{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T22:16:22Z","timestamp":1791929782926,"version":"4.3.4"},"reference-count":28,"publisher":"MIT Press - Journals","issue":"2","content-domain":{"domain":["direct.mit.edu"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2021,5,14]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper studies the wild bootstrap\u2013based test proposed in Cameron, Gelbach, and Miller (2008). Existing analyses of its properties require that number of clusters is \u201clarge.\u201d In an asymptotic framework in which the number of clusters is \u201csmall,\u201d we provide conditions under which an unstudentized version of the test is valid. These conditions include homogeneity-like restrictions on the distribution of covariates. We further establish that a studentized version of the test may only overreject the null hypothesis by a \u201csmall\u201d amount that decreases exponentially with the number of clusters. We obtain a qualitatively similar result for \u201cscore\u201d bootstrap-based tests, which permit testing in nonlinear models.<\/jats:p>","DOI":"10.1162\/rest_a_00887","type":"journal-article","created":{"date-parts":[[2019,12,16]],"date-time":"2019-12-16T22:35:19Z","timestamp":1576535719000},"page":"346-363","update-policy":"https:\/\/doi.org\/10.1162\/mitpressjournals.corrections.policy","source":"Crossref","is-referenced-by-count":96,"title":["The Wild Bootstrap with a \u201cSmall\u201d Number of \u201cLarge\u201d Clusters"],"prefix":"10.1162","volume":"103","author":[{"given":"Ivan A.","family":"Canay","sequence":"first","affiliation":[{"name":"Northwestern University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andres","family":"Santos","sequence":"additional","affiliation":[{"name":"UCLA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Azeem M.","family":"Shaikh","sequence":"additional","affiliation":[{"name":"University of Chicago"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"281","published-online":{"date-parts":[[2021,5,10]]},"reference":[{"key":"2021051022215786500_B1","doi-asserted-by":"crossref","first-page":"3286","DOI":"10.1257\/aer.101.7.3286","article-title":"The Consequences of Radical Reform: The French Revolution","volume":"101","author":"Acemoglu","year":"2011","journal-title":"American Economic Review"},{"key":"2021051022215786500_B2","author":"Amemiya","year":"1985","journal-title":"Advanced Econometrics"},{"key":"2021051022215786500_B3","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1162\/003355304772839588","article-title":"How Much Should We Trust Differences-in-Differences Estimates?","volume":"119","author":"Bertrand","year":"2004","journal-title":"Quarterly Journal of Economics"},{"key":"2021051022215786500_B4","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.jeconom.2011.01.007","article-title":"Inference with Dependent Data Using Cluster Covariance Estimators","volume":"165","author":"Bester","year":"2011","journal-title":"Journal of Econometrics"},{"key":"2021051022215786500_B5","first-page":"414","author":"Cameron","year":"2008","journal-title":"Bootstrap-Based Improvements for Inference with Clustered Errors"},{"key":"2021051022215786500_B6","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.3982\/ECTA13081","article-title":"Randomization Tests under an Approximate Symmetry Assumption","volume":"85","author":"Canay","year":"2017","journal-title":"Econometrica"},{"key":"2021051022215786500_B7","first-page":"698","article-title":"Asymptotic Behavior of a t Test Robust to Cluster Heterogeneity","author":"Carter","year":"2017"},{"key":"2021051022215786500_B8","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1016\/j.jeconom.2008.08.003","article-title":"The Wild Bootstrap, Tamed at Last","volume":"146","author":"Davidson","year":"2008","journal-title":"Journal of Econometrics"},{"key":"2021051022215786500_B9","first-page":"361","article-title":"The Size Distortion of Bootstrap Tests","author":"Davidson","year":"1999","journal-title":"Econometric Theory"},{"key":"2021051022215786500_B10","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.jeconom.2019.04.035","article-title":"Asymptotic Theory and Wild Bootstrap Inference with Clustered Errors","volume":"212","author":"Djogbenou","year":"2019","journal-title":"Journal of Econometrics"},{"key":"2021051022215786500_B11","first-page":"221","article-title":"Inference with Difference-in-Differences and Other Panel Data","author":"Donald","year":"2007"},{"key":"2021051022215786500_B12","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1093\/restud\/rdt040","article-title":"Growing Up in a Recession","volume":"81","author":"Giuliano","year":"2014","journal-title":"Review of Economic