{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,3]],"date-time":"2026-10-03T22:58:08Z","timestamp":1791068288672,"version":"4.1.0"},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2008,1,22]],"date-time":"2008-01-22T00:00:00Z","timestamp":1200960000000},"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":[[2008,6]]},"abstract":"<jats:p>Based on an idea of Granger (1986,<jats:italic>Oxford Bulletin of Economics and Statistics<\/jats:italic>48, 213\u2013228), we analyze a new vector autoregressive model defined from the fractional lag operator 1 \u2212 (1 \u2212<jats:italic>L<\/jats:italic>)<jats:sup><jats:italic>d<\/jats:italic><\/jats:sup>. We first derive conditions in terms of the coefficients for the model to generate processes that are fractional of order zero. We then show that if there is a unit root, the model generates a fractional process<jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>of order<jats:italic>d<\/jats:italic>,<jats:italic>d<\/jats:italic>&gt; 0, for which there are vectors \u03b2 so that \u03b2\u203c<jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>is fractional of order<jats:italic>d<\/jats:italic>\u2212<jats:italic>b<\/jats:italic>, 0 &lt;<jats:italic>b<\/jats:italic>\u2264<jats:italic>d<\/jats:italic>. We find a representation of the solution that demonstrates the fractional properties. Finally we suggest a model that allows for a polynomial fractional vector, that is, the process<jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>is fractional of order<jats:italic>d<\/jats:italic>, \u03b2\u203c<jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>is fractional of order<jats:italic>d<\/jats:italic>\u2212<jats:italic>b<\/jats:italic>, and a linear combination of \u03b2\u203c<jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>and \u0394<jats:sup><jats:italic>b<\/jats:italic><\/jats:sup><jats:italic>X<\/jats:italic><jats:sub><jats:italic>t<\/jats:italic><\/jats:sub>is fractional of order<jats:italic>d<\/jats:italic>\u2212 2<jats:italic>b<\/jats:italic>. The representations and conditions are analogous to the well-known conditions for<jats:italic>I<\/jats:italic>(0),<jats:italic>I<\/jats:italic>(1), and<jats:italic>I<\/jats:italic>(2) variables.<\/jats:p>","DOI":"10.1017\/s0266466608080274","type":"journal-article","created":{"date-parts":[[2008,1,22]],"date-time":"2008-01-22T10:16:08Z","timestamp":1200996968000},"page":"651-676","source":"Crossref","is-referenced-by-count":219,"title":["A REPRESENTATION THEORY FOR A CLASS OF VECTOR AUTOREGRESSIVE MODELS FOR FRACTIONAL PROCESSES"],"prefix":"10.1017","volume":"24","author":[{"given":"S\u00d8ren","family":"Johansen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2008,1,22]]},"reference":[{"key":"S0266466608080274_ref019","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/S0304-4076(01)00076-8","article-title":"Semiparametric fractional cointegration analysis","volume":"105","author":"Robinson","year":"2001","journal-title":"Journal of Econometrics"},{"key":"S0266466608080274_ref014","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9469.00074"},{"key":"S0266466608080274_ref011","volume-title":"Statistical Analysis of Stationary Time Series","author":"Grenander","year":"1956"},{"key":"S0266466608080274_ref010","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.1980.tb00297.x"},{"key":"S0266466608080274_ref007","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1093\/oso\/9780198296836.003.0008","volume-title":"Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger","author":"Engsted","year":"1999"},{"key":"S0266466608080274_ref009","doi-asserted-by":"publisher","DOI":"10.1111\/j.1468-0084.1986.mp48003002.x"},{"key":"S0266466608080274_ref017","unstructured":"Lyhagen J. (1998) Maximum Likelihood Estimation of the Multivariate Fractional Cointegrating Model. Working paper, Stockholm School of Economics."},{"key":"S0266466608080274_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/1913236"},{"key":"S0266466608080274_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-0320-4"},{"key":"S0266466608080274_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4076(02)00091-X"},{"key":"S0266466608080274_ref001","volume-title":"Statistics for Long-Memory Processes","author":"Beran","year":"1994"},{"key":"S0266466608080274_ref008","volume-title":"Tables of Integrals, Sums, Series and Products","author":"Gradshteyn","year":"1971"},{"key":"S0266466608080274_ref013","volume-title":"Likelihood Based Inference on Cointegration in the Vector Autoregressive Model","author":"Johansen","year":"1996"},{"key":"S0266466608080274_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4076(02)00092-1"},{"key":"S0266466608080274_ref018","volume-title":"Functions of a Complex Variable with Applications","author":"Phillips","year":"1958"},{"key":"S0266466608080274_ref012","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/68.1.165"},{"key":"S0266466608080274_ref015","article-title":"Representation of cointegrated autoregressive processes with application to fractional processes","author":"Johansen","journal-title":"Econometric Review"},{"key":"S0266466608080274_ref016","unstructured":"Lasak K. (2005) Likelihood Based Testing for Fractional Cointegration. Discussion paper, University of Barcelona."},{"key":"S0266466608080274_ref005","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.2004.00335.x"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466608080274","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,21]],"date-time":"2024-02-21T14:33:51Z","timestamp":1708526031000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466608080274\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1,22]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,6]]}},"alternative-id":["S0266466608080274"],"URL":"https:\/\/doi.org\/10.1017\/s0266466608080274","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1,22]]}}}