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Integer Programming for Generalized Causal Bootstrap Designs
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Proposes an integer program that numerically finds the variance-maximizing (worst-case) joint distribution of potential outcomes, used as input to a causal bootstrap for design-based confidence intervals across general assignments and linear/quadratic-in-treatment estimators.
Problem
Quantifying design uncertainty (from random treatment assignment) for estimators under non-standard designs (e.g., matched pairs, covariate-balancing) where closed-form worst-case couplings for the causal bootstrap are unavailable, especially in small geographical experiments.
Approach
Formulate the search for the variance-maximizing joint distribution of potential outcomes as a binary integer program (quadratic objective linearized) with marginal-matching constraints, then feed the resulting least-favorable copula into the causal bootstrap; validity proven for unconfounded, conditionally unconfounded, and bounded-confoundedness individualistic assignments.
Key results
- For matched-pairs with doubly-robust estimator on GDP, optimal causal bootstrap yields CI widths about 10%+ narrower than the sampling bootstrap across effect sizes (e.g., 97 vs 109 at 2.5% effect), with 100% coverage in the matched-pairs A/A setting.
- Under complete randomization the optimal causal bootstrap recovers the isotone copula results exactly; sampling bootstrap and optimal causal bootstrap both achieve roughly 87-90% coverage in complete randomization A/A tests.
- Isotone copula bootstrap under matched pairs gives zero coverage (CIs never include zero) in the A/A setting, while the integer-program approach retains satisfactory coverage.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the optimal causal bootstrap instead of the sampling bootstrap | GDP A/A, complete randomization without covariates · GDP A/A | median CI width: 3967 -> 3348 | Helped |
| Use the optimal causal bootstrap instead of the sampling bootstrap | GDP A/A, matched pairs without covariates · GDP A/A | median CI width: 4110 -> 2233 | Helped |
| Use the optimal causal bootstrap instead of the sampling bootstrap | GDP A/A, matched pairs without covariates · GDP A/A | coverage: 100% -> 100% | No clear effect |
| Use the optimal causal bootstrap instead of the sampling bootstrap | GDP multiplicative-effect experiments with 2018 GDP covariates · GDP, matched pairs, 2.5% effect | median CI width: 109 -> 97 | Helped |
| Use the optimal causal bootstrap instead of the sampling bootstrap | GDP multiplicative-effect experiments with 2018 GDP covariates · GDP, matched pairs, 10% effect | median CI width: 235 -> 203 | Helped |
| Use the optimal causal bootstrap instead of the sampling bootstrap | GDP, complete randomization, 2018 GDP covariate · GDP, multiplicative 2.5% effect | power: 0.32 -> 0.39 | Helped |
| Use the optimal causal bootstrap instead of the conservative variance interval | GDP, matched pairs, 2018 GDP covariate · GDP, multiplicative 10% effect | median CI width: 310 -> 203 | Helped |
| Use the optimal causal bootstrap instead of the isotone copula bootstrap | GDP A/A, matched pairs without covariates · GDP A/A | coverage: 0% -> 100% | Helped |
| Use the optimal causal bootstrap instead of the isotone copula bootstrap | GDP complete-randomization, difference-in-means experiments · GDP, complete randomization | median CI width | No clear effect |
| Add 2018 GDP as a covariate | Opt. Causal Boot. under complete randomization · GDP A/A | median CI width: 3348 -> 133 | Helped |
| Use matched pairs instead of complete randomization | Opt. Causal Boot. without covariates · GDP A/A | median CI width: 3348 -> 2233 | Helped |
| Use matched pairs instead of complete randomization | Isotone Copula bootstrap · GDP A/A without covariates | coverage: 87% -> 0% | Hurt |
Surprising findings
- The isotone copula bootstrap, used beyond complete randomization, yields 0% coverage under matched-pairs designs because imputed within-pair effects are identical, yielding near-zero estimated variance.
- Conservative Neyman-style variance estimates can yield narrower CIs than the sampling bootstrap for small effects but wider ones for large effects.
Limitations
- Computational scaling limited to small datasets (hundreds of units).
- Asymptotic validity for general confounded assignments is not established; only individualistic bounded-confoundedness case is covered.
- Continuous outcomes must be discretized to the observed support, and the slackness epsilon is needed for feasibility.
Experimental setup
Datasets: IMF World Economic Outlook GDP country-level data 2017-2019 (top 50 countries)
Benchmarks: simulated geographical experiments on IMF GDP data (top 50 countries)
Metrics: 95% CI coverage CI width power runtime
Baselines: Sampling bootstrap Conservative variance (Neyman-style) Isotone copula causal bootstrap
Builds on: causal bootstrap isotone copula (Fréchet-Hoeffding bound) Neyman variance estimator Horvitz-Thompson estimator
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