{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T13:12:04Z","timestamp":1791637924288,"version":"4.3.4"},"reference-count":32,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2023,4,7]],"date-time":"2023-04-07T00:00:00Z","timestamp":1680825600000},"content-version":"vor","delay-in-days":5799,"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-0624841"],"award-info":[{"award-number":["DMS-0624841"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2007,6,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We introduce a general formulation for dimension reduction and coefficient estimation in the multivariate linear model. We argue that many of the existing methods that are commonly used in practice can be formulated in this framework and have various restrictions. We continue to propose a new method that is more flexible and more generally applicable. The method proposed can be formulated as a novel penalized least squares estimate. The penalty that we employ is the coefficient matrix's Ky Fan norm. Such a penalty encourages the sparsity among singular values and at the same time gives shrinkage coefficient estimates and thus conducts dimension reduction and coefficient estimation simultaneously in the multivariate linear model. We also propose a generalized cross-validation type of criterion for the selection of the tuning parameter in the penalized least squares. Simulations and an application in financial econometrics demonstrate competitive performance of the new method. An extension to the non-parametric factor model is also discussed.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2007.00591.x","type":"journal-article","created":{"date-parts":[[2007,5,23]],"date-time":"2007-05-23T18:25:51Z","timestamp":1179944751000},"page":"329-346","source":"Crossref","is-referenced-by-count":209,"title":["Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression"],"prefix":"10.1093","volume":"69","author":[{"given":"Ming","family":"Yuan","sequence":"first","affiliation":[{"name":"Georgia Institute of Technology , Atlanta , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ali","family":"Ekici","sequence":"additional","affiliation":[{"name":"Georgia Institute of Technology , Atlanta , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaosong","family":"Lu","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University , Pittsburgh , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Renato","family":"Monteiro","sequence":"additional","affiliation":[{"name":"Georgia Institute of Technology , Atlanta , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2007,5,22]]},"reference":[{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1214\/aoms\/1177729580","article-title":"Estimating linear restrictions on regression coefficients for multivariate normal distributions","volume":"22","author":"Anderson","year":"1951","journal-title":"Ann. Math. Statist."},{"key":"2023040800083544200_","volume-title":"An Introduction to Multivariate Statistical Analysis","author":"Anderson","year":"2003","edition":"3"},{"key":"2023040800083544200_","volume-title":"Adaptive regression and model selection in data mining problems. PhD Thesis","author":"Bakin","year":"1999"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"226","DOI":"10.2307\/2533213","article-title":"Model selection for multivariate regression in small samples","volume":"50","author":"Bedrick","year":"1994","journal-title":"Biometrics"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718829","volume-title":"Lectures on Modern Convex Optimization: Analysis, Algorithms, Engineering Applications","author":"Ben-Tal","year":"2001"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.1214\/aos\/1032181158","article-title":"Heuristics of instability and stabilization in model selection","volume":"24","author":"Breiman","year":"1996","journal-title":"Ann. Statist."},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1111\/1467-9868.00054","article-title":"Predicting multivariate responses in multiple linear regression (with discussion)","volume":"59","author":"Breiman","year":"1997","journal-title":"J. R. Statist. Soc. B"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"1374","DOI":"10.1080\/01621459.1994.10476876","article-title":"Joint continuum regression for multiple predictands","volume":"89","author":"Brooks","year":"1994","journal-title":"J. Am. Statist. Ass."},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1093\/biomet\/86.3.635","article-title":"The choice of variables in multivariate regression: a non-conjugate Bayesian decision theory approach","volume":"86","author":"Brown","year":"1999","journal-title":"Biometrika"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1111\/1467-9868.00144","article-title":"Multivariate Bayesian variable selection and prediction","volume":"60","author":"Brown","year":"1998","journal-title":"J. R. Statist. Soc. B"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1111\/1467-9868.00348","article-title":"Bayes model averaging with selection of regressors","volume":"64","author":"Brown","year":"2002","journal-title":"J. R. Statist. Soc. B"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1214\/ss\/1038425655","article-title":"Flexible smoothing with B-splines and penalties (with discussion)","volume":"11","author":"Eilers","year":"1996","journal-title":"Statist. Sci."