{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T20:45:58Z","timestamp":1786999558886,"version":"build-2736575974"},"reference-count":35,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T00:00:00Z","timestamp":1683849600000},"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":[[2024,12]]},"abstract":"<jats:p>In GARCH-mixed-data sampling models, the volatility is decomposed into the product of two factors which are often interpreted as \u201cshort-run\u201d (high-frequency) and \u201clong-run\u201d (low-frequency) components. While two-component volatility models are widely used in applied works, some of their theoretical properties remain unexplored. We show that the strictly stationary solutions of such models do not admit any small-order finite moment, contrary to classical GARCH. It is shown that the strong consistency and the asymptotic normality of the quasi-maximum likelihood estimator hold despite the absence of moments. Tests for the presence of a long-run volatility relying on the asymptotic theory and a bootstrap procedure are proposed. Our results are illustrated via Monte Carlo experiments and real financial data.<\/jats:p>","DOI":"10.1017\/s0266466623000142","type":"journal-article","created":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T02:53:31Z","timestamp":1683860011000},"page":"1422-1455","source":"Crossref","is-referenced-by-count":4,"title":["INFERENCE ON GARCH-MIDAS MODELS WITHOUT ANY SMALL-ORDER MOMENT"],"prefix":"10.1017","volume":"40","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\/0000-0002-6449-0024","authenticated-orcid":false,"given":"Baye Matar","family":"Kandji","sequence":"additional","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":[[2023,5,12]]},"reference":[{"key":"S0266466623000142_r32","doi-asserted-by":"publisher","DOI":"10.1007\/BF00532266"},{"key":"S0266466623000142_r33","volume-title":"Asymptotic Statistics","author":"van der Vaart","year":"2000"},{"key":"S0266466623000142_r5","first-page":"788","article-title":"The Lindeberg-L\u00e9vy theorem for martingales","volume":"12","author":"Billingsley","year":"1961","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0266466623000142_r4","unstructured":"Beutner, E. , Heinemann, A. , & Smeekes, S. (2020) A residual bootstrap for conditional value-at-risk. Preprint, arXiv:1808.0912. https:\/\/EconPapers.repec.org\/RePEc:arx:papers:1808.09125."},{"key":"S0266466623000142_r26","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392040"},{"key":"S0266466623000142_r34","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466614000334"},{"key":"S0266466623000142_r6","first-page":"211","article-title":"with stationary coefficients","volume":"18","author":"Brandt","year":"1986","journal-title":"Advances in Applied Probability"},{"key":"S0266466623000142_r2","doi-asserted-by":"publisher","DOI":"10.1111\/1468-0262.00210"},{"key":"S0266466623000142_r31","doi-asserted-by":"publisher","DOI":"10.1524\/strm.2013.1088"},{"key":"S0266466623000142_r35","doi-asserted-by":"publisher","DOI":"10.2307\/1912526"},{"key":"S0266466623000142_r11","doi-asserted-by":"publisher","DOI":"10.1080\/07350015.2018.1482759"},{"key":"S0266466623000142_r9","doi-asserted-by":"publisher","DOI":"10.2139\/ssrn.3793571"},{"key":"S0266466623000142_r1","doi-asserted-by":"publisher","DOI":"10.1080\/07474938.2014.977064"},{"key":"S0266466623000142_r8","doi-asserted-by":"publisher","DOI":"10.3390\/jrfm11020023"},{"key":"S0266466623000142_r22","doi-asserted-by":"publisher","DOI":"10.2307\/2171789"},{"key":"S0266466623000142_r18","doi-asserted-by":"publisher","DOI":"10.1198\/jasa.2009.0117"},{"key":"S0266466623000142_r10","doi-asserted-by":"publisher","DOI":"10.1002\/jae.2404"},{"key":"S0266466623000142_r25","article-title":"Iterated function systems driven by non independent sequences: Structure and inference","author":"Kandji","year":"2023","journal-title":"Journal of Applied Probability"},{"key":"S0266466623000142_r20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2020.05.009"},{"key":"S0266466623000142_r29","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1015957401"},{"key":"S0266466623000142_r12","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176349937"},{"key":"S0266466623000142_r7","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2020.05.006"},{"key":"S0266466623000142_r13","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(95)01737-2"},{"key":"S0266466623000142_r27","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOS900"},{"key":"S0266466623000142_r28","doi-asserted-by":"publisher","DOI":"10.1111\/sjos.12158"},{"key":"S0266466623000142_r15","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1093\/oso\/9780198296836.003.0020","volume-title":"Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive WJ Granger","author":"Engle","year":"1999"},{"key":"S0266466623000142_r19","doi-asserted-by":"publisher","DOI":"10.1002\/9781119313472"},{"key":"S0266466623000142_r24","unstructured":"Ibragimov, R. , Pedersen, R.S. , & Skrobotov, A. (2021) New approaches to robust inference on market (non-)efficiency volatility clustering and non linear dependence. Preprint, arXiv:2006.01212v3."},{"key":"S0266466623000142_r3","doi-asserted-by":"publisher","DOI":"10.3150\/bj\/1068128975"},{"key":"S0266466623000142_r16","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466609990430"},{"key":"S0266466623000142_r30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2010.03.034"},{"key":"S0266466623000142_r14","doi-asserted-by":"publisher","DOI":"10.1162\/REST_a_00300"},{"key":"S0266466623000142_r17","doi-asserted-by":"publisher","DOI":"10.3150\/bj\/1093265632"},{"key":"S0266466623000142_r21","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466612000655"},{"key":"S0266466623000142_r23","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466623000117"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466623000142","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,23]],"date-time":"2024-12-23T01:31:56Z","timestamp":1734917516000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466623000142\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,12]]},"references-count":35,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["S0266466623000142"],"URL":"https:\/\/doi.org\/10.1017\/s0266466623000142","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,12]]}}}