Research map / Probabilistic and scientific modeling / Treatment effects
Triple Changes Estimator for Targeted Policies
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Extends the changes-in-changes (CiC) framework to a triple-difference-style setting by replacing CiC's no-drift assumption with state-independent drift maps, yielding point and partial identification, a finite-sample estimator, and an application to Medicaid expansion and children's preventive care.
Problem
Difference-in-differences relies on parallel trends, and triple differences only identify average effects and is biased under scale differences; CiC identifies full counterfactual distributions but assumes no drift between treated and control groups, which fails for targeted policies.
Approach
Combine triple differencing with CiC: assume the non-linear drift map between control and treated groups is identical across two states (state-independent drift), allowing group-specific production functions and identifying the full counterfactual CDF of the treated group in the policy state.
Key results
- In the linear simulation, DiD and CiC show persistent bias while triple difference and triple changes estimators converge to zero bias as sample size grows.
- In the nonlinear simulation, the triple difference estimator is biased while the triple changes estimators remain asymptotically unbiased.
- On NSCH data, the triple difference estimate of the Medicaid expansion effect on preventive visits was 0.170 (90% CI [-0.004, 0.344]) and triple changes was 0.145 (90% CI [-0.01, 0.331]).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use triple difference instead of DiD or CiC | Synthetic linear model · Synthetic data | Relative bias | Helped |
| Use triple changes instead of triple difference | Synthetic nonlinear model · Synthetic data | Relative bias | Helped |
| Apply Medicaid expansion in Louisiana | Children in eligible households · NSCH preventive doctor visits (2016–2017) | ATT: 0.145, 90% confidence interval [−0.01, 0.331] | Helped |
Surprising findings
- Triple changes estimates on the NSCH application have confidence intervals that include zero, so the positive effect is not statistically distinguishable from zero at 90% level.
Limitations
- Point identification relies on state-independent drift and monotonicity assumptions that are untestable; relaxing monotonicity yields only partial identification.
- Empirical application uses only two states and two time periods and relies on a bootstrap with wide intervals.
- Estimation is only derived for the average treatment effect on the treated under smoothness and bounded-density assumptions.
Experimental setup
Datasets: National Survey on Children's Health (NSCH) 2016-2017 synthetic data
Benchmarks: synthetic simulations (relative bias) NSCH Medicaid expansion application
Metrics: relative bias ATT point estimates bootstrap 90% confidence intervals
Baselines: DiD triple difference CiC
Builds on: changes-in-changes triple difference difference-in-differences Brenier's theorem
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