},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1080\/00401706.1993.10485033","article-title":"A statistical view of some chemometrics regression tools (with discussion)","volume":"35","author":"Frank","year":"1993","journal-title":"Technometrics"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1093\/biomet\/84.3.707","article-title":"Modified AIC and Cp in multivariate linear regression","volume":"84","author":"Fujikoshi","year":"1997","journal-title":"Biometrika"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1080\/00401706.1979.10489751","article-title":"Generalized cross validation as a method for choosing a good ridge parameter","volume":"21","author":"Golub","year":"1979","journal-title":"Technometrics"},{"key":"2023040800083544200_","volume-title":"Generalized Additive Models","author":"Hastie","year":"1990"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511840371","volume-title":"Topics in Matrix Analysis","author":"Horn","year":"1991"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1037\/h0058165","article-title":"The most predictable criterion","volume":"26","author":"Hotelling","year":"1935","journal-title":"J. Educ. Psychol."},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1093\/biomet\/28.3-4.321","article-title":"Relations between two sets of variables","volume":"28","author":"Hotelling","year":"1936","journal-title":"Biometrika"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/0047-259X(75)90042-1","article-title":"Reduced-rank regression for the multivariate linear model","volume":"5","author":"Izenman","year":"1975","journal-title":"J. Multiv. Anal."},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1214\/aos\/1009210544","article-title":"On the distribution of the largest eigenvalue in principal components analysis","volume":"29","author":"Johnstone","year":"2001","journal-title":"Ann. Statist."},{"key":"2023040800083544200_","first-page":"471","article-title":"Boosting for high-multivariate responses in high-dimensional linear regression","volume":"16","author":"Lutz","year":"2006","journal-title":"Statist. Sin."},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1080\/01621459.1965.10480787","article-title":"Principal components regression with exploratory statistical research","volume":"60","author":"Massy","year":"1965","journal-title":"J. Am. Statist. Ass."},{"key":"2023040800083544200_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0679-8","volume-title":"Elements of Multivariate Time Series Analysis","author":"Reinsel","year":"1997","edition":"2"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2853-8","volume-title":"Multivariate Reduced-rank Regression: Theory and Applications","author":"Reinsel","year":"1998"},{"key":"2023040800083544200_","article-title":"Penalized regression splines","author":"Ruppert","year":"1997","journal-title":"Technical Report"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"22","DOI":"10.2307\/2527708","article-title":"Multivariate analysis of variance (ANOVA)","volume":"18","author":"Smith","year":"1962","journal-title":"Biometrics"},{"key":"2023040800083544200_","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. B"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1198\/004017005000000139","article-title":"Simultaneous variable selection","volume":"47","author":"Turlach","year":"2005","journal-title":"Technometrics"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s10107-002-0347-5","article-title":"Solving semidefinite-quadratic-linear programs using SDPT3","volume":"95","author":"T\u00fct\u00fcnc\u00fc","year":"2003","journal-title":"Math. Progrmmng"},{"key":"2023040800083544200_","volume-title":"Perspectives in Probability and Statistics: Papers in Honour of M. S. Bartlett","author":"Wold","year":"1975"},{"key":"2023040800083544200_","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1111\/j.1467-9868.2005.00532.x","article-title":"Model selection and estimation in regression with grouped variables","volume":"68","author":"Yuan","year":"2006","journal-title":"J. R. Statist. Soc. B"}],"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%2Fj.1467-9868.2007.00591.x","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/69\/3\/329\/49795441\/jrsssb_69_3_329.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/69\/3\/329\/49795441\/jrsssb_69_3_329.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,16]],"date-time":"2025-01-16T15:32:42Z","timestamp":1737041562000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/69\/3\/329\/7109373"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,5,22]]},"references-count":32,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2007,5,22]]},"published-print":{"date-parts":[[2007,6,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.1467-9868.2007.00591.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2007,6]]},"published":{"date-parts":[[2007,5,22]]}}}