{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T22:13:53Z","timestamp":1777414433779,"version":"3.51.4"},"reference-count":31,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2021,3,23]],"date-time":"2021-03-23T00:00:00Z","timestamp":1616457600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:p>The paper examines the effect of conditional heteroskedasticity on least squares inference in stochastic regression models of unknown integration order and proposes an inference procedure that is robust to models within the (near) I(0)\u2013(near) I(1) range with GARCH innovations. We show that a regressor signal of exact order<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000086_inline1.png\"\/><jats:tex-math>$O_{p}\\left ( n\\kappa _{n}\\right ) $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>for arbitrary<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000086_inline2.png\"\/><jats:tex-math>$\\,\\kappa _{n}\\rightarrow \\infty $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is sufficient to eliminate stationary GARCH effects from the limit distributions of least squares based estimators and self-normalized test statistics. The above order dominates the<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000086_inline3.png\"\/><jats:tex-math>$O_{p}\\left ( n\\right ) $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>signal of stationary regressors but may be dominated by the<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000086_inline4.png\"\/><jats:tex-math>$O_{p}\\left ( n^{2}\\right ) $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>signal of I(1) regressors, thereby showing that least squares invariance to GARCH effects is not an exclusively I(1) phenomenon but extends to processes with persistence degree arbitrarily close to stationarity. The theory validates standard inference for self normalized test statistics based on the ordinary least squares estimator when<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000086_inline5.png\"\/><jats:tex-math>$\\kappa _{n}\\rightarrow \\infty $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000086_inline6.png\"\/><jats:tex-math>$\\kappa _{n}\/n\\rightarrow 0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and the IVX estimator (Phillips and Magdalinos (2009a), Econometric Inference in the Vicinity of Unity. Working paper, Singapore Management University; Kostakis, Magdalinos, and Stamatogiannis, 2015a,<jats:italic>Review of Financial Studies<\/jats:italic>28(5), 1506\u20131553.) when<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000086_inline7.png\"\/><jats:tex-math>$\\kappa _{n}\\rightarrow \\infty $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and the innovation sequence of the system is a covariance stationary vec-GARCH process. An adjusted version of the IVX\u2013Wald test is shown to also accommodate GARCH effects in purely stationary regressors, thereby extending the procedure\u2019s validity over the entire (near) I(0)\u2013(near) I(1) range of regressors under conditional heteroskedasticity in the innovations. It is hoped that the wide range of applicability of this adjusted IVX\u2013Wald test, established in Theorem 4.4, presents an advantage for the procedure\u2019s suitability as a tool for applied research.<\/jats:p>","DOI":"10.1017\/s0266466621000086","type":"journal-article","created":{"date-parts":[[2021,3,23]],"date-time":"2021-03-23T08:40:56Z","timestamp":1616488856000},"page":"875-912","source":"Crossref","is-referenced-by-count":4,"title":["LEAST SQUARES AND IVX LIMIT THEORY IN SYSTEMS OF PREDICTIVE REGRESSIONS WITH GARCH INNOVATIONS"],"prefix":"10.1017","volume":"38","author":[{"given":"Tassos","family":"Magdalinos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,3,23]]},"reference":[{"key":"S0266466621000086_r8","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350492"},{"key":"S0266466621000086_r11","doi-asserted-by":"publisher","DOI":"10.1142\/p591"},{"key":"S0266466621000086_r19","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466608090154"},{"key":"S0266466621000086_r17","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1997.10474076"},{"key":"S0266466621000086_r22","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/74.3.535"},{"key":"S0266466621000086_r26","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511493157.008"},{"key":"S0266466621000086_r1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810800"},{"key":"S0266466621000086_r18","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466603192092"},{"key":"S0266466621000086_r7","doi-asserted-by":"publisher","DOI":"10.1016\/j.spa.2011.06.001"},{"key":"S0266466621000086_r15","doi-asserted-by":"publisher","DOI":"10.2139\/ssrn.2517662"},{"key":"S0266466621000086_r6","first-page":"275","article-title":"Ergodicit\u00e9 des cha\u00eenes de Markov \u00e0 valeurs dans une vari\u00e9t\u00e9 alg\u00e9brique: application aux mod\u00e8les GARCH multivari\u00e9s","volume":"343","author":"Boussama","year":"2006","journal-title":"Comptes Rendus de l\u2019 Academie des Sciences Paris"},{"key":"S0266466621000086_r24","doi-asserted-by":"publisher","DOI":"10.2307\/1911357"},{"key":"S0266466621000086_r10","doi-asserted-by":"publisher","DOI":"10.1002\/9780470670057"},{"key":"S0266466621000086_r30","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600011397"},{"key":"S0266466621000086_r13","volume-title":"Matrix Analysis","author":"Horn","year":"2013"},{"key":"S0266466621000086_r12","volume-title":"Martingale Limit Theory and its Application","author":"Hall","year":"1980"},{"key":"S0266466621000086_r27","unstructured":"Phillips, P.C.B. & Magdalinos, T. 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