{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T14:16:02Z","timestamp":1779286562263,"version":"3.51.4"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2011,11,25]],"date-time":"2011-11-25T00:00:00Z","timestamp":1322179200000},"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":[[2012,6]]},"abstract":"<jats:p>We show that the empirical distribution of the roots of the vector autoregression (VAR) of order <jats:italic>p<\/jats:italic> fitted to <jats:italic>T<\/jats:italic> observations of a general stationary or nonstationary process converges to the uniform distribution over the unit circle on the complex plane, when both <jats:italic>T<\/jats:italic> and <jats:italic>p<\/jats:italic> tend to infinity so that (ln <jats:italic>T<\/jats:italic>)\/<jats:italic>p<\/jats:italic> \u2192 0 and <jats:italic>p<\/jats:italic><jats:sup>3<\/jats:sup>\/<jats:italic>T<\/jats:italic> \u2192 0. In particular, even if the process is a white noise, nearly all roots of the estimated VAR will converge by absolute value to unity. For fixed <jats:italic>p<\/jats:italic>, we derive an asymptotic approximation to the expected empirical distribution of the estimated roots as <jats:italic>T<\/jats:italic> \u2192 \u221e. The approximation is concentrated in a circular region in the complex plane for various data generating processes and sample sizes.<\/jats:p>","DOI":"10.1017\/s0266466611000636","type":"journal-article","created":{"date-parts":[[2011,11,25]],"date-time":"2011-11-25T10:43:46Z","timestamp":1322217826000},"page":"485-508","source":"Crossref","is-referenced-by-count":6,"title":["UNIT ROOTS IN WHITE NOISE"],"prefix":"10.1017","volume":"28","author":[{"given":"Alexei","family":"Onatski","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Harald","family":"Uhlig","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2011,11,25]]},"reference":[{"key":"S0266466611000636_ref19","doi-asserted-by":"publisher","DOI":"10.1002\/jae.3950060411"},{"key":"S0266466611000636_ref21","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600007179"},{"key":"S0266466611000636_ref22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-02-06340-2"},{"key":"S0266466611000636_ref3","first-page":"141","volume-title":"NBER Macroeconomics Annual 1991","volume":"6","author":"Campbell","year":"1991"},{"key":"S0266466611000636_ref16","doi-asserted-by":"crossref","unstructured":"Nielsen B. & Nielsen H.B. (2008) Properties of Estimated Characteristic Roots. Discussion paper 08\u201313. University of Copenhagen, Department of Economics.","DOI":"10.2139\/ssrn.1146453"},{"key":"S0266466611000636_ref1","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176342709"},{"key":"S0266466611000636_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511840371"},{"key":"S0266466611000636_ref5","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.2006.00468.x"},{"key":"S0266466611000636_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(85)90027-2"},{"key":"S0266466611000636_ref4","doi-asserted-by":"publisher","DOI":"10.2307\/1969500"},{"key":"S0266466611000636_ref11","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.2003.00328.x"},{"key":"S0266466611000636_ref13","first-page":"25","article-title":"Forecasting with Bayesian vector autoregressions: Five years of experience","volume":"4","author":"Litterman","year":"1986","journal-title":"Journal of Business and Economic Statistics"},{"key":"S0266466611000636_ref10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02422945"},{"key":"S0266466611000636_ref14","first-page":"168","volume-title":"Cointegration, Causality, and Forecasting. A Festschrift in Honour of Clive W.J. Granger","author":"L\u00fctkepohl","year":"1999"},{"key":"S0266466611000636_ref9","doi-asserted-by":"publisher","DOI":"10.1112\/S0010437X07003302"},{"key":"S0266466611000636_ref17","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/80.4.919"},{"key":"S0266466611000636_ref20","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600010720"},{"key":"S0266466611000636_ref23","first-page":"40","article-title":"On the distribution of roots of random polynomials","volume":"3","author":"Shparo","year":"1962","journal-title":"Vestnik Moskovskogo Universiteta, Series 1: Mathematics and Mechanics"},{"key":"S0266466611000636_ref24","first-page":"59","article-title":"\u00dcber die Nullstellen der Polynomen, die in einem Kreise gleichm\u00e4ssig konvergieren","volume":"21","author":"Szeg\u00f6","year":"1922","journal-title":"Sitzungsberichte der Berliner Mathematischen Gesellschaft"},{"key":"S0266466611000636_ref2","first-page":"588","article-title":"Effects of not knowing the order of an autoregressive process on the mean squared error of prediction\u2014I","volume":"76","author":"Bhansali","year":"1981","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466611000636_ref18","doi-asserted-by":"publisher","DOI":"10.2307\/2938169"},{"key":"S0266466611000636_ref7","volume-title":"The Statistical Theory of Linear Systems","author":"Hannan","year":"1988"},{"key":"S0266466611000636_ref6","first-page":"89","volume-title":"Proceedings of the Third Berkeley Symposium on Probability and Statistics","volume":"II","author":"Hammersley","year":"1956"},{"key":"S0266466611000636_ref15","doi-asserted-by":"crossref","first-page":"979","DOI":"10.3982\/ECTA6814","article-title":"Testing models of low-frequency variability","volume":"76","author":"M\u00fcller","year":"2008","journal-title":"Econometrica"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466611000636","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T20:55:48Z","timestamp":1556225748000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466611000636\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,11,25]]},"references-count":24,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,6]]}},"alternative-id":["S0266466611000636"],"URL":"https:\/\/doi.org\/10.1017\/s0266466611000636","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,11,25]]}}}