Studies"},{"key":"2021051022215786500_B13","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.2307\/1912775","article-title":"Large Sample Properties of Generalized Method of Moments Estimators","volume":"50","author":"Hansen","year":"1982","journal-title":"Econometrica"},{"key":"2021051022215786500_B14","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1198\/jbes.2009.08046","article-title":"t-Statistic Based Correlation and Heterogeneity Robust Inference","volume":"28","author":"Ibragimov","year":"2010","journal-title":"Journal of Business and Economic Statistics"},{"key":"2021051022215786500_B15","first-page":"83","article-title":"Inference with Few Heterogeneous Clusters","author":"Ibragimov","year":"2016"},{"key":"2021051022215786500_B16","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1515\/2156-6674.1006","article-title":"A Score Based Approach to Wild Bootstrap Inference","volume":"1","author":"Kline","year":"2012","journal-title":"Journal of Econometric Methods"},{"key":"2021051022215786500_B17","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1257\/aer.20120700","article-title":"Leader Punishment and Cooperation in Groups: Experimental Field Evidence from Commons Management in Ethiopia","volume":"105","author":"Kosfeld","year":"2015","journal-title":"American Economic Review"},{"key":"2021051022215786500_B18","author":"Lehmann","year":"2005","journal-title":"Testing Statistical Hypotheses"},{"key":"2021051022215786500_B19","doi-asserted-by":"crossref","first-page":"1696","DOI":"10.1214\/aos\/1176351062","article-title":"Bootstrap Procedures under Some Non-IID models","volume":"16","author":"Liu","year":"1988","journal-title":"Annals of Statistics"},{"key":"2021051022215786500_B20","article-title":"Bootstrap and Asymptotic Inference with Multiway Clustering","volume":"1415","author":"MacKinnon","year":"2019","journal-title":"Queen's University Economics Department working paper"},{"key":"2021051022215786500_B21","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1002\/jae.2508","article-title":"Wild Bootstrap Inference for Wildly Different Cluster Sizes","volume":"32","author":"MacKinnon","year":"2017","journal-title":"Journal of Applied Econometrics"},{"key":"2021051022215786500_B22","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1214\/aos\/1176349025","article-title":"Bootstrap and Wild Bootstrap for High Dimensional Linear Models","volume":"21","author":"Mammen","year":"1993","journal-title":"Annals of Statistics"},{"key":"2021051022215786500_B23","doi-asserted-by":"crossref","first-page":"470","DOI":"10.1137\/0905034","article-title":"Generating Correlation Matrices","volume":"5","author":"Marsaglia","year":"1984","journal-title":"SIAM Journal on Scientific and Statistical Computing"},{"key":"2021051022215786500_B24","doi-asserted-by":"crossref","first-page":"1568","DOI":"10.1093\/restud\/rdv016","article-title":"The Institutional Causes of China's Great Famine, 1959\u20131961","volume":"82","author":"Meng","year":"2015","journal-title":"Review of Economic Studies"},{"key":"2021051022215786500_B25","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/0304-4076(86)90021-7","article-title":"Random Group Effects and the Precision of Regression Estimates","volume":"32","author":"Moulton","year":"1986","journal-title":"Journal of Econometrics"},{"key":"2021051022215786500_B26","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2545-2","author":"van der Vaart","year":"1996","journal-title":"Weak Convergence and Empirical Processes"},{"key":"2021051022215786500_B27","author":"Webb","year":"2013"},{"key":"2021051022215786500_B28","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1257\/000282803321946930","article-title":"Cluster-Sample Methods in Applied Econometrics","volume":"93","author":"Wooldridge","year":"2003","journal-title":"American Economic Review"}],"container-title":["The Review of Economics and Statistics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/direct.mit.edu\/rest\/article-pdf\/103\/2\/346\/1915882\/rest_a_00887.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/direct.mit.edu\/rest\/article-pdf\/103\/2\/346\/1915882\/rest_a_00887.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T23:52:33Z","timestamp":1620690753000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/rest\/article\/103\/2\/346\/97650\/The-Wild-Bootstrap-with-a-Small-Number-of-Large"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5]]},"references-count":28,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,5,10]]},"published-print":{"date-parts":[[2021,5,14]]}},"URL":"https:\/\/doi.org\/10.1162\/rest_a_00887","relation":{},"ISSN":["0034-6535","1530-9142"],"issn-type":[{"value":"0034-6535","type":"print"},{"value":"1530-9142","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2021,5]]},"published":{"date-parts":[[2021,5]]}}}