{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T06:00:41Z","timestamp":1770962441400,"version":"3.50.1"},"reference-count":21,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,9,15]],"date-time":"2014-09-15T00:00:00Z","timestamp":1410739200000},"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":[[2015,6]]},"abstract":"<jats:p>The purpose of this paper is to differentiate between several asymptotically valid methods for confidence set construction for the autoregressive coefficient in AR(1) models. We show that the nonparametric grid bootstrap procedure suggested by Hansen (1999,<jats:italic>Review of Economics and Statistics<\/jats:italic>81, 594\u2013607) achieves a second order refinement in the local-to-unity asymptotic approach when compared with a modified version of Stock\u2019s (1991,<jats:italic>Journal of Monetary Economics<\/jats:italic>28, 435\u2013459) and Andrews\u2019 (1993,<jats:italic>Econometrica<\/jats:italic>61, 139\u2013165) grid testing procedures. We establish a second order expansion of the<jats:italic>t<\/jats:italic>-statistic in an AR(1) model in the local-to-unity asymptotic approach, which differs drastically from the usual Edgeworth-type expansions by approximating the statistic around a nonstandard and nonpivotal limit.<\/jats:p>","DOI":"10.1017\/s0266466614000383","type":"journal-article","created":{"date-parts":[[2014,9,15]],"date-time":"2014-09-15T14:08:36Z","timestamp":1410790116000},"page":"426-448","source":"Crossref","is-referenced-by-count":7,"title":["SECOND ORDER EXPANSION OF THE<i>T<\/i>-STATISTIC IN AR(1) MODELS"],"prefix":"10.1017","volume":"31","author":[{"given":"Anna","family":"Mikusheva","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,9,15]]},"reference":[{"key":"S0266466614000383_ref17","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1093\/biomet\/74.3.535","article-title":"Toward a unified asymptotic theory for autoregression","volume":"74","author":"Phillips","year":"1987","journal-title":"Biometrika"},{"key":"S0266466614000383_ref12","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1017\/S0266466611000636","article-title":"Unit roots in white noise","volume":"28","author":"Onatski","year":"2012","journal-title":"Econometric Theory"},{"key":"S0266466614000383_ref11","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1214\/154957804100000060","article-title":"The Skorokhod embedding problem and its offspring","volume":"1","author":"Obloj","year":"2004","journal-title":"Probability Surveys"},{"key":"S0266466614000383_ref15","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1017\/S0266466600004126","article-title":"Asymptotic expansions in nonstationary vector autoregressions","volume":"3","author":"Phillips","year":"1987","journal-title":"Econometric Theory"},{"key":"S0266466614000383_ref2","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1080\/07350015.1994.10510007","article-title":"Approximately median-unbiased estimation of autoregressive models","volume":"12","author":"Andrews","year":"1994","journal-title":"Journal of Business and Economic Statistics"},{"key":"S0266466614000383_ref18","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1002\/jae.3950060411","article-title":"Bayesian routes and unit roots: De Rebus Prioribus Semper Est Disputandum","volume":"6","author":"Phillips","year":"1991","journal-title":"Journal of Applied Econometrics"},{"key":"S0266466614000383_ref3","doi-asserted-by":"crossref","first-page":"721","DOI":"10.3982\/ECTA7015","article-title":"Hybrid and size-corrected subsample methods","volume":"77","author":"Andrews","year":"2009","journal-title":"Econometrica"},{"key":"S0266466614000383_ref5","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1214\/aos\/1176348142","article-title":"Bootstrapping unstable first-order autoregressive processes","volume":"19","author":"Basawa","year":"1991","journal-title":"Annals of Statistics"},{"key":"S0266466614000383_ref9","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.1214\/aop\/1176990334","article-title":"Weak limit theorems for stochastic integrals and stochastic differential equations","volume":"19","author":"Kurtz","year":"1991","journal-title":"Annals of Probability"},{"key":"S0266466614000383_ref4","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1017\/S0266466609100051","article-title":"Asymptotic size and a problem with subsampling and with the m out of n bootstrap","volume":"26","author":"Andrews","year":"2010","journal-title":"Econometric Theory"},{"key":"S0266466614000383_ref21","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/0304-3932(91)90034-L","article-title":"Confidence intervals for the largest autoregressive root in US macroeconomic time series","volume":"28","author":"Stock","year":"1991","journal-title":"Journal of Monetary Economics"},{"key":"S0266466614000383_ref13","doi-asserted-by":"crossref","first-page":"1845","DOI":"10.1111\/1468-0262.00471","article-title":"Bootstrap unit root tests","volume":"71","author":"Park","year":"2003","journal-title":"Econometrica"},{"key":"S0266466614000383_ref19","doi-asserted-by":"crossref","unstructured":"Phillips, P.C.B . (2012) On confidence intervals for autoregressive roots and predictive regression. Unpublished manuscript.","DOI":"10.2139\/ssrn.2150172"},{"key":"S0266466614000383_ref20","volume-title":"Studies in the Theory of Random Processes","author":"Skorokhod","year":"1965"},{"key":"S0266466614000383_ref14","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1016\/j.jeconom.2005.06.009","article-title":"A bootstrap theory for weakly integrated processes","volume":"133","author":"Park","year":"2006","journal-title":"Journal of Econometrics"},{"key":"S0266466614000383_ref10","doi-asserted-by":"crossref","first-page":"1411","DOI":"10.1111\/j.1468-0262.2007.00798.x","article-title":"Uniform inference in autoregressive models","volume":"75","author":"Mikusheva","year":"2007","journal-title":"Econometrica"},{"key":"S0266466614000383_ref1","doi-asserted-by":"crossref","first-page":"139","DOI":"10.2307\/2951781","article-title":"Exactly median-unbiased estimation of first order autoregressive\/unit root models","volume":"61","author":"Andrews","year":"1993","journal-title":"Econometrica"},{"key":"S0266466614000383_ref6","doi-asserted-by":"crossref","first-page":"1709","DOI":"10.1214\/aos\/1176351063","article-title":"Edgeworth correction by bootstrap in autoregressions","volume":"16","author":"Bose","year":"1988","journal-title":"Annals of Statistics"},{"key":"S0266466614000383_ref8","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1162\/003465399558463","article-title":"The grid bootstrap and the autoregressive model","volume":"81","author":"Hansen","year":"1999","journal-title":"Review of Economics and Statistics"},{"key":"S0266466614000383_ref16","doi-asserted-by":"crossref","first-page":"277","DOI":"10.2307\/1913237","article-title":"Time series regression with a unit root","volume":"55","author":"Phillips","year":"1987","journal-title":"Econometrica"},{"key":"S0266466614000383_ref7","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971286","volume-title":"Probability","author":"Breiman","year":"1992"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466614000383","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,2]],"date-time":"2024-06-02T14:00:23Z","timestamp":1717336823000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466614000383\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,15]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,6]]}},"alternative-id":["S0266466614000383"],"URL":"https:\/\/doi.org\/10.1017\/s0266466614000383","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,9,15]]}}}