{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T18:20:14Z","timestamp":1789669214068,"version":"4.0.1"},"reference-count":33,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":4882,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1995,10]]},"abstract":"<jats:p>The paper develops a statistical theory for regressions with integrated regressors of unknown order and unknown cointegrating dimension. In practice, we are often unsure whether unit roots or cointegration is present in time series data, and we are also uncertain about the order of integration in some cases. This paper addresses issues of estimation and inference in cases of such uncertainty. Phillips (1995,<jats:italic>Econometrica<\/jats:italic>63, 1023\u20131078) developed a theory for time series regressions with an unknown mixture of 1(0) and 1(1) variables and established that the method of<jats:italic>fully modified ordinary least squares<\/jats:italic>(FM-OLS) is applicable to models (including vector autoregressions) with some unit roots and unknown cointegrating rank. This paper extends these results to models that contain some I(0), I(1), and I(2) regressors. The theory and methods here are applicable to cointegrating regressions that include unknown numbers of I(0), I(1), and I(2) variables and an unknown degree of cointegration. Such models require a somewhat different approach than that of Phillips (1995). The paper proposes a residual-based fully modified ordinary least-squares (RBFMOLS) procedure, which employs residuals from a first-order autoregression of the first differences of the entire regressor set in the construction of the FMOLS estimator. The asymptotic theory for the RBFM-OLS estimator is developed and is shown to be normal for all the stationary coefficients and mixed normal for all the nonstationary coefficients. Under Gaussian assumptions, estimation of the cointegration space by RBFM-OLS is optimal even though the dimension of the space is unknown.<\/jats:p>","DOI":"10.1017\/s0266466600009968","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T15:40:44Z","timestamp":1236699644000},"page":"1033-1094","source":"Crossref","is-referenced-by-count":13,"title":["Time Series Regression with Mixtures of Integrated Processes"],"prefix":"10.1017","volume":"11","author":[{"given":"Yoosoon","family":"Chang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter C.B.","family":"Phillips","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600009968_ref032","volume-title":"Estimation of Cointegration Vectors with Linear Restrictions","author":"Saikkonen","year":"1992"},{"key":"S0266466600009968_ref031","volume-title":"Spectral Analysis and Time Series","volume":"I and II","author":"Priestley","year":"1981"},{"key":"S0266466600009968_ref030","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176348666"},{"key":"S0266466600009968_ref029","doi-asserted-by":"publisher","DOI":"10.2307\/2298004"},{"key":"S0266466600009968_ref021","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177706962"},{"key":"S0266466600009968_ref019","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600013402"},{"key":"S0266466600009968_ref016","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600009014"},{"key":"S0266466600009968_ref012","doi-asserted-by":"publisher","DOI":"10.2307\/2938278"},{"key":"S0266466600009968_ref015","first-page":"819","article-title":"Stochastic trends and economic fluctuations","volume":"81","author":"King","year":"1991","journal-title":"American Economic Review"},{"key":"S0266466600009968_ref013","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600012755"},{"key":"S0266466600009968_ref010","doi-asserted-by":"publisher","DOI":"10.1017\/CCOL0521345529.002"},{"key":"S0266466600009968_ref009","first-page":"87","volume-title":"Nonstationary Time Series and Cointegration","author":"Hargreaves","year":"1993"},{"key":"S0266466600009968_ref026","first-page":"967","article-title":"Fully modified least squares in I(2) regression","volume":"10","author":"Phillips","year":"1994","journal-title":"Econometric Theory"},{"key":"S0266466600009968_ref008","first-page":"225","article-title":"Estimation and inference in models of cointegration: A simulation study","volume":"8","author":"Hansen","year":"1991","journal-title":"Advances in Econometrics"},{"key":"S0266466600009968_ref006","article-title":"Some properties of time series data and their use in econometric model specification","volume":"2","author":"Granger","year":"1981","journal-title":"Journal of Econometrics"},{"key":"S0266466600009968_ref002","volume-title":"Fully Modified Estimation of Cointegrated Systems with 1(2) Processes","author":"Chang","year":"1993"},{"key":"S0266466600009968_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/2938229"},{"key":"S0266466600009968_ref025","doi-asserted-by":"publisher","DOI":"10.2307\/2171721"},{"key":"S0266466600009968_ref022","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4076(86)90001-1"},{"key":"S0266466600009968_ref014","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600009026"},{"key":"S0266466600009968_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/2951679"},{"key":"S0266466600009968_ref004","doi-asserted-by":"publisher","DOI":"10.2307\/1913236"},{"key":"S0266466600009968_ref003","volume-title":"Fully Modified Regressions with an Unknown Mixture of I(0) and I(2) Regressors","author":"Chang","year":"1994"},{"key":"S0266466600009968_ref023","doi-asserted-by":"publisher","DOI":"10.2307\/2938258"},{"key":"S0266466600009968_ref017","volume-title":"Fully Modified IV, GIVE, and GMM Estimation with Possibly Nonstationary Regressors and Instruments","author":"Kitamura","year":"1992"},{"key":"S0266466600009968_ref007","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316429"},{"key":"S0266466600009968_ref028","doi-asserted-by":"publisher","DOI":"10.2307\/2297545"},{"key":"S0266466600009968_ref024","first-page":"413","volume-title":"Nonparametric and Semiparameteric Methods in Econometrics and Statistics","author":"Phillips","year":"1991"},{"key":"S0266466600009968_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1889(88)90041-3"},{"key":"S0266466600009968_ref005","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1093\/oso\/9780198283393.003.0012","volume-title":"Long-Run Economic Relations: Readings in Cointegration","author":"Engle","year":"1991"},{"key":"S0266466600009968_ref033","doi-asserted-by":"publisher","DOI":"10.2307\/2951763"},{"key":"S0266466600009968_ref027","doi-asserted-by":"publisher","DOI":"10.2307\/2297602"},{"key":"S0266466600009968_ref020","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600012287"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600009968","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,11]],"date-time":"2024-03-11T09:26:36Z","timestamp":1710149196000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600009968\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,10]]},"references-count":33,"journal-issue":{"issue":"5","published-print":{"date-parts":[[1995,10]]}},"alternative-id":["S0266466600009968"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600009968","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,10]]}}}