{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T21:27:35Z","timestamp":1770672455970,"version":"3.49.0"},"reference-count":43,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T00:00:00Z","timestamp":1259625600000},"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":[[2009,12]]},"abstract":"<jats:p>This paper presents a family of simple nonparametric unit root tests indexed by one parameter,<jats:italic>d<\/jats:italic>, and containing the Breitung (2002,<jats:italic>Journal of Econometrics<\/jats:italic>108, 342\u2013363) test as the special case<jats:italic>d<\/jats:italic>= 1. It is shown that (a) each member of the family with<jats:italic>d<\/jats:italic>&gt; 0 is consistent, (b) the asymptotic distribution depends on<jats:italic>d<\/jats:italic>and thus reflects the parameter chosen to implement the test, and (c) because the asymptotic distribution depends on<jats:italic>d<\/jats:italic>and the test remains consistent for all<jats:italic>d<\/jats:italic>&gt; 0, it is possible to analyze the power of the test for different values of<jats:italic>d<\/jats:italic>. The usual Phillips\u2013Perron and Dickey\u2013Fuller type tests are indexed by bandwidth, lag length, etc., but have none of these three properties.<\/jats:p><jats:p>It is shown that members of the family with<jats:italic>d<\/jats:italic>&lt; 1 have higher asymptotic local power than the Breitung (2002) test, and when<jats:italic>d<\/jats:italic>is small the asymptotic local power of the proposed nonparametric test is relatively close to the parametric power envelope, particularly in the case with a linear time trend. Furthermore, generalized least squares (GLS) detrending is shown to improve power when<jats:italic>d<\/jats:italic>is small, which is not the case for the Breitung (2002) test. Simulations demonstrate that when applying a sieve bootstrap procedure, the proposed variance ratio test has very good size properties, with finite-sample power that is higher than that of the Breitung (2002) test and even rivals the (nearly) optimal parametric GLS detrended augmented Dickey\u2013Fuller test with lag length chosen by an information criterion.<\/jats:p>","DOI":"10.1017\/s0266466609990247","type":"journal-article","created":{"date-parts":[[2009,8,12]],"date-time":"2009-08-12T08:53:25Z","timestamp":1250067205000},"page":"1515-1544","source":"Crossref","is-referenced-by-count":23,"title":["A POWERFUL TEST OF THE AUTOREGRESSIVE UNIT ROOT HYPOTHESIS BASED ON A TUNING PARAMETER FREE STATISTIC"],"prefix":"10.1017","volume":"25","author":[{"given":"Morten \u00d8rregaard","family":"Nielsen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,12,1]]},"reference":[{"key":"S0266466609990247_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2007.01.019"},{"key":"S0266466609990247_ref27","first-page":"107","volume-title":"Advances in Econometrics","volume":"8","author":"Park","year":"1990"},{"key":"S0266466609990247_ref40","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2003.12.012"},{"key":"S0266466609990247_ref38","doi-asserted-by":"crossref","first-page":"2843","DOI":"10.1016\/S1573-4412(05)80016-9","volume-title":"Handbook of Econometrics","volume":"4","author":"Stock","year":"1994"},{"key":"S0266466609990247_ref37","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1765(92)90023-R"},{"key":"S0266466609990247_ref36","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1093\/oso\/9780199257294.003.0002","volume-title":"Time Series with Long Memory","author":"Robinson","year":"2003"},{"key":"S0266466609990247_ref35","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1994.10476881"},{"key":"S0266466609990247_ref14","doi-asserted-by":"publisher","DOI":"10.2307\/2171846"},{"key":"S0266466609990247_ref41","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1080\/07350015.1998.10524736","article-title":"Testing for a shift in mean without having to estimate serial-correlation parameters","volume":"16","author":"Vogelsang","year":"1998","journal-title":"Journal of Business and Economic Statistics"},{"key":"S0266466609990247_ref7","doi-asserted-by":"publisher","DOI":"10.1214\/009053607000000136"},{"key":"S0266466609990247_ref34","doi-asserted-by":"publisher","DOI":"10.1111\/1467-6419.00064"},{"key":"S0266466609990247_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600165028"},{"key":"S0266466609990247_ref8","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350492"},{"key":"S0266466609990247_ref42","doi-asserted-by":"publisher","DOI":"10.2307\/2998543"},{"key":"S0266466609990247_ref21","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-4149(99)00088-5"},{"key":"S0266466609990247_ref2","unstructured":"Akonom J. & Gourieroux C. 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