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Doubly Robust Estimation of Treatment Effects in Staggered Difference-in-Differences with Time-Varying Covariates

Yuhao Deng, Le Kang

Abstract

The difference-in-differences (DiD) design is a quasi-experimental method for estimating treatment effects. In staggered DiD with multiple treatment groups and periods, estimation based on the two-way fixed effects model yields negative weights when averaging heterogeneous group-period treatment effects into an overall effect. To address this issue, we first define group-period average treatment effects on the treated (ATT), and then define groupwise, periodwise, dynamic, and overall ATTs nonparametrically, so that the estimands are model-free. We propose doubly robust estimators for these types of ATTs in the form of augmented inverse variance weighting (AIVW). The proposed framework allows time-varying covariates that partially explain the time trends in outcomes. Even if part of the working models is misspecified, the proposed estimators still consistently estimate the parameter of interest. The asymptotic variance can be explicitly computed from influence functions. Under a homoskedastic working model, the AIVW estimator is simplified to an augmented inverse probability weighting (AIPW) estimator. We demonstrate the desirable properties of the proposed estimators through simulation and an application that compares the effects of a parallel admission mechanism with immediate admission on the China National College Entrance Examination.

Doubly Robust Estimation of Treatment Effects in Staggered Difference-in-Differences with Time-Varying Covariates

Abstract

The difference-in-differences (DiD) design is a quasi-experimental method for estimating treatment effects. In staggered DiD with multiple treatment groups and periods, estimation based on the two-way fixed effects model yields negative weights when averaging heterogeneous group-period treatment effects into an overall effect. To address this issue, we first define group-period average treatment effects on the treated (ATT), and then define groupwise, periodwise, dynamic, and overall ATTs nonparametrically, so that the estimands are model-free. We propose doubly robust estimators for these types of ATTs in the form of augmented inverse variance weighting (AIVW). The proposed framework allows time-varying covariates that partially explain the time trends in outcomes. Even if part of the working models is misspecified, the proposed estimators still consistently estimate the parameter of interest. The asymptotic variance can be explicitly computed from influence functions. Under a homoskedastic working model, the AIVW estimator is simplified to an augmented inverse probability weighting (AIPW) estimator. We demonstrate the desirable properties of the proposed estimators through simulation and an application that compares the effects of a parallel admission mechanism with immediate admission on the China National College Entrance Examination.
Paper Structure (11 sections, 3 theorems, 37 equations, 3 figures, 2 tables)

This paper contains 11 sections, 3 theorems, 37 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Causal difference-in-differences
  3. Average treatment effects on the treated
  4. Identification
  5. Estimation
  6. Regression and weighted estimators
  7. Proposed semiparametric estimator
  8. Connection with linear outcome regression models
  9. Simulation Studies
  10. Application to NCEE (Gaokao) data
  11. Discussion

Key Result

Theorem 1

Under Assumptions 1--4, the group-period ATT is identifiable for evert $1\leq g \leq t \leq T$,

Figures (3)

  • Figure 1: Groupwise ATT, periodwise ATT, dynamic ATT, and overall ATT aggregated from group-cell ATTs.
  • Figure 2: Average estimate, standard deviation (SD), and average standard error (SE) for the group-period ATTs, groupwise ATTs, periodwise ATTs, and dynamic ATTs over time based on AIPW and AIVW. The x-axis is the time label, and the y-axis is the group label.
  • Figure 3: Estimated periodwise ATTs for justified envy based on AIPW and AIVW in the NCEE dataset.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Remark 2