{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T15:44:22Z","timestamp":1785426262523,"version":"3.56.0"},"reference-count":54,"publisher":"Oxford University Press (OUP)","issue":"4","license":[{"start":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T00:00:00Z","timestamp":1676851200000},"content-version":"vor","delay-in-days":1755,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1502437"],"award-info":[{"award-number":["DMS-1502437"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1208982"],"award-info":[{"award-number":["DMS-1208982"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1403708"],"award-info":[{"award-number":["DMS-1403708"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["SES-1260782"],"award-info":[{"award-number":["SES-1260782"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["R01 CA127334"],"award-info":[{"award-number":["R01 CA127334"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,9,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>A major challenge in instrumental variable (IV) analysis is to find instruments that are valid, or have no direct effect on the outcome and are ignorable. Typically one is unsure whether all of the putative IVs are in fact valid. We propose a general inference procedure in the presence of invalid IVs, called two-stage hard thresholding with voting. The procedure uses two hard thresholding steps to select strong instruments and to generate candidate sets of valid IVs. Voting takes the candidate sets and uses majority and plurality rules to determine the true set of valid IVs. In low dimensions with invalid instruments, our proposal correctly selects valid IVs, consistently estimates the causal effect, produces valid confidence intervals for the causal effect and has oracle optimal width, even if the so-called 50% rule or the majority rule is violated. In high dimensions, we establish nearly identical results without oracle optimality. In simulations, our proposal outperforms traditional and recent methods in the invalid IV literature. We also apply our method to reanalyse the causal effect of education on earnings.<\/jats:p>","DOI":"10.1111\/rssb.12275","type":"journal-article","created":{"date-parts":[[2018,5,3]],"date-time":"2018-05-03T08:06:39Z","timestamp":1525334799000},"page":"793-815","source":"Crossref","is-referenced-by-count":79,"title":["Confidence Intervals for Causal Effects with Invalid Instruments by Using Two-Stage Hard Thresholding with Voting"],"prefix":"10.1093","volume":"80","author":[{"given":"Zijian","family":"Guo","sequence":"first","affiliation":[{"name":"Rutgers University , Piscataway , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hyunseung","family":"Kang","sequence":"additional","affiliation":[{"name":"University of Wisconsin\u2014Madison , Madison , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"T.","family":"Tony Cai","sequence":"additional","affiliation":[{"name":"University of Pennsylvania , Philadelphia , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dylan S.","family":"Small","sequence":"additional","affiliation":[{"name":"University of Pennsylvania , Philadelphia , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2018,5,2]]},"reference":[{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.jeconom.2006.06.007","article-title":"Performance of conditional Wald tests in IV regression with weak instruments","volume":"139","author":"Andrews","year":"2007","journal-title":"J. Econmetr."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1080\/01621459.1996.10476902","article-title":"Identification of causal effects using instrumental variables","volume":"91","author":"Angrist","year":"1996","journal-title":"J. Am. Statist. Ass."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"979","DOI":"10.2307\/2937954","article-title":"Does compulsory school attendance affect schooling and earnings?","volume":"106","author":"Angrist","year":"1991","journal-title":"Q. J. Econ."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"2297","DOI":"10.1002\/sim.6128","article-title":"Instrumental variable methods for causal inference","volume":"33","author":"Baiocchi","year":"2014","journal-title":"Statist. Med."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"2369","DOI":"10.3982\/ECTA9626","article-title":"Sparse models and methods for optimal instruments with an application to eminent domain","volume":"80","author":"Belloni","year":"2012","journal-title":"Econometrica"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1093\/restud\/rdt044","article-title":"Inference on treatment effects after selection among high-dimensional controls","volume":"81","author":"Belloni","year":"2014","journal-title":"Rev. Econ. Stud."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1093\/biomet\/asr043","article-title":"Square-root lasso: pivotal recovery of sparse signals via conic programming","volume":"98","author":"Belloni","year":"2011","journal-title":"Biometrika"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1705","DOI":"10.1214\/08-AOS620","article-title":"Simultaneous analysis of lasso and Dantzig selector","volume":"37","author":"Bickel","year":"2009","journal-title":"Ann. Statist."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1111\/j.1467-985X.2004.00360.x","article-title":"Evaluating the effect of education on earnings: models, methods and results from the National Child Development Survey","volume":"168","author":"Blundell","year":"2005","journal-title":"J. R. Statist. Soc."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1093\/ije\/dyv080","article-title":"Mendelian randomization with invalid instruments: effect estimation and bias detection through egger regression","volume":"44","author":"Bowden","year":"2015","journal-title":"Int. J. Epidem."