{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,9]],"date-time":"2026-09-09T04:07:58Z","timestamp":1788926878144,"version":"build-2803163510"},"reference-count":29,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2022,3,21]],"date-time":"2022-03-21T00:00:00Z","timestamp":1647820800000},"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":[[2023,10]]},"abstract":"<jats:p>The paper establishes the local asymptotic normality property for general conditionally heteroskedastic time series models of multiplicative form, \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466622000093_inline1.png\"\/><jats:tex-math>\n$\\epsilon _t=\\sigma _t(\\boldsymbol {\\theta }_0)\\eta _t$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where the volatility \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466622000093_inline2.png\"\/><jats:tex-math>\n$\\sigma _t(\\boldsymbol {\\theta }_0)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is a parametric function of \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466622000093_inline3.png\"\/><jats:tex-math>\n$\\{\\epsilon _{s}, s&lt; t\\}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466622000093_inline4.png\"\/><jats:tex-math>\n$(\\eta _t)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is a sequence of i.i.d. random variables with common density \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466622000093_inline5.png\"\/><jats:tex-math>\n$f_{\\boldsymbol {\\theta }_0}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In contrast with earlier results, the finite dimensional parameter \n<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466622000093_inline6.png\"\/><jats:tex-math>\n$\\boldsymbol {\\theta }_0$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> enters in both the volatility and the density specifications. To deal with nondifferentiable functions, we introduce a conditional notion of the familiar quadratic mean differentiability condition which takes into account parameter variation in both the volatility and the errors density. Our results are illustrated on two particular models: the APARCH with asymmetric Student-<jats:italic>t<\/jats:italic> distribution, and the Beta-<jats:italic>t<\/jats:italic>-GARCH model, and are extended to handle a conditional mean.<\/jats:p>","DOI":"10.1017\/s0266466622000093","type":"journal-article","created":{"date-parts":[[2022,3,21]],"date-time":"2022-03-21T07:29:43Z","timestamp":1647847783000},"page":"1067-1092","source":"Crossref","is-referenced-by-count":5,"title":["LOCAL ASYMPTOTIC NORMALITY OF GENERAL CONDITIONALLY HETEROSKEDASTIC AND SCORE-DRIVEN TIME-SERIES MODELS"],"prefix":"10.1017","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1528-8652","authenticated-orcid":false,"given":"Christian","family":"Francq","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7344-1849","authenticated-orcid":false,"given":"Jean-Michel","family":"Zakoian","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2022,3,21]]},"reference":[{"key":"S0266466622000093_r11","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1007\/BF00773401","article-title":"Local asymptotic normality of multivariate ARMA processes with a linear trend","volume":"47","author":"Garel","year":"1995","journal-title":"Annals of the Institute of Statistical Mathematics"},{"key":"S0266466622000093_r12","first-page":"2054","article-title":"Local asymptotic normality for regression models with long-memory disturbance","volume":"27","author":"Hallin","year":"1999","journal-title":"Annals of Statistics"},{"key":"S0266466622000093_r10","doi-asserted-by":"publisher","DOI":"10.3150\/bj\/1093265632"},{"key":"S0266466622000093_r29","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511802256"},{"key":"S0266466622000093_r25","first-page":"1049","article-title":"Pseudo-maximum likelihood estimation of ARCH(1) models","volume":"34","author":"Robinson","year":"2006","journal-title":"Annals of Statistic"},{"key":"S0266466622000093_r17","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1032526954"},{"key":"S0266466622000093_r18","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350256"},{"key":"S0266466622000093_r28","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(85)90051-X"},{"key":"S0266466622000093_r30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2010.01.013"},{"key":"S0266466622000093_r9","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1031833674"},{"key":"S0266466622000093_r6","doi-asserted-by":"publisher","DOI":"10.2139\/ssrn.1297183"},{"key":"S0266466622000093_r3","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316962"},{"key":"S0266466622000093_r5","unstructured":"Blasques, F. , Koopman, S.J. , & Lucas, A. 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