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Difference-in-Differences with Sample Selection

Gayani Rathnayake, Akanksha Negi, Otavio Bartalotti, Xueyan Zhao

Abstract

We consider the identification of average treatment effects on the treated (ATT) in difference-in-differences (DiD) settings in the presence of endogenous sample selection. We first establish that the conventional DiD estimand generally fails to recover causally meaningful treatment effects, even if selection and treatment assignment are independent. We then partially identify the ATT for individuals whose outcomes would be observed post-treatment under either counterfactual treatment state, and derive sharp bounds on this parameter under different sets of assumptions on the relationship between sample selection and treatment assignment. These identification results are extended to allow for covariates, repeated cross-section data, and two-by-two comparisons in staggered adoption designs. Furthermore, we present identification results for the ATT of three additional empirically relevant latent groups by imposing outcome mean dominance assumptions that have intuitive appeal in applications. Finally, two empirical illustrations demonstrate the approach's usefulness by revisiting (i) the effect of a job training program on earnings and (ii) the effect of a working-from-home policy on employee performance.

Difference-in-Differences with Sample Selection

Abstract

We consider the identification of average treatment effects on the treated (ATT) in difference-in-differences (DiD) settings in the presence of endogenous sample selection. We first establish that the conventional DiD estimand generally fails to recover causally meaningful treatment effects, even if selection and treatment assignment are independent. We then partially identify the ATT for individuals whose outcomes would be observed post-treatment under either counterfactual treatment state, and derive sharp bounds on this parameter under different sets of assumptions on the relationship between sample selection and treatment assignment. These identification results are extended to allow for covariates, repeated cross-section data, and two-by-two comparisons in staggered adoption designs. Furthermore, we present identification results for the ATT of three additional empirically relevant latent groups by imposing outcome mean dominance assumptions that have intuitive appeal in applications. Finally, two empirical illustrations demonstrate the approach's usefulness by revisiting (i) the effect of a job training program on earnings and (ii) the effect of a working-from-home policy on employee performance.

Paper Structure

This paper contains 83 sections, 26 theorems, 428 equations, 3 figures, 24 tables.

Table of Contents

  1. Introduction
  2. Model Framework
  3. What does DiD identify if we ignore sample selection?
  4. Identification of ATT for OOO
  5. Identification without Monotonicity
  6. Identification with Monotonicity
  7. Identification under Monotonicity and Assumption \ref{['Partialselection']}(a)
  8. Generalization with Covariates
  9. Identification of ATT for Other Latent Groups
  10. Identification of ATT for ONO Group
  11. Identification of ATT for NNO Group
  12. Identification of ATT for NOO Group
  13. Identification of Overall ATT
  14. Estimation and Inference
  15. Simulation
  16. ...and 68 more sections

Key Result

Lemma 1

Under Assumptions no anti and PT_OOO, the DiD estimand for the observed group in a two-period panel, $\tau_{\textup{DiDs}} \equiv \mathbbm{E}[Y_1-Y_0|D=1, S_0=1, S_1=1] - \mathbbm{E}[Y_1-Y_0|D=0,S_0=1, S_1=1]$, can be decomposed as:

Figures (3)

  • Figure 1: Distribution of estimated mixture proportion, $\hat{p}_{OOO1}$, with monotonicity
  • Figure G.1: Distribution of estimated mixture proportion, $\hat{p}_{ONO0}$, with monotonicity
  • Figure G.2: Distribution of estimated mixture proportion, $\hat{p}_{NNO1}$, with monotonicity

Theorems & Definitions (51)

  • Lemma 1: Bias of $\tau_{\textup{DiDs}}$
  • Lemma 2
  • Theorem 1: Bounds for $\tau_{OOO}$
  • Lemma 3
  • Theorem 2: Bounds for $\tau_{OOO}$ under positive monotonicity
  • Remark 1
  • Remark 2
  • Theorem 3: Bounds for $\tau_{ONO}$ under positive monotonicity
  • Theorem 4: Bounds for $\tau_{NNO}$ under positive monotonicity
  • Theorem 5: Bounds for $\tau_{NOO}$ under positive monotonicity
  • ...and 41 more