{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T20:25:57Z","timestamp":1777407957464,"version":"3.51.4"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2005,7,19]],"date-time":"2005-07-19T00:00:00Z","timestamp":1121731200000},"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":[[2005,8]]},"abstract":"<jats:p>In this paper, we prove the validity of an Edgeworth expansion to the \ndistribution of the Whittle maximum likelihood estimator for stationary \nlong-memory Gaussian models with unknown parameter \n<jats:inline-formula>\n\t      <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:href=\"S0266466605050383ffm001.gif\"\/>\n\t    <\/jats:inline-formula>. \nThe error of the (<jats:italic>s<\/jats:italic> \u2212 2)-order expansion is shown to be \n<jats:italic>o<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>\u2212(<jats:italic>s<\/jats:italic>\u22122)\/2<\/jats:sup>)\u2014the \nusual independent and identically distributed rate\u2014for a wide range \nof models, including the popular ARFIMA(<jats:italic>p<\/jats:italic>,<jats:italic>d<\/jats:italic>,<jats:italic>q<\/jats:italic>) \nmodels. The expansion is valid under mild assumptions on the behavior of \nthe spectral density and its derivatives in the neighborhood of the \norigