{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,14]],"date-time":"2023-12-14T14:10:16Z","timestamp":1702563016864},"reference-count":43,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2021,9,27]],"date-time":"2021-09-27T00:00:00Z","timestamp":1632700800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:p>We study linear subset regression in the context of the high-dimensional overall model <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline1.png\" \/><jats:tex-math>\n$y = \\vartheta +\\theta ' z + \\epsilon $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with univariate response <jats:italic>y<\/jats:italic> and a <jats:italic>d<\/jats:italic>-vector of random regressors <jats:italic>z<\/jats:italic>, independent of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline2.png\" \/><jats:tex-math>\n$\\epsilon $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Here, \u201chigh-dimensional\u201d means that the number <jats:italic>d<\/jats:italic> of available explanatory variables is much larger than the number <jats:italic>n<\/jats:italic> of observations. We consider simple linear submodels where <jats:italic>y<\/jats:italic> is regressed on a set of <jats:italic>p<\/jats:italic> regressors given by <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline3.png\" \/><jats:tex-math>\n$x = M'z$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, for some <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline4.png\" \/><jats:tex-math>\n$d \\times p$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> matrix <jats:italic>M<\/jats:italic> of full rank <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline5.png\" \/><jats:tex-math>\n$p &lt; n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. The corresponding simple model, that is, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline6.png\" \/><jats:tex-math>\n$y=\\alpha +\\beta ' x + e$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, is usually justified by imposing appropriate restrictions on the unknown parameter <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline7.png\" \/><jats:tex-math>\n$\\theta $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in the overall model; otherwise, this simple model can be grossly misspecified in the sense that relevant variables may have been omitted. In this paper, we establish asymptotic validity of the standard <jats:italic>F<\/jats:italic>-test on the surrogate parameter <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline8.png\" \/><jats:tex-math>\n$\\beta $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, in an appropriate sense, even when the simple model is misspecified, that is, without any restrictions on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S026646662100044X_inline9.png\" \/><jats:tex-math>\n$\\theta $\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> whatsoever and without assuming Gaussian data.<\/jats:p>","DOI":"10.1017\/s026646662100044x","type":"journal-article","created":{"date-parts":[[2021,9,27]],"date-time":"2021-09-27T14:00:56Z","timestamp":1632751256000},"page":"1249-1272","source":"Crossref","is-referenced-by-count":0,"title":["STATISTICAL INFERENCE WITH <i>F<\/i>-STATISTICS WHEN FITTING SIMPLE MODELS TO HIGH-DIMENSIONAL DATA"],"prefix":"10.1017","volume":"39","author":[{"given":"Hannes","family":"Leeb","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lukas","family":"Steinberger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,9,27]]},"reference":[{"key":"S026646662100044X_r37","doi-asserted-by":"publisher","DOI":"10.1214\/18-AOS1719"},{"key":"S026646662100044X_r8","doi-asserted-by":"publisher","DOI":"10.1016\/0167-7152(94)00112-L"},{"key":"S026646662100044X_r3","volume-title":"An Introduction to Multivariate Analysis","author":"Anderson","year":"1958"},{"key":"S026646662100044X_r4","doi-asserted-by":"publisher","DOI":"10.1214\/18-AOS1721"},{"key":"S026646662100044X_r6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2005.08.003"},{"key":"S026646662100044X_r27","doi-asserted-by":"publisher","DOI":"10.1016\/j.ecosta.2018.01.001"},{"key":"S026646662100044X_r33","doi-asserted-by":"publisher","DOI":"10.1109\/6.67285"},{"key":"S026646662100044X_r7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jspi.2004.03.020"},{"key":"S026646662100044X_r20","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176349155"},{"key":"S026646662100044X_r38","doi-asserted-by":"publisher","DOI":"10.1198\/016214502388618960"},{"key":"S026646662100044X_r41","doi-asserted-by":"publisher","DOI":"10.2307\/1912934"},{"key":"S026646662100044X_r17","doi-asserted-by":"publisher","DOI":"10.1214\/17-AOS1601"},{"key":"S026646662100044X_r31","first-page":"97","article-title":"Misspecified \n\n\n\n${t}^2$\n\n\n tests. 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