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1002\/gepi.21965","article-title":"Consistent estimation in Mendelian randomization with some invalid instruments using a weighted median estimator","volume":"40","author":"Bowden","year":"2016","journal-title":"Genet. Epidem."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-20192-9","volume-title":"Statistics for High-dimensional Data: Methods, Theory and Applications","author":"B\u00fchlmann","year":"2011"},{"key":"2023022101104358700_","article-title":"Robust instrumental variable methods using multiple candidate instruments with application to Mendelian randomization","volume-title":"arXiv Preprint","author":"Burgess","year":"2016"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1093\/ije\/dyv108","article-title":"Mendelian randomization: where are we now and where are we going?","volume":"44","author":"Burgess","year":"2015","journal-title":"Int. J. Epidem."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1080\/02331888.2016.1265969","article-title":"Confidence intervals for high-dimensional linear regression: minimax rates and adaptivity","volume":"45","author":"Cai","year":"2017","journal-title":"Ann. Statist."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","DOI":"10.3386\/w4483","article-title":"Using geographic variation in college proximity to estimate the return to schooling","volume-title":"Working Paper 4483","author":"Card","year":"1993"},{"key":"2023022101104358700_","first-page":"1801","article-title":"The causal effect of education on earnings","volume-title":"Handbook of Labor Economics","author":"Card","year":"1999"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/j.jeconom.2015.02.019","article-title":"Select the valid and relevant moments: an information-based lasso for GMM with many moments","volume":"186","author":"Cheng","year":"2015","journal-title":"J. Econmetr."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1257\/aer.p20151022","article-title":"Post-selection and post-regularization inference in linear models with many controls and instruments","volume":"105","author":"Chernozhukov","year":"2015","journal-title":"Am. Econ. Rev."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1093\/ije\/dyg070","article-title":"Mendelian randomization: can genetic epidemiology contribute to understanding environmental determinants of disease?","volume":"32","author":"Davey Smith","year":"2003","journal-title":"Int. J. Epidem."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1093\/ije\/dyh132","article-title":"Mendelian randomization: prospects, potentials, and limitations","volume":"33","author":"Davey Smith","year":"2004","journal-title":"Int. J. Epidem."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1109\/18.382009","article-title":"De-noising by soft-thresholding","volume":"41","author":"Donoho","year":"1995","journal-title":"IEEE Trans. Inform. Theory"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1093\/biomet\/81.3.425","article-title":"Ideal spatial adaptation by wavelet shrinkage","volume":"81","author":"Donoho","year":"1994","journal-title":"Biometrika"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"872","DOI":"10.1214\/13-AOS1202","article-title":"Endogeneity in high dimensions","volume":"42","author":"Fan","year":"2014","journal-title":"Ann. Statist."},{"key":"2023022101104358700_","article-title":"Birth order and sibship sex composition as instruments in the study of education and earnings","volume-title":"Discussion Paper 5514","author":"Gary-Bobo","year":"2006"},{"key":"2023022101104358700_","article-title":"High-dimensional instrumental variables regression and confidence sets","volume-title":"Preprint arXiv:1105.2454","author":"Gautier","year":"2011"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1214\/14-AOS1221","article-title":"On asymptotically optimal confidence regions and tests for high-dimensional models","volume":"42","author":"van de Geer","year":"2014","journal-title":"Ann. Statist."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/j.econlet.2008.09.004","article-title":"Detecting invalid instruments using l 1-gmm","volume":"101","author":"Han","year":"2008","journal-title":"Econ. Lett."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1093\/ije\/dyx102","article-title":"Robust inference in summary data Mendelian randomization via the zero modal pleiotropy assumption","volume":"46","author":"Hartwig","year":"2017","journal-title":"Int. J. Epidem."},{"key":"2023022101104358700_","volume-title":"The Elements of Statistical Learning: Data Mining, Inference, and Prediction","author":"Hastie","year":"2016","edition":"3rd edn"},{"key":"2023022101104358700_","first-page":"3","article-title":"Survey response in the long run: the Wisconsin longitudinal study","volume":"17","author":"Hauser","year":"2005","journal-title":"Fld Meth."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1097\/01.ede.0000222409.00878.37","article-title":"Instruments for causal inference: an epidemiologist's dream?","volume":"17","author":"Hern\u00e1n","year":"2006","journal-title":"Epidemiology"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"449","DOI":"10.2307\/271055","article-title":"Causal inference, path analysis, and recursive structural equations models","volume":"18","author":"Holland","year":"1988","journal-title":"Sociol. Methodol."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1214\/14-STS480","article-title":"Instrumental variables: an econometrician's perspective","volume":"29","author":"Imbens","year":"2014","journal-title":"Statist. Sci."