{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,3]],"date-time":"2026-10-03T22:58:06Z","timestamp":1791068286464,"version":"4.1.0"},"reference-count":0,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[1999,8,1]],"date-time":"1999-08-01T00:00:00Z","timestamp":933465600000},"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":[[1999,8]]},"abstract":"<jats:p>This paper deals with a scalar <jats:italic>I<\/jats:italic>(<jats:italic>d<\/jats:italic>) process \n{<jats:italic>y<\/jats:italic><jats:sub><jats:italic>j<\/jats:italic><\/jats:sub>}, where the integration order \n<jats:italic>d<\/jats:italic> is any real number. Under this setting, we first explore \nasymptotic properties of various statistics associated with \n{<jats:italic>y<\/jats:italic><jats:sub><jats:italic>j<\/jats:italic><\/jats:sub>}, assuming that <jats:italic>d<\/jats:italic> is known \nand is greater than or equal to \u00bd. Note that \n{<jats:italic>y<\/jats:italic><jats:sub><jats:italic>j<\/jats:italic><\/jats:sub>} becomes stationary when \n<jats:italic>d<\/jats:italic> &lt; \u00bd, whose case is not our concern here. It turns \nout that the case of <jats:italic>d<\/jats:italic> = \u00bd needs a separate treatment \nfrom <jats:italic>d<\/jats:italic> &gt; \u00bd. We then consider, under the normality \nassumption, testing and estimation for <jats:italic>d<\/jats:italic>, allowing for any \nvalue of <jats:italic>d<\/jats:italic>. The tests suggested here are asymptotically \nuniformly most powerful invariant, whereas the maximum likelihood \nestimator is asymptotically efficient. The asymptotic theory for \nthese results will not assume normality. Unlike in the usual unit \nroot problem based on autoregressive models, standard asymptotic \nresults hold for test statistics and estimators, where <jats:italic>d<\/jats:italic> \nneed not be restricted to <jats:italic>d<\/jats:italic> \u2265 \u00bd. Simulation \nexperiments are conducted to examine the finite sample performance \nof both the tests and estimators.<\/jats:p>","DOI":"10.1017\/s0266466699154045","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T13:47:49Z","timestamp":1027777669000},"page":"549-582","source":"Crossref","is-referenced-by-count":147,"title":["THE NONSTATIONARY FRACTIONAL UNIT ROOT"],"prefix":"10.1017","volume":"15","author":[{"given":"Katsuto","family":"Tanaka","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[1999,8,1]]},"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466699154045","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,2]],"date-time":"2019-09-02T16:16:37Z","timestamp":1567440997000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466699154045\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,8]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1999,8]]}},"alternative-id":["S0266466699154045"],"URL":"https:\/\/doi.org\/10.1017\/s0266466699154045","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,8]]}}}