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On Efficient Estimation of Distributional Treatment Effects under Covariate-Adaptive Randomization
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Proposes an AIPW-style regression-adjusted estimator of distributional treatment effects (DTE) and probability treatment effects under covariate-adaptive randomization, proves its asymptotic Gaussian process limit and that it attains the semiparametric efficiency bound.
Problem
Estimating distributional treatment effects precisely in randomized experiments with covariate-adaptive randomization (stratified block randomization, Efron's biased-coin design), where extra covariates beyond strata are available but the usual i.i.d. assignment assumptions fail.
Approach
Treat the conditional CDF of the outcome given treatment, stratum and covariates as a distribution regression (binary outcome indicator 1{Y<=y}) and plug ML estimates into a Neyman-orthogonal AIPW moment with cross-fitting, stratum-level assignment probabilities, yielding a DTE estimator that attains the semiparametric efficiency bound under CAR.
Key results
- In simulations with n=1,000 (1,000 replications), ML (gradient boosting) regression adjustment reduced RMSE and CI length relative to the empirical estimator, with linear adjustment in between; ML coverage 0.97-0.99.
- On the microcredit data (n=611), regression adjustment reduced standard errors by 1% to 13% (average 7%) across revenue levels.
- Regression adjustment reduced standard error by about 10% for the probability of zero revenue, which decreased by 10 percentage points (SE 4.6 pp).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use linear regression adjustment instead of the empirical estimator | DTE estimation · Simulation study | RMSE | Helped |
| Use gradient boosting adjustment instead of the empirical estimator | DTE estimation · Simulation study | RMSE | Helped |
| Use gradient boosting adjustment instead of the empirical estimator | DTE estimation · Simulation study | 95% CI coverage: close to the nominal level of 0.95; 0.97 to 0.99 | Mixed |
| Add gradient boosting covariate adjustment | DTE estimation · Microfinance experiment, enterprise revenue | standard error: 1% to 13% reduction across revenue levels; average reduction of 7% | Helped |
| Add gradient boosting covariate adjustment | PTE estimation · Microfinance experiment, enterprise revenue | standard error | Mixed |
| Add gradient boosting covariate adjustment | PTE estimation · Microfinance experiment, probability of zero revenue | probability treatment effect: decreases by 10 percentage points (pp), with a standard error of 4.6 pp | Helped |
Surprising findings
- For the probability treatment effect on the microcredit data, regression adjustment did not consistently reduce standard errors relative to the empirical estimator, attributed to the small sample size.
Limitations
- Assumes perfect compliance and no interference between units.
- Variance reduction depends on covariates being predictive of the outcome; diminishes with weak covariates.
- Precision gains are harder to achieve with many strata and modest sample sizes.
Experimental setup
Datasets: simulated data (n=1,000 and 5,000) Mongolian microcredit field experiment (Attanasio et al. 2015), n=611
Benchmarks: simulated DTE estimation (RMSE, CI length, coverage) Mongolian microcredit enterprise revenue DTE/PTE
Metrics: RMSE average CI length coverage probability of 95% CI
Baselines: empirical (unadjusted) DTE estimator linear regression adjustment
Builds on: AIPW doubly robust estimation Neyman-orthogonal moment conditions cross-fitting (double machine learning) Bugni, Canay and Shaikh covariate-adaptive randomization inference
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