{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T17:07:43Z","timestamp":1786122463194,"version":"build-2736575974"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","license":[{"start":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T00:00:00Z","timestamp":1786060800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["72495124,72525001"],"award-info":[{"award-number":["72495124,72525001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100005089","name":"Natural Science Foundation of Beijing Municipality","doi-asserted-by":"publisher","award":["JQ25030,Z240004"],"award-info":[{"award-number":["JQ25030,Z240004"]}],"id":[{"id":"10.13039\/501100005089","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"abstract":"<jats:p>\n                    Asymptotic optimality is a key theoretical property in model averaging. Due to technical difficulties, existing studies rely on restricted weight sets or on the assumption that there is no true model with fixed dimensions in the candidate set. The focus of this article is to overcome these difficulties by studying a class of least squares model averaging methods with nested candidate models, which includes Mallows model averaging (MMA) of Hansen (2007,\n                    <jats:italic>Econometrica<\/jats:italic>\n                    75, 1175\u20131189) and parsimonious model averaging (PMA) of Zhang et al. (2020,\n                    <jats:italic>Journal of the American Statistical Association<\/jats:italic>\n                    115, 972\u2013984) as special cases. The main results are as follows. First, MMA with fixed model dimensions is neither asymptotically loss optimal nor asymptotically risk optimal. Second, when the true model dimension diverges, MMA is asymptotically both loss and risk optimal. Third, when the true model dimension is fixed, PMA is asymptotically risk optimal but not asymptotically loss optimal, which differs from the corresponding result in Fang, Yuan, and Tian (2023,\n                    <jats:italic>Econometric Theory<\/jats:italic>\n                    39, 412\u2013441). This finding also reveals that the discrete weight set of Hansen (2007) and the usual weight set can lead to different asymptotic properties: asymptotic loss optimality holds under the former but fails under the latter. Finally, when the true model dimension diverges, PMA is asymptotically both loss and risk optimal. Simulation studies illustrate the theoretical findings in a variety of settings.\n                  <\/jats:p>","DOI":"10.1017\/s0266466626100553","type":"journal-article","created":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T09:44:28Z","timestamp":1786095868000},"page":"1-16","source":"Crossref","is-referenced-by-count":0,"title":["ON ASYMPTOTIC OPTIMALITY OF LEAST SQUARES MODEL AVERAGING WHEN TRUE MODEL IS\u00a0INCLUDED"],"prefix":"10.1017","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4563-3605","authenticated-orcid":false,"given":"Wenchao","family":"Xu","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/031t68441","id-type":"ROR","asserted-by":"publisher"}],"name":"Shanghai University of International Business and Economics"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinyu","family":"Zhang","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/011xvna82","id-type":"ROR","asserted-by":"publisher"}],"name":"Fuzhou University and Academy of Mathematics and Systems Science, Chinese Academy of Sciences"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2026,8,7]]},"reference":[{"key":"S0266466626100553_r4","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466622000032"},{"key":"S0266466626100553_r13","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2024.2402566"},{"key":"S0266466626100553_r6","doi-asserted-by":"publisher","DOI":"10.1111\/j.1468-0262.2007.00785.x"},{"key":"S0266466626100553_r15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2009.10.030"},{"key":"S0266466626100553_r18","doi-asserted-by":"publisher","DOI":"10.1016\/j.econlet.2015.03.017"},{"key":"S0266466626100553_r21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2013.01.004"},{"key":"S0266466626100553_r9","doi-asserted-by":"publisher","DOI":"10.1198\/jasa.2011.tm09478"},{"key":"S0266466626100553_r22","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2015.1115762"},{"key":"S0266466626100553_r10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2020.10.004"},{"key":"S0266466626100553_r1","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2013.838168"},{"key":"S0266466626100553_r11","doi-asserted-by":"publisher","DOI":"10.1111\/ectj.12009"},{"key":"S0266466626100553_r7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2011.06.019"},{"key":"S0266466626100553_r24","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.2019.1604363"},{"key":"S0266466626100553_r5","doi-asserted-by":"publisher","DOI":"10.1080\/07350015.2020.1870477"},{"key":"S0266466626100553_r14","first-page":"221","article-title":"An asymptotic theory for linear model selection","volume":"7","author":"Shao","year":"1997","journal-title":"Statistica Sinica"},{"key":"S0266466626100553_r17","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466624000355"},{"key":"S0266466626100553_r8","first-page":"1","article-title":"Functional martingale residual process for high-dimensional Cox regression with model averaging","volume":"21","author":"He","year":"2020","journal-title":"Journal of Machine Learning Research"},{"key":"S0266466626100553_r20","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466618000269"},{"key":"S0266466626100553_r16","doi-asserted-by":"publisher","DOI":"10.1016\/j.econlet.2015.03.027"},{"key":"S0266466626100553_r23","first-page":"1583","article-title":"Model averaging based on Kullback\u2013Leibler distance","volume":"25","author":"Zhang","year":"2015","journal-title":"Statistica Sinica"},{"key":"S0266466626100553_r19","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466620000055"},{"key":"S0266466626100553_r3","doi-asserted-by":"publisher","DOI":"10.1016\/j.econlet.2022.110683"},{"key":"S0266466626100553_r2","doi-asserted-by":"publisher","DOI":"10.1016\/j.econlet.2020.109586"},{"key":"S0266466626100553_r12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2014.11.005"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466626100553","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T09:44:29Z","timestamp":1786095869000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466626100553\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,7]]},"references-count":24,"alternative-id":["S0266466626100553"],"URL":"https:\/\/doi.org\/10.1017\/s0266466626100553","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,7]]}}}