{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T17:48:12Z","timestamp":1782755292479,"version":"3.54.5"},"reference-count":46,"publisher":"Oxford University Press (OUP)","license":[{"start":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T00:00:00Z","timestamp":1782691200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12371270"],"award-info":[{"award-number":["12371270"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100012543","name":"Shanghai Science and Technology Development Funds","doi-asserted-by":"publisher","award":["23JC1402100"],"award-info":[{"award-number":["23JC1402100"]}],"id":[{"id":"10.13039\/100012543","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shanghai Engineering Research Center of Finance Intelligence","award":["19DZ2254600"],"award-info":[{"award-number":["19DZ2254600"]}]},{"name":"MOE Project of Humanities and Social Sciences","award":["25YJCZH291"],"award-info":[{"award-number":["25YJCZH291"]}]},{"name":"Youth Innovation Fund of the University of Science and Technology of China","award":["WK2040250119"],"award-info":[{"award-number":["WK2040250119"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In this paper, we study online estimation and inference of regression coefficients in the presence of hidden confounders by leveraging over-parameterized models. Unlike existing offline approaches that rely on factor and sparse models, our closed-form estimator simultaneously removes hidden-confounder bias and is directly applicable to streaming data. Using tools from random matrix theory, we analyse phase transition phenomena in the variance of the coefficient estimator that arise as the sample size transitions from being smaller than to larger than the number of predictors. Notably, we show that adding more covariates only slightly affects the estimator\u2019s variance, mitigating concerns about variance inflation in over-parameterized settings. We validate the effectiveness of our method for both individual coefficient inference and multiple testing through simulations and applications to two real datasets.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkag110","type":"journal-article","created":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T12:25:41Z","timestamp":1781612741000},"source":"Crossref","is-referenced-by-count":0,"title":["Online inference under over-parameterized models with hidden confounders"],"prefix":"10.1093","author":[{"given":"Shuyan","family":"Chen","sequence":"first","affiliation":[{"name":"University of Science and Technology of China School of Management, , Hefei, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xingdong","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yeheng","family":"Ge","sequence":"additional","affiliation":[{"name":"The Hong Kong Polytechnic University, Hong Kong SAR, China Department of Data Science and Artificial Intelligence,"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mengyun","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2026,6,29]]},"reference":[{"issue":"1","key":"2026062913325201600_qkag110-B1","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1137\/23M1558781","article-title":"High-dimensional analysis of double descent for linear regression with random projections","volume":"6","author":"Bach","year":"2024","journal-title":"SIAM Journal on Mathematics of Data Science"},{"issue":"2","key":"2026062913325201600_qkag110-B2","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1162\/REST_a_00519","article-title":"Maximum likelihood estimation and inference for approximate factor models of high dimension","volume":"98","author":"Bai","year":"2016","journal-title":"Review of Economics and Statistics"},{"issue":"1","key":"2026062913325201600_qkag110-B3","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1111\/j.2517-6161.1995.tb02031.x","article-title":"Controlling the false discovery rate: A practical and powerful approach to multiple testing","volume":"57","author":"Benjamini","year":"1995","journal-title":"Journal of the Royal Statistical Society: Series B"},{"issue":"4","key":"2026062913325201600_qkag110-B4","doi-asserted-by":"publisher","first-page":"1212","DOI":"10.3150\/12-BEJSP11","article-title":"Statistical significance in high-dimensional linear models","volume":"19","author":"B\u00fchlmann","year":"2013","journal-title":"Bernoulli"},{"issue":"5","key":"2026062913325201600_qkag110-B5","doi-asserted-by":"publisher","first-page":"41","DOI":"10.2469\/faj.v56.n5.2389","article-title":"The increasing importance of industry factors","volume":"56","author":"Cavaglia","year":"2000","journal-title":"Financial Analysts Journal"},{"issue":"1","key":"2026062913325201600_qkag110-B6","doi-asserted-by":"publisher","first-page":"C1","DOI":"10.1111\/ectj.12097","article-title":"Double\/debiased machine learning for treatment and structural parameters","volume":"21","author":"Chernozhukov","year":"2018","journal-title":"The Econometrics Journal"},{"issue":"6","key":"2026062913325201600_qkag110-B7","doi-asserted-by":"publisher","first-page":"3320","DOI":"10.1214\/11-AOS944","article-title":"High dimensional covariance matrix estimation in approximate factor models","volume":"39","author":"Fan","year":"2011","journal-title":"Annals of Statistics"},{"issue":"4","key":"2026062913325201600_qkag110-B8","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1111\/rssb.12016","article-title":"Large covariance estimation by thresholding principal orthogonal complements","volume":"75","author":"Fan","year":"2013","journal-title":"Journal of the Royal Statistical Society: Series B"},{"issue":"4","key":"2026062913325201600_qkag110-B9","doi-asserted-by":"publisher","first-page":"1692","DOI":"10.1214\/23-AOS2304","article-title":"Bridging