The Triple Difference Estimator
Causal Question / Estimand
Methodological: what does the widely used Triple-Difference (DDD) estimator actually identify, and under what assumption? The paper derives the identifying conditions for the DDD estimand rather than estimating a substantive effect.
Identification Strategy
Olden & Møen give a complete formal treatment of DDD, which is computed as the difference between two difference-in-differences estimators. Their central result: DDD does not require both underlying DiDs to satisfy Parallel-Trends. Since the triple difference nets the two DiDs against each other, any bias that is common to both is differenced out. Identification therefore needs only one parallel-trends assumption — stated on the differential between the two subgroups (a parallel trend “in ratios”/differences): the treatment–control trend gap must have evolved identically for the affected and unaffected subgroups absent treatment. They also formalize the regression implementation, the estimand under two outcome variables, and inference (standard-error) issues.
Key Assumptions
- Triple-Difference — the object of study; DDD as the difference of two DiDs.
- Parallel-Trends — a single DDD parallel-trends condition on the between-subgroup differential suffices; two separate DiD parallel-trends assumptions are not required.
Threats to Validity
n/a — theoretical/methodological. The paper’s contribution is precisely to pin down when DDD is unbiased: it fails if the DiD bias differs across the two subgroups (so the biases do not cancel), i.e. if the single differential-parallel-trend assumption is violated.
Setting / Data
n/a — theoretical, with illustrative reference to applied triple-difference papers in the AER/JPE/QJE (32 articles, 2010–2017) whose identifying assumptions were left informal.
Key Claims
- DDD requires only one parallel-trends assumption, not two — the difference of two biased DiDs is unbiased when the bias is common to both.
- Much applied use of triple difference rests on intuition; the identifying assumption had been neither formally derived nor consistently stated.
- Recommends unifying terminology on “triple difference” / “difference-in-difference-in- differences,” crediting Gruber1994-IncidenceMandatedMaternity as the originator.
Connections
- Formalizes the Triple-Difference design introduced by Gruber1994-IncidenceMandatedMaternity.
- Refines the Parallel-Trends discussion at the center of the modern DiD literature (RothSantAnna2023-WhatsTrendingInDiD, RambachanRoth2023-MoreCredibleParallelTrends).
- See also DiD.
Citation
Olden, A., & Møen, J. (2022). The Triple Difference Estimator. Econometrics Journal, 25(3), 531–553.