{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T18:42:05Z","timestamp":1785350525249,"version":"3.55.0"},"reference-count":31,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2019,11,22]],"date-time":"2019-11-22T00:00:00Z","timestamp":1574380800000},"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":[[2020,10]]},"abstract":"<jats:p>This article defines the class of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466619000306_inline1.png\"\/><jats:tex-math>${\\cal H}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-valued autoregressive (AR) processes with a unit root of finite type, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466619000306_inline2.png\"\/><jats:tex-math>${\\cal H}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is a possibly infinite-dimensional separable Hilbert space, and derives a generalization of the Granger\u2013Johansen Representation Theorem valid for any integration order <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466619000306_inline3.png\"\/><jats:tex-math>$d = 1,2, \\ldots$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. An existence theorem shows that the solution of an AR process with a unit root of finite type is necessarily integrated of some finite integer order <jats:italic>d<\/jats:italic>, displays a common trends representation with a finite number of common stochastic trends, and it possesses an infinite-dimensional cointegrating space when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466619000306_inline4.png\"\/><jats:tex-math>${\\rm{dim}}{\\cal H} = \\infty$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. A characterization theorem clarifies the connections between the structure of the AR operators and (<jats:italic>i<\/jats:italic>) the order of integration, (<jats:italic>ii<\/jats:italic>) the structure of the attractor space and the cointegrating space, (<jats:italic>iii<\/jats:italic>) the expression of the cointegrating relations, and (<jats:italic>iv<\/jats:italic>) the triangular representation of the process. Except for the fact that the dimension of the cointegrating space is infinite when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466619000306_inline5.png\"\/><jats:tex-math>${\\rm{dim}}{\\cal H} = \\infty$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, the representation of AR processes with a unit root of finite type coincides with the one of finite-dimensional VARs, which can be obtained setting <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466619000306_inline6.png\"\/><jats:tex-math>${\\cal H} = ^p $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in the present results.<\/jats:p>","DOI":"10.1017\/s0266466619000306","type":"journal-article","created":{"date-parts":[[2019,11,22]],"date-time":"2019-11-22T10:41:16Z","timestamp":1574419276000},"page":"803-839","source":"Crossref","is-referenced-by-count":20,"title":["COINTEGRATION IN FUNCTIONAL AUTOREGRESSIVE PROCESSES"],"prefix":"10.1017","volume":"36","author":[{"given":"Massimo","family":"Franchi","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paolo","family":"Paruolo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2019,11,22]]},"reference":[{"key":"S0266466619000306_ref27","doi-asserted-by":"publisher","DOI":"10.2307\/2938258"},{"key":"S0266466619000306_ref9","doi-asserted-by":"publisher","DOI":"10.1257\/0002828053828581"},{"key":"S0266466619000306_ref29","doi-asserted-by":"publisher","DOI":"10.4159\/9780674369542"},{"key":"S0266466619000306_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-53858-1.00007-7"},{"key":"S0266466619000306_ref24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2008.03.001"},{"key":"S0266466619000306_ref17","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466612000345"},{"key":"S0266466619000306_ref1","first-page":"133","article-title":"The Chang-Kim-Park model of cointegrated density-valued time series cannot accommodate a stochastic trend","volume":"14","author":"Beare","year":"2017","journal-title":"Econ Journal Watch"},{"key":"S0266466619000306_ref25","doi-asserted-by":"publisher","DOI":"10.1201\/9781315117416"},{"key":"S0266466619000306_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7980-4"},{"key":"S0266466619000306_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1154-9"},{"key":"S0266466619000306_ref3","article-title":"Representation of I(1) and I(2) autoregressive Hilbertian processes","author":"Beare","year":"2019","journal-title":"Econometric Theory"},{"key":"S0266466619000306_ref16","doi-asserted-by":"publisher","DOI":"10.1017\/S026646669915401X"},{"key":"S0266466619000306_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2008.11.074"},{"key":"S0266466619000306_ref30","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2018.11.032"},{"key":"S0266466619000306_ref23","volume-title":"Likelihood-based Inference in Cointegrated Vector Auto-Regressive Models","author":"Johansen","year":"1996"},{"key":"S0266466619000306_ref28","first-page":"413","volume-title":"Nonparametric and Semiparametric Methods in Economics and Statistics","author":"Phillips","year":"1991b"},{"key":"S0266466619000306_ref12","doi-asserted-by":"publisher","DOI":"10.1515\/jtse-2012-0006"},{"key":"S0266466619000306_ref20","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-3655-3"},{"key":"S0266466619000306_ref7","unstructured":"Chang, Y. , Hu, B. , & Park, J. 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