{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T03:48:55Z","timestamp":1785556135684,"version":"3.56.0"},"reference-count":37,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T00:00:00Z","timestamp":1698969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,9,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We consider the efficient estimation of total causal effects in the presence of unmeasured confounding using conditional instrumental sets. Specifically, we consider the two-stage least-squares estimator in the setting of a linear structural equation model with correlated errors that is compatible with a known acyclic directed mixed graph. To set the stage for our results, we characterize the class of linearly valid conditional instrumental sets that yield consistent two-stage least-squares estimators for the target total effect and derive a new asymptotic variance formula for these estimators. Equipped with these results, we provide three graphical tools for selecting more efficient linearly valid conditional instrumental sets: first, a graphical criterion that, for certain pairs of linearly valid conditional instrumental sets, identifies which of the two corresponding estimators has the smaller asymptotic variance second, an algorithm that greedily adds covariates that reduce the asymptotic variance to a given linearly valid conditional instrumental set and, third, a linearly valid conditional instrumental set for which the corresponding estimator has the smallest asymptotic variance that can be ensured with a graphical criterion.<\/jats:p>","DOI":"10.1093\/biomet\/asad066","type":"journal-article","created":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T00:59:35Z","timestamp":1699405175000},"page":"771-788","source":"Crossref","is-referenced-by-count":6,"title":["Graphical tools for selecting conditional instrumental sets"],"prefix":"10.1093","volume":"111","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1000-3622","authenticated-orcid":false,"given":"L","family":"Henckel","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, University College Dublin , Belfield, Dublin 4, Ireland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M","family":"Buttenschoen","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Oxford , 24\u201329 St Giles\u2019, Oxford OX1 3LB, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M H","family":"Maathuis","sequence":"additional","affiliation":[{"name":"Seminar for Statistics, ETH Zurich , R\u00e4mistrasse 101 , 8092 Zurich, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2023,11,3]]},"reference":[{"key":"2024090418520420300_asad066-B1","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1257\/aer.91.5.1369","article-title":"The colonial origins of comparative development: an empirical investigation","volume":"91","author":"Acemoglu","year":"2001","journal-title":"Am. 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