{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T16:57:50Z","timestamp":1787504270328,"version":"build-2736575974"},"reference-count":36,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2023,3,11]],"date-time":"2023-03-11T00:00:00Z","timestamp":1678492800000},"content-version":"vor","delay-in-days":3802,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-0804937"],"award-info":[{"award-number":["DMS-0804937"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1104545"],"award-info":[{"award-number":["DMS-1104545"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Science and Engineering Research Council of Canada"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2013,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>The paper is concerned with inference for linear models with fixed regressors and weakly dependent stationary time series errors. Theoretically, we obtain asymptotic normality for the M-estimator of the regression parameter under mild conditions and establish a uniform Bahadur representation for recursive M-estimators. Methodologically, we extend the recently proposed self-normalized approach of Shao from stationary time series to the regression set-up, where the sequence of response variables is typically non-stationary in mean. Since the limiting distribution of the self-normalized statistic depends on the design matrix and its corresponding critical values are case dependent, we develop a simulation-based approach to approximate the critical values consistently. Through a simulation study, we demonstrate favourable finite sample performance of our method in comparison with a block-bootstrap-based approach. Empirical illustrations using two real data sets are also provided.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2012.01044.x","type":"journal-article","created":{"date-parts":[[2012,10,12]],"date-time":"2012-10-12T06:32:51Z","timestamp":1350023571000},"page":"323-343","source":"Crossref","is-referenced-by-count":37,"title":["Inference for Linear Models with Dependent Errors"],"prefix":"10.1093","volume":"75","author":[{"given":"Zhou","family":"Zhou","sequence":"first","affiliation":[{"name":"University of Toronto , Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaofeng","family":"Shao","sequence":"additional","affiliation":[{"name":"University of Illinois at Urbana\u2013Champaign , Champaign , USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2012,10,12]]},"reference":[{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"547","DOI":"10.1007\/BF01845702","article-title":"Strong representations for LAD estimators in linear models","volume":"83","author":"Babu","year":"1989","journal-title":"Probab. Theor. Reltd Flds"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316962","volume-title":"Convergence of Probability Measures, 2nd edn","author":"Billingsley","year":"1999"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1198\/073500104000000631","article-title":"Powerful trend function tests that are robust to strong serial correlation, with an application to the Prebisch: Singer hypothesis","volume":"23","author":"Bunzel","year":"2005","journal-title":"J. Bus. Econ. Statist."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.spl.2003.10.018","article-title":"M-estimation for linear models with spatially-correlated errors","volume":"66","author":"Cui","year":"2004","journal-title":"Statist. Probab. Lett."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198522065.001.0001","volume-title":"Time Series, a Biostatistical Introduction","author":"Diggle","year":"1990"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/S0304-4076(97)00058-4","article-title":"The moving blocks bootstrap and robust inference for linear least squares and quantile regressions","volume":"82","author":"Fitzenberger","year":"1997","journal-title":"J. Econmetr."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1017\/S0266466609090719","article-title":"Nonstandard quantile-regression inference","volume":"25","author":"Goh","year":"2009","journal-title":"Econometr. Theor."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1198\/016214504000002087","article-title":"Bootstrap standard error estimates for linear regression","volume":"100","author":"Goncalves","year":"2005","journal-title":"J. Am. Statist. Ass."},{"key":"2023031201060674000_","first-page":"1","article-title":"Primary commodity prices, manufactured goods prices, and the terms of trade of developing countries","volume":"2","author":"Grilli","year":"1988","journal-title":"Wrld Bnk Econ Rev."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1111\/j.2517-6161.1996.tb02111.x","article-title":"On sample reuse methods for dependent data","volume":"58","author":"Hall","year":"1996","journal-title":"J. R. Statist. Soc. B"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"1302","DOI":"10.1016\/j.jeconom.2007.02.005","article-title":"A simple, robust and powerful test of the trend hypothesis","volume":"141","author":"Harvey","year":"2007","journal-title":"J. Econmetr."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0304-4076(03)00154-4","article-title":"An alternative bootstrap to moving blocks for time series regression models","volume":"117","author":"Hidalgo","year":"2003","journal-title":"J. Econmetr."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1111\/j.1468-0262.2004.00517.x","article-title":"The error rejection probability of simple autocorrelation robust tests","volume":"72","author":"Jansson","year":"2004","journal-title":"Econometrica"},{"key":"2023031201060674000_","first-page":"1130","article-title":"A new asymptotic theory for heteroskedasticity-autocorrelation robust tests","volume":"21","author":"Kiefer","year":"2005","journal-title":"Econmetr. Theor."