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Counterfactual Identification Under Monotonicity Constraints
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Extends counterfactual identification to settings with local monotonicity constraints on causal graphs, introducing a monotonicity reduction lemma and the M-ID algorithm for arbitrary counterfactual queries.
Problem
Counterfactual quantities that are not identifiable non-parametrically (e.g., LATE, PoCs, post-treatment conditioning) may become identifiable under shape constraints such as monotonicity, but prior work hard-codes these assumptions for specific settings.
Approach
Encode local monotonicity as signed edges in a causal graph (CGMA), prove a monotonicity reduction lemma that rewrites inconsistent ctf-factors into consistent ones using monotonicity, and extend the Ctf-ID algorithm into M-ID that identifies arbitrary counterfactual queries from observational and experimental data.
Key results
- On the 401(k) SCM, the proposed identification tracks ground-truth LATE across all four income quartiles, while classic-LATE and cond-LATE deviate substantially (e.g., cond-LATE gives a negative effect in middle quartiles where truth is positive).
- In the synthetic fairness SCM, the monotonicity-based estimator for (z,m)-TE matches ground truth, whereas the (z,m)-TV difference does not equal the total effect.
- M-ID is proved sound (Thm 1), and M-REDUCE failure implies non-identifiability (Thm 2), making the reduction complete for removing inconsistencies.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the monotonicity-aware graphical estimator instead of classic-LATE or cond(M)-LATE | LATE estimation on the 401(k) dataset · 401(k) dataset, income quartiles | agreement with ground-truth LATE | Helped |
| Use classic-LATE or cond(M)-LATE instead of the monotonicity-aware graphical estimator | LATE estimation on the 401(k) dataset · 401(k) dataset, income quartiles | agreement with ground-truth LATE | Hurt |
| Use the observational conditional difference (z,m)-TV in place of the identified counterfactual (z,m)-TE | Total-effect estimation in the fairness SCM · Synthetic fairness graph, age and education subgroups | agreement with ground-truth (z,m)-TE | Hurt |
Surprising findings
- Conditioning on the mediator before applying the classical LATE estimator (cond(M)-LATE) can yield the wrong sign of effect, not just a biased magnitude.
- Authors argue the prior belief that causal diagrams cannot encode shape restrictions like monotonicity is false.
Limitations
- Monotonicity reduction lemma is stated for binary variables; extension to non-binary domains is only discussed in an appendix.
- Completeness of general identification under monotonicity is an open question; the algorithm is sound but not shown complete.
- Only discrete variables are treated; continuous-variable monotonicity is left to future work.
Experimental setup
Datasets: 401(k) dataset from SIPP 1991 (Abadie 2003)
Benchmarks: 401(k) LATE by income quartile synthetic fairness SCM (z,m)-TE
Metrics: estimated LATE vs ground truth estimated (z,m)-TE vs ground truth with 95% bootstrap CI
Baselines: classic LATE estimator conditional LATE (cond(M)-LATE) z-TV / (z,m)-TV conditional difference
Builds on: Ctf-ID ID algorithm LATE (Imbens and Angrist) Pearl probabilities of causation
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