factor and sparse models","volume":"51","author":"Fan","year":"2023","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026062913325201600_qkag110-B10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v033.i01","article-title":"Regularization paths for generalized linear models via coordinate descent","volume":"33","author":"Friedman","year":"2010","journal-title":"Journal of Statistical Software"},{"issue":"4","key":"2026062913325201600_qkag110-B11","doi-asserted-by":"publisher","first-page":"387","DOI":"10.2307\/1907330","article-title":"Partial time regressions as compared with individual trends","volume":"1","author":"Frisch","year":"1933","journal-title":"Econometrica: Journal of the Econometric Society"},{"issue":"6","key":"2026062913325201600_qkag110-B12","doi-asserted-by":"publisher","first-page":"756","DOI":"10.1002\/ana.26339","article-title":"Neurological events reported after COVID-19 vaccines: An analysis of vaccine adverse event reporting system","volume":"91","author":"Frontera","year":"2022","journal-title":"Annals of Neurology"},{"issue":"3","key":"2026062913325201600_qkag110-B13","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1080\/09636412.2010.505865","article-title":"Protecting \u201cthe prize\u201d: Oil and the us national interest","volume":"19","author":"Gholz","year":"2010","journal-title":"Security Studies"},{"issue":"5","key":"2026062913325201600_qkag110-B14","doi-asserted-by":"publisher","first-page":"1079","DOI":"10.1111\/rssb.12152","article-title":"Tests for high dimensional generalized linear models","volume":"78","author":"Guo","year":"2016","journal-title":"Journal of the Royal Statistical Society: Series B"},{"issue":"3","key":"2026062913325201600_qkag110-B15","doi-asserted-by":"publisher","first-page":"1320","DOI":"10.1214\/21-aos2152","article-title":"Doubly debiased lasso: High-dimensional inference under hidden confounding","volume":"50","author":"Guo","year":"2022","journal-title":"Annals of Statistics"},{"issue":"4","key":"2026062913325201600_qkag110-B16","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1111\/rssb.12275","article-title":"Confidence intervals for causal effects with invalid instruments by using two\u2013stage hard thresholding with voting","volume":"80","author":"Guo","year":"2018","journal-title":"Journal of the Royal Statistical Society: Series B"},{"issue":"2","key":"2026062913325201600_qkag110-B17","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1137\/1031049","article-title":"Updating the inverse of a matrix","volume":"31","author":"Hager","year":"1989","journal-title":"SIAM Review"},{"issue":"337","key":"2026062913325201600_qkag110-B18","first-page":"1","article-title":"Inference on high-dimensional single-index models with streaming data","volume":"25","author":"Han","year":"2024","journal-title":"Journal of Machine Learning Research"},{"issue":"1","key":"2026062913325201600_qkag110-B19","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1093\/biomet\/asad046","article-title":"Online inference with debiased stochastic gradient descent","volume":"111","author":"Han","year":"2024","journal-title":"Biometrika"},{"issue":"1","key":"2026062913325201600_qkag110-B20","doi-asserted-by":"publisher","first-page":"1212","DOI":"10.1214\/19-EJS1554","article-title":"False discovery rate control via debiased lasso","volume":"13","author":"Javanmard","year":"2019","journal-title":"Electronic Journal of Statistics"},{"issue":"2","key":"2026062913325201600_qkag110-B21","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1214\/17-AOS1559","article-title":"Online rules for control of false discovery rate and false discovery exceedance","volume":"46","author":"Javanmard","year":"2018","journal-title":"Annals of Statistics"},{"issue":"513","key":"2026062913325201600_qkag110-B22","doi-asserted-by":"publisher","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":"Journal of the American Statistical Association"},{"issue":"10","key":"2026062913325201600_qkag110-B23","doi-asserted-by":"publisher","first-page":"733","DOI":"10.1038\/nrg2825","article-title":"Tackling the widespread and critical impact of batch effects in high-throughput data","volume":"11","author":"Leek","year":"2010","journal-title":"Nature Reviews: Genetics"},{"issue":"9","key":"2026062913325201600_qkag110-B24","doi-asserted-by":"publisher","first-page":"e161","DOI":"10.1371\/journal.pgen.0030161","article-title":"Capturing heterogeneity in gene expression studies by surrogate variable analysis","volume":"3","author":"Leek","year":"2007","journal-title":"PLoS Genetics"},{"issue":"2","key":"2026062913325201600_qkag110-B25","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1093\/biomet\/asz074","article-title":"Estimation of error variance via ridge regression","volume":"107","author":"Liu","year":"2020","journal-title":"Biometrika"},{"key":"2026062913325201600_qkag110-B26","article-title":"A residual bootstrap for high-dimensional regression with near low-rank designs","volume":"27","author":"Lopes","year":"2014","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"1","key":"2026062913325201600_qkag110-B27","doi-asserted-by":"publisher","first-page":"88","DOI":"10.3200\/JECE.39.1.88-91","article-title":"A simple proof of the FWL theorem","volume":"39","author":"Lovell","year":"2008","journal-title":"The Journal of Economic Education"},{"issue":"14","key":"2026062913325201600_qkag110-B28","doi-asserted-by":"publisher","first-page":"2734","DOI":"10.1002\/sim.10095","article-title":"Online causal inference with application to near real-time post-market vaccine safety surveillance","volume":"43","author":"Luo","year":"2024","journal-title":"Statistics in