{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T20:20:41Z","timestamp":1790713241843,"version":"4.1.0"},"reference-count":42,"publisher":"The Econometric Society","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ECTA"],"abstract":"<jats:p>We consider estimation and inference on average treatment effects under unconfoundedness conditional on the realizations of the treatment variable and covariates. Given nonparametric smoothness and\/or shape restrictions on the conditional mean of the outcome variable, we derive estimators and confidence intervals (CIs) that are optimal in finite samples when the regression errors are normal with known variance. In contrast to conventional CIs, our CIs use a larger critical value that explicitly takes into account the potential bias of the estimator. When the error distribution is unknown, feasible versions of our CIs are valid asymptotically, even when \n            <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n              <a:msqrt>\n                <a:mi>n<\/a:mi>\n              <\/a:msqrt>\n            <\/a:math>\u2010inference is not possible due to lack of overlap, or low smoothness of the conditional mean. We also derive the minimum smoothness conditions on the conditional mean that are necessary for \n            <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n              <c:msqrt>\n                <c:mi>n<\/c:mi>\n              <\/c:msqrt>\n            <\/c:math>\u2010inference. When the conditional mean is restricted to be Lipschitz with a large enough bound on the Lipschitz constant, the optimal estimator reduces to a matching estimator with the number of matches set to one. We illustrate our methods in an application to the National Supported Work Demonstration.\n          <\/jats:p>","DOI":"10.3982\/ecta16907","type":"journal-article","created":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T12:20:13Z","timestamp":1620044413000},"page":"1141-1177","source":"Crossref","is-referenced-by-count":23,"title":["Finite\u2010Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness"],"prefix":"10.3982","volume":"89","author":[{"given":"Timothy B.","family":"Armstrong","sequence":"first","affiliation":[{"name":"Department of Economics, Yale University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michal","family":"Koles\u00e1r","sequence":"additional","affiliation":[{"name":"Department of Economics, Princeton University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2359","published-online":{"date-parts":[[2021]]},"reference":[{"issue":"1","key":"ecta200300-bib-0001","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1111\/j.1468-0262.2006.00655.x","volume":"74","author":"Abadie","year":"2006","journal-title":"Econometrica"},{"issue":"1","key":"ecta200300-bib-0002","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1198\/jbes.2009.07333","volume":"29","author":"Abadie","year":"2011","journal-title":"Journal of Business & Economic Statistics"},{"issue":"498","key":"ecta200300-bib-0003","doi-asserted-by":"crossref","first-page":"833","DOI":"10.1080\/01621459.2012.682537","volume":"107","author":"Abadie","year":"2012","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"ecta200300-bib-0004","doi-asserted-by":"crossref","first-page":"265","DOI":"10.3982\/ECTA12675","volume":"88","author":"Abadie","year":"2020","journal-title":"Econometica"},{"issue":"508","key":"ecta200300-bib-0005","doi-asserted-by":"crossref","first-page":"1601","DOI":"10.1080\/01621459.2014.928218","volume":"109","author":"Abadie","year":"2014","journal-title":"Journal of the American Statistical Association"},{"key":"ecta200300-bib-0006","doi-asserted-by":"crossref","unstructured":"Armstrong, T. B., and M. Koles\u00e1r (2018a): \u201cFinite-Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness,\u201d arXiv:1712.04594v2.","DOI":"10.2139\/ssrn.3309772"},{"issue":"2","key":"ecta200300-bib-0007","doi-asserted-by":"crossref","first-page":"655","DOI":"10.3982\/ECTA14434","volume":"86","author":"Armstrong","year":"2018","journal-title":"Econometrica"},{"issue":"1","key":"ecta200300-bib-0008","doi-asserted-by":"crossref","first-page":"77","DOI":"10.3982\/QE1609","volume":"12","author":"Armstrong","year":"2021","journal-title":"Quantitative Economics"},{"key":"ecta200300-bib-0009","doi-asserted-by":"crossref","unstructured":"Armstrong, T. B., M. Koles\u00e1r (2021): \u201cSupplement to \u2018Finite-Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness\u2019,\u201d Econometrica Supplemental Material, 89, https:\/\/doi.org\/10.3982\/ECTA16907.","DOI":"10.3982\/ECTA16907"},{"issue":"3","key":"ecta200300-bib-0010","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.1257\/aer.20120070","volume":"105","author":"Bailey","year":"2015","journal-title":"American Economic Review"},{"issue":"1","key":"ecta200300-bib-0011","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.cam.2005.08.011","volume":"196","author":"Beliakov","year":"2006","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"ecta200300-bib-0012","volume-title":"Probability and Measure","author":"Billingsley","year":"1995","edition":"Third"},{"issue":"5","key":"ecta200300-bib-0013","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1162\/REST_a_00431","volume":"96","author":"Busso","year":"2014","journal-title":"The Review of Economics and Statistics"},{"issue":"5","key":"ecta200300-bib-0014","doi-asserted-by":"crossref","first-page":"1805","DOI":"10.1214\/009053604000000049","volume":"32","author":"Cai","year":"2004","journal-title":"The Annals of Statistics"},{"issue":"2","key":"ecta200300-bib-0015","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1214\/009053607000000947","volume":"36","author":"Chen","year":"2008","journal-title":"The Annals of Statistics"},{"issue":"1","key":"ecta200300-bib-0016","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1093\/biomet\/asn055","volume":"96","author":"Crump","year":"2009","journal-title":"Biometrika"},{"issue":"448","key":"ecta200300-bib-0017","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1080\/01621459.1999.10473858","volume":"94","author":"Dehejia","year":"1999","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"ecta200300-bib-0018","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1214\/aos\/1176325367","volume":"22","author":"Donoho","year":"1994","journal-title":"The Annals of Statistics"},{"issue":"3","key":"ecta200300-bib-0019","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1214\/aos\/1024691081","volume":"26","author":"Donoho","year":"1998","journal-title":"The Annals of Statistics"},{"issue":"3","key":"ecta200300-bib-0020","doi-asserted-by":"crossref","first-page":"1416","DOI":"10.1214\/aos\/1176347758","volume":"18","author":"Donoho","year":"1990","journal-title":"The Annals of Statistics"},{"issue":"2","key":"ecta200300-bib-0021","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1214\/009053604000000067","volume":"32","author":"Efron","year":"2004","journal-title":"The Annals of Statistics"},{"issue":"1","key":"ecta200300-bib-0022","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1086\/426041","volume":"113","author":"Galiani","year":"2005","journal-title":"Journal of Political Economy"},{"issue":"2","key":"ecta200300-bib-0023","doi-asserted-by":"crossref","first-page":"315","DOI":"10.2307\/2998560","volume":"66","author":"Hahn","year":"1998","journal-title":"Econometrica"},{"issue":"2","key":"ecta200300-bib-0024","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1111\/1467-937X.00044","volume":"65","author":"Heckman","year":"1998","journal-title":"The Review of Economic Studies"},{"issue":"4","key":"ecta200300-bib-0025","doi-asserted-by":"crossref","first-page":"605","DOI":"10.2307\/2971733","volume":"64","author":"Heckman","year":"1997","journal-title":"Review of Economic Studies"},{"issue":"4","key":"ecta200300-bib-0026","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.1111\/1468-0262.00442","volume":"71","author":"Hirano","year":"2003","journal-title":"Econometrica"},{"issue":"62","key":"ecta200300-bib-0027","first-page":"1","volume":"21","author":"Kallus","year":"2020","journal-title":"Journal of Machine Learning Research"},{"key":"ecta200300-bib-0028","unstructured":"Khan, S., and D. 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