{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T11:19:57Z","timestamp":1788520797841,"version":"build-2803163510"},"reference-count":19,"publisher":"Oxford University Press (OUP)","license":[{"start":{"date-parts":[[2026,8,11]],"date-time":"2026-08-11T00:00:00Z","timestamp":1786406400000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["72503208"],"award-info":[{"award-number":["72503208"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In cluster randomized controlled trials (CRCTs) with finite populations, the exact design-based variance of the Horvitz\u2013Thompson (HT) estimator of the average treatment effect (ATE) depends on the joint distribution of unobserved, cluster aggregates of potential outcomes and is therefore not point-identified. We study a two-stage sampling design in which clusters are sampled at random, units are then sampled within selected clusters, and treatment is assigned at the cluster level. We first derive the exact design-based variance of the HT estimator of the ATE, accounting for cluster sampling, within-cluster sampling, and treatment assignment. We then extend Aronow et al. (Annals of Statistics 42, 850\u201371 (2014)) to derive sharp bounds for an estimated variance based on estimated cluster totals and propose a feasible consistent estimator of the upper bound. In simulations and an empirical application, confidence intervals based on the proposed upper-bound estimator are typically narrower than those based on conventional cluster-robust standard errors.<\/jats:p>","DOI":"10.1093\/ectj\/utag026","type":"journal-article","created":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T11:22:35Z","timestamp":1786360955000},"source":"Crossref","is-referenced-by-count":0,"title":["The exact variance of the average treatment effect estimator in cluster randomized controlled trials"],"prefix":"10.1093","author":[{"given":"Yue","family":"Fang","sequence":"first","affiliation":[{"name":"School of Management and Economics, The Chinese University of Hong Kong , Shenzhen, 2001 Longxiangdadao TxD 913, Shenzhen, Guangdong, 518172 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Geert","family":"Ridder","sequence":"additional","affiliation":[{"name":"Department of Economics, University of Southern California, Kaprielian Hall, University Park Campus , Los Angeles, CA-90089 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,8,11]]},"reference":[{"key":"2026090407061888300_bib1","doi-asserted-by":"crossref","first-page":"265","DOI":"10.3982\/ECTA12675","article-title":"Sampling-based versus design-based uncertainty in regression analysis","volume":"88","author":"Abadie","year":"2020","journal-title":"Econometrica"},{"key":"2026090407061888300_bib2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/qje\/qjac038","article-title":"When should you adjust standard errors for clustering?","volume":"138","author":"Abadie","year":"2023","journal-title":"Quarterly Journal of Economics"},{"key":"2026090407061888300_bib3","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1214\/13-AOS1200","article-title":"Sharp bounds on the variance in randomized 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