Medicine"},{"issue":"1","key":"2026062913325201600_qkag110-B29","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1111\/rssb.12352","article-title":"Renewable estimation and incremental inference in generalized linear models with streaming data sets","volume":"82","author":"Luo","year":"2020","journal-title":"Journal of the Royal Statistical Society: Series B"},{"issue":"534","key":"2026062913325201600_qkag110-B30","doi-asserted-by":"publisher","first-page":"984","DOI":"10.1080\/01621459.2019.1699421","article-title":"Global and simultaneous hypothesis testing for high-dimensional logistic regression models","volume":"116","author":"Ma","year":"2021","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2026062913325201600_qkag110-B31","first-page":"1","article-title":"High-dimensional inference for generalized linear models with hidden confounding","volume":"24","author":"Ouyang","year":"2023","journal-title":"Journal of Machine Learning Research"},{"issue":"2","key":"2026062913325201600_qkag110-B32","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1108\/eb025476","article-title":"Competitive strategy","volume":"1","author":"Porter","year":"1997","journal-title":"Measuring Business Excellence"},{"issue":"4","key":"2026062913325201600_qkag110-B33","doi-asserted-by":"publisher","first-page":"931","DOI":"10.2307\/1912705","article-title":"Root-n-consistent semiparametric regression","volume":"56","author":"Robinson","year":"1988","journal-title":"Econometrica: Journal of the Econometric Society"},{"issue":"3","key":"2026062913325201600_qkag110-B34","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1080\/00401706.2016.1142900","article-title":"Online updating of statistical inference in the big data setting","volume":"58","author":"Schifano","year":"2016","journal-title":"Technometrics: A Journal of Statistics for the Physical, Chemical, and Engineering Sciences"},{"issue":"3","key":"2026062913325201600_qkag110-B35","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1111\/j.1540-6261.1964.tb02865.x","article-title":"Capital asset prices: A theory of market equilibrium under conditions of risk","volume":"19","author":"Sharpe","year":"1964","journal-title":"Journal of Finance"},{"issue":"535","key":"2026062913325201600_qkag110-B36","doi-asserted-by":"publisher","first-page":"1307","DOI":"10.1080\/01621459.2019.1710154","article-title":"Statistical inference for high-dimensional models via recursive online-score estimation","volume":"116","author":"Shi","year":"2021","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2026062913325201600_qkag110-B37","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1111\/j.1467-9868.2004.00439.x","article-title":"Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: A unified approach","volume":"66","author":"Storey","year":"2004","journal-title":"Journal of the Royal Statistical Society: Series B"},{"issue":"548","key":"2026062913325201600_qkag110-B38","doi-asserted-by":"publisher","first-page":"2857","DOI":"10.1080\/01621459.2023.2283938","article-title":"A decorrelating and debiasing approach to simultaneous inference for high-dimensional confounded models","volume":"119","author":"Sun","year":"2024","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"2026062913325201600_qkag110-B39","doi-asserted-by":"publisher","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":"Journal of the Royal Statistical Society: Series B"},{"issue":"527","key":"2026062913325201600_qkag110-B40","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1080\/01621459.2018.1498346","article-title":"On the use of the lasso for instrumental variables estimation with some invalid instruments","volume":"114","author":"Windmeijer","year":"2019","journal-title":"Journal of the American Statistical Association"},{"key":"2026062913325201600_qkag110-B41","author":"Xu","year":"2024"},{"key":"2026062913325201600_qkag110-B42","article-title":"A framework for multi-A (rmed)\/B (andit) testing with online FDR control","volume":"30","author":"Yang","year":"2017","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"1","key":"2026062913325201600_qkag110-B43","doi-asserted-by":"publisher","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":"Journal of the Royal Statistical Society: Series B"},{"issue":"518","key":"2026062913325201600_qkag110-B44","doi-asserted-by":"publisher","first-page":"757","DOI":"10.1080\/01621459.2016.1166114","article-title":"Simultaneous inference for high-dimensional linear models","volume":"112","author":"Zhang","year":"2017","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"2026062913325201600_qkag110-B45","doi-asserted-by":"publisher","first-page":"1401","DOI":"10.1214\/21-AOS2156","article-title":"Ridge regression revisited: Debiasing, thresholding and bootstrap","volume":"50","author":"Zhang","year":"2022","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026062913325201600_qkag110-B46","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1287\/opre.2020.2005","article-title":"Nonsparse learning with latent variables","volume":"69","author":"Zheng","year":"2021","journal-title":"Operations Research"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article-pdf\/doi\/10.1093\/jrsssb\/qkag110\/68646199\/qkag110.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article-pdf\/doi\/10.1093\/jrsssb\/qkag110\/68646199\/qkag110.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T17:32:59Z","timestamp":1782754379000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/advance-article\/doi\/10.1093\/jrsssb\/qkag110\/8721608"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,29]]},"references-count":46,"URL":"https:\/\/doi.org\/10.1093\/jrsssb\/qkag110","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,29]]},"article-number":"qkag110"}}