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1111\/1468-0262.00128","article-title":"Simple robust testing of regression hypotheses","volume":"68","author":"Kiefer","year":"2000","journal-title":"Econometrica"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511754098","volume-title":"Quantile Regression","author":"Koenker","year":"2005"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1214\/aos\/1176343892","article-title":"Behavior of robust estimators in the regression model with dependent errors","volume":"5","author":"Koul","year":"1977","journal-title":"Ann. Statist."},{"key":"2023031201060674000_","volume-title":"Robust tests of hypotheses in models with m-estimation","author":"Lee","year":"2006"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"1066","DOI":"10.1198\/016214501753208726","article-title":"Testing that a dependent process is uncorrelated","volume":"96","author":"Lobato","year":"2001","journal-title":"J. Am. Statist. Ass."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1111\/1467-9361.00050","article-title":"A general test of the Prebisch-Singer hypothesis","volume":"3","author":"Lutz","year":"1999","journal-title":"Rev. Devlpmnt Econ."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.jeconom.2009.03.011","article-title":"Estimating deterministic trends with an integrated or stationary noise component","volume":"151","author":"Perron","year":"2009","journal-title":"J. Econmetr."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1017\/S0266466600004709","article-title":"A shortcut to LAD estimator asymptotics","volume":"7","author":"Phillips","year":"1991","journal-title":"Econmetr. Theor."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1214\/aos\/1176343738","article-title":"Robust estimation in dependent situations","volume":"5","author":"Portnoy","year":"1977","journal-title":"Ann. Statist."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1214\/aos\/1176344568","article-title":"Further remarks on robust estimation in dependent situations","volume":"7","author":"Portnoy","year":"1979","journal-title":"Ann. Statist."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/0047-259X(91)90034-Y","article-title":"Asymptotic behavior of regression quantiles in non-stationary, dependence cases","volume":"38","author":"Portnoy","year":"1991","journal-title":"J. Multiv. Anal."},{"key":"2023031201060674000_","first-page":"367","article-title":"Asymptotic behavior of M-estimators for the linear model with dependent errors","volume":"9","author":"Prakasa Rao","year":"1981","journal-title":"Bull. Inst. Math. Acad. Sin."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1111\/j.1467-9892.1980.tb00300.x","article-title":"State-dependent models: a general approach to non-linear time series analysis","volume":"1","author":"Priestley","year":"1980","journal-title":"J. Time Ser. Anal."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1080\/10485250600687812","article-title":"Improved nonparametric confidence intervals in time series regression","volume":"18","author":"Romano","year":"2006","journal-title":"J. Nonparam. Statist."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1111\/j.1467-9868.2009.00737.x","article-title":"A self-normalized approach to confidence interval construction in time series","volume":"72","author":"Shao","year":"2010","journal-title":"J. R. Statist. Soc. B"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1080\/01621459.1997.10474060","article-title":"Subseries methods in regression","volume":"92","author":"Sherman","year":"1997","journal-title":"J. Am. Statist. Ass."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1111\/j.0012-9682.2008.00822.x","article-title":"Optimal bandwidth selection in heteroscedasticity-autocorrelation robust testing","volume":"76","author":"Sun","year":"2008","journal-title":"Econometrica"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"123","DOI":"10.2307\/2998543","article-title":"Trend function hypothesis testing in the presence of serial correlation","volume":"66","author":"Vogelsang","year":"1998","journal-title":"Econometrica"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"14150","DOI":"10.1073\/pnas.0506715102","article-title":"Nonlinear system theory: another look at dependence","volume":"102","author":"Wu","year":"2005","journal-title":"Proc. Natn. Acad. Sci. USA"},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1214\/009053606000001406","article-title":"M-estimation of linear models with dependent errors","volume":"35","author":"Wu","year":"2007","journal-title":"Ann. Statist."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"2294","DOI":"10.1214\/009117907000000060","article-title":"Strong invariance principles for dependent random variables","volume":"35","author":"Wu","year":"2007","journal-title":"Ann. Probab."},{"key":"2023031201060674000_","doi-asserted-by":"crossref","first-page":"2696","DOI":"10.1214\/08-AOS636","article-title":"Local linear quantile estimation for non-stationary time series","volume":"37","author":"Zhou","year":"2009","journal-title":"Ann. Statist."}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fj.1467-9868.2012.01044.x","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/75\/2\/323\/49510373\/jrsssb_75_2_323.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/75\/2\/323\/49510373\/jrsssb_75_2_323.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T03:04:20Z","timestamp":1714446260000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/75\/2\/323\/7075418"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,12]]},"references-count":36,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2012,10,12]]},"published-print":{"date-parts":[[2013,3,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.1467-9868.2012.01044.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2013,3]]},"published":{"date-parts":[[2012,10,12]]}}}