{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T23:45:37Z","timestamp":1792021537027,"version":"4.4.0"},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":6738,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:p>This paper establishes necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process. As a special case, it is shown that the IGARCH(1,1) process with no drift converges almost surely to zero, while IGARCH(1,1) with a positive drift is strictly stationary and ergodic. We examine the persistence of shocks to conditional variance in the GARCH(l.l) model, and show that whether these shocks \"persist\" or not depends crucially on the definition of persistence. We also develop necessary and sufficient conditions for the finiteness of absolute moments of any (including fractional) order.<\/jats:p>","DOI":"10.1017\/s0266466600005296","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T15:47:33Z","timestamp":1236700053000},"page":"318-334","source":"Crossref","is-referenced-by-count":743,"title":["Stationarity and Persistence in the GARCH(1,1) Model"],"prefix":"10.1017","volume":"6","author":[{"given":"Daniel B.","family":"Nelson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600005296_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(86)90063-1"},{"key":"S0266466600005296_ref007","doi-asserted-by":"publisher","DOI":"10.1080\/07474938608800101"},{"key":"S0266466600005296_ref020","volume-title":"An Atlas of Functions","author":"Spanier","year":"1987"},{"key":"S0266466600005296_ref009","doi-asserted-by":"publisher","DOI":"10.1080\/07474938608800097"},{"key":"S0266466600005296_ref021","volume-title":"Almost Sure Convergence","author":"Stout","year":"1974"},{"key":"S0266466600005296_ref008","unstructured":"8. GAUSS: The GAUSS System Version 2.0. Kent, WA: Aptech Systems, Inc., 1988."},{"key":"S0266466600005296_ref023","unstructured":"23. Zucker R. Elementary transcendental functions: Logarithmic, exponential, circular and hyperbolic functions. Chapter 4 and pp. 65\u2013226 in [1]."},{"key":"S0266466600005296_ref012","volume-title":"Introductory Real Analysis","author":"Kolmogorov"},{"key":"S0266466600005296_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/1912773"},{"key":"S0266466600005296_ref001","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz","year":"1965"},{"key":"S0266466600005296_ref004","volume-title":"Real Analysis and Probability","author":"Dudley","year":"1989"},{"key":"S0266466600005296_ref010","volume-title":"Table of Integrals, Series and Products","author":"Gradshteyn","year":"1980"},{"key":"S0266466600005296_ref011","volume-title":"Inequalities","author":"Hardy","year":"1952"},{"key":"S0266466600005296_ref022","volume-title":"Asymptotic Theory for Econometricians","author":"White","year":"1984"},{"key":"S0266466600005296_ref013","volume-title":"Special Functions and Their Applications","author":"Lebedev","year":"1972"},{"key":"S0266466600005296_ref014","unstructured":"14. Nelson D.B. ARCH models as diffusion approximations. Working Paper Series in Economics and Econometrics #89\u201377, Graduate School of Business, University of Chicago, 1989. Forthcoming, Journal of Econometrics."},{"key":"S0266466600005296_ref015","first-page":"1142","article-title":"The persistence of volatility and stock market fluctuations","volume":"76","author":"Poterba","year":"1986","journal-title":"American Economic Review"},{"key":"S0266466600005296_ref018","unstructured":"18. Sampson M. A stationarity condition for the GARCH(l.l) process. Department of Economics, Concordia University (mimeo), 1988."},{"key":"S0266466600005296_ref003","unstructured":"3. Davis P.J. The gamma function and related functions. Chapter 6 and pp. 253\u2013294 in [1]."},{"key":"S0266466600005296_ref019","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3324-9"},{"key":"S0266466600005296_ref006","doi-asserted-by":"publisher","DOI":"10.1080\/07474938608800095"},{"key":"S0266466600005296_ref016","volume-title":"Integrals and Series: Volume 1, Elementary Functions","author":"Prudnikov","year":"1986"},{"key":"S0266466600005296_ref017","volume-title":"Real Analysis","author":"Royden"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600005296","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T20:21:11Z","timestamp":1558210871000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600005296\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0266466600005296"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600005296","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}