1
  • Theorem 1
  • Theorem 2