{"paper":{"title":"Stacked Triple Differences","license":"http://creativecommons.org/licenses/by/4.0/","headline":"A linear regression on stacked four-cell triple-difference data identifies a cell-size-weighted average of treatment effects.","cross_cats":[],"primary_cat":"econ.EM","authors_text":"Meng Hsuan Hsieh","submitted_at":"2026-04-24T19:51:17Z","abstract_excerpt":"Triple differences (DDD) is a workhorse quasi-experimental design in applied economics. But, under staggered adoption, its conventional three-way fixed-effects (3WFE) implementation inherits the interpretation issues now well understood in the difference-in-differences literature. I introduce stacked DDD. I extend the stacked difference-in-differences approach to the DDD setting by creating self-contained stacks, each consisting of four cells over an event window: treated and clean comparison cohorts, each with treatment-eligible and treatment-ineligible units. Appending these stacks yields a "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"I prove that, at each post-treatment event-time, a linear regression with fully saturated fixed-effects applied to the stacked dataset identifies a strictly positive, cell-size-weighted average of stack-level conditional average treatment effects, with stack weights proportional to stack-level cell sizes.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that clean comparison cohorts satisfy pairwise parallel trends with treated cohorts within each stack, and that treatment-ineligible units provide valid counterfactuals without anticipation effects or spillovers.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Stacked DDD appends self-contained stacks of treated and clean comparison cohorts to identify a cell-size-weighted average of stack-level conditional average treatment effects via saturated fixed-effects regression.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A linear regression on stacked four-cell triple-difference data identifies a cell-size-weighted average of treatment effects.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"eda5cc99ddef192acbfc14ed7ef4fe28c01c32416d66db9ac4025c8ff6583d84"},"source":{"id":"2604.22982","kind":"arxiv","version":2},"verdict":{"id":"dc2a4400-59d3-4540-888f-da3c2e678834","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T08:53:35.417505Z","strongest_claim":"I prove that, at each post-treatment event-time, a linear regression with fully saturated fixed-effects applied to the stacked dataset identifies a strictly positive, cell-size-weighted average of stack-level conditional average treatment effects, with stack weights proportional to stack-level cell sizes.","one_line_summary":"Stacked DDD appends self-contained stacks of treated and clean comparison cohorts to identify a cell-size-weighted average of stack-level conditional average treatment effects via saturated fixed-effects regression.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that clean comparison cohorts satisfy pairwise parallel trends with treated cohorts within each stack, and that treatment-ineligible units provide valid counterfactuals without anticipation effects or spillovers.","pith_extraction_headline":"A linear regression on stacked four-cell triple-difference data identifies a cell-size-weighted average of treatment effects."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.22982/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T23:35:10.145192Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"5945e0b001cc7b9d8a76293d7a40d1e3f25e600cbf00cb30e6ab84f0a618dc27"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}