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Regression Adjustment for Estimating Distributional Treatment Effects in Randomized Controlled Trials

Tatsushi Oka, Shota Yasui, Yuta Hayakawa, Undral Byambadalai

Abstract

In this paper, we address the issue of estimating and inferring distributional treatment effects in randomized experiments. The distributional treatment effect provides a more comprehensive understanding of treatment heterogeneity compared to average treatment effects. We propose a regression adjustment method that utilizes distributional regression and pre-treatment information, establishing theoretical efficiency gains without imposing restrictive distributional assumptions. We develop a practical inferential framework and demonstrate its advantages through extensive simulations. Analyzing water conservation policies, our method reveals that behavioral nudges systematically shift consumption from high to moderate levels. Examining health insurance coverage, we show the treatment reduces the probability of zero doctor visits by 6.6 percentage points while increasing the likelihood of 3-6 visits. In both applications, our regression adjustment method substantially improves precision and identifies treatment effects that were statistically insignificant under conventional approaches.

Regression Adjustment for Estimating Distributional Treatment Effects in Randomized Controlled Trials

Abstract

In this paper, we address the issue of estimating and inferring distributional treatment effects in randomized experiments. The distributional treatment effect provides a more comprehensive understanding of treatment heterogeneity compared to average treatment effects. We propose a regression adjustment method that utilizes distributional regression and pre-treatment information, establishing theoretical efficiency gains without imposing restrictive distributional assumptions. We develop a practical inferential framework and demonstrate its advantages through extensive simulations. Analyzing water conservation policies, our method reveals that behavioral nudges systematically shift consumption from high to moderate levels. Examining health insurance coverage, we show the treatment reduces the probability of zero doctor visits by 6.6 percentage points while increasing the likelihood of 3-6 visits. In both applications, our regression adjustment method substantially improves precision and identifies treatment effects that were statistically insignificant under conventional approaches.
Paper Structure (23 sections, 9 theorems, 92 equations, 14 figures, 1 algorithm)

This paper contains 23 sections, 9 theorems, 92 equations, 14 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Treatment Effect and Estimation Method
  3. Setup and Treatment Effects
  4. Distributional Regression
  5. Regression-Adjusted Treatment Effect Estimator
  6. Theoretical Results
  7. Oracle Estimator and Efficiency
  8. Asymptotic Distribution of DTE Estimator
  9. Monte-Carlo Simulation Results
  10. Empirical Applications
  11. Nudge and Water Consumption
  12. Insurance Coverage and Healthcare Utilization
  13. Conclusion
  14. Theoretical Results
  15. Proof of Theorem \ref{['theorem:distribution']}
  16. ...and 8 more sections

Key Result

Theorem 1

(a) Suppose that Assumption as:as1 holds and that $\hat{\pi}_{k} = \pi_{k} + o(1)$ as $n\to \infty$ for every $k \in \mathcal{K}$. Then, for each $k \in \mathcal{K}$ and for any $y \in \mathcal{Y}$, we have in $\ell^{\infty}(\mathcal{Y})$, provided that $\mathbb{E} [ ( 1\!{\rm l}_{ \{Y(k) \le \cdot\} } - F_{Y(k)} )^2 ] \ge \mathbb{E} [ ( 1\!{\rm l}_{ \{Y(k) \le \cdot\} } - G_{Y(k)|\bm{X}} )^2 ]$.

Figures (14)

  • Figure 1: Performance metrics of simple and regression-adjusted DTE estimators
  • Figure 2: Performance metrics of simple and regression-adjusted DTE estimators
  • Figure 3: Nudge Effect on Water Consumption (thousands of gallons) Distributional and Probability Treatment Effect
  • Figure 4: Effect of Health Insurance Coverage on Doctor Visits Distributional and Probability Treatment Effect
  • Figure 5: Performance metrics of simple and regression-adjusted DTE estimators
  • ...and 9 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma A.1
  • proof
  • proof : Proof of Theorem \ref{['theorem:distribution']}
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • ...and 9 more