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"467","DOI":"10.2307\/2951620","article-title":"Identification and estimation of local average treatment effects","volume":"62","author":"Imbens","year":"1994","journal-title":"Econometrica"},{"key":"2023022101104358700_","first-page":"2869","article-title":"Confidence intervals and hypothesis testing for high-dimensional regression","volume":"15","author":"Javanmard","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"2023022101104358700_","article-title":"A simple and robust confidence interval for causal effects with possibly invalid instruments","volume-title":"Preprint arXiv:1504.03718","author":"Kang","year":"2016"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1080\/01621459.2014.994705","article-title":"Instrumental variables estimation with some invalid instruments and its application to Mendelian randomization","volume":"111","author":"Kang","year":"2016","journal-title":"J. Am. Statist. Ass."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1080\/07350015.2014.978175","article-title":"Identification and inference with many invalid instruments","volume":"33","author":"Koles\u00e1r","year":"2015","journal-title":"J. Bus. Econ. Statist."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1002\/sim.3034","article-title":"Mendelian randomization: using genes as instruments for making causal inferences in epidemiology","volume":"27","author":"Lawlor","year":"2008","journal-title":"Statist. Med."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0266466605050036","article-title":"Model selection and inference: facts and fiction","volume":"21","author":"Leeb","year":"2005","journal-title":"Econmetr. Theory"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1027","DOI":"10.1111\/1468-0262.00438","article-title":"A conditional likelihood ratio test for structural models","volume":"71","author":"Moreira","year":"2003","journal-title":"Econometrica"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1257\/jep.20.4.111","article-title":"Avoiding invalid instruments and coping with weak instruments","volume":"20","author":"Murray","year":"2006","journal-title":"J. Econ. Perspect."},{"key":"2023022101104358700_","first-page":"465","article-title":"On the application of probability theory to agricultural experiments: Essay on principles, section 9","volume":"5","author":"Neyman","year":"1923","journal-title":"Statist. Sci."},{"key":"2023022101104358700_","first-page":"e000944","article-title":"Multigene interactions and the prediction of depression in the Wisconsin Longitudinal Study","volume":"2","author":"Roetker","year":"2012","journal-title":"Br. Med. J. Open"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1037\/h0037350","article-title":"Estimating causal effects of treatments in randomized and nonrandomized studies","volume":"66","author":"Rubin","year":"1974","journal-title":"J. Educ. Psychol."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1198\/016214507000000608","article-title":"Sensitivity analysis for instrumental variables regression with overidentifying restrictions","volume":"102","author":"Small","year":"2007","journal-title":"J. Am. Statist. Ass."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1198\/073500102288618658","article-title":"A survey of weak instruments and weak identification in generalized method of moments","volume":"20","author":"Stock","year":"2002","journal-title":"J. Bus. Econ. Statist."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"370","DOI":"10.1097\/EDE.0b013e31828d0590","article-title":"Commentary: How to report instrumental variables analyses (suggestions welcome)","volume":"24","author":"Swanson","year":"2013","journal-title":"Epidemiology"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression shrinkage and selection via the lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Statist. Soc."},{"key":"2023022101104358700_","article-title":"On the use of the lasso for instrumental variables estimation with some invalid instruments","volume-title":"Technical Report","author":"Windmeijer","year":"2016"},{"key":"2023022101104358700_","volume-title":"Econometric Analysis of Cross Section and Panel Data","author":"Wooldridge","year":"2010","edition":"2nd edn"},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"1734","DOI":"10.1093\/ije\/dyx034","article-title":"Mendelianrandomization: an R package for performing Mendelian randomization analyses using summarized data","volume":"46","author":"Yavorska","year":"2017","journal-title":"Int. J. Epidem."},{"key":"2023022101104358700_","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1111\/rssb.12026","article-title":"Confidence intervals for low dimensional parameters in high dimensional linear models","volume":"76","author":"Zhang","year":"2014","journal-title":"J. R. Statist. Soc."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Frssb.12275","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1111\/rssb.12275","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1111\/rssb.12275","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/am-pdf\/10.1111\/rssb.12275","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/80\/4\/793\/49270637\/jrsssb_80_4_793.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/80\/4\/793\/49270637\/jrsssb_80_4_793.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T02:36:11Z","timestamp":1751596571000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/80\/4\/793\/7048331"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,2]]},"references-count":54,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,5,2]]},"published-print":{"date-parts":[[2018,9,1]]}},"URL":"https:\/\/doi.org\/10.1111\/rssb.12275","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2018,9]]},"published":{"date-parts":[[2018,5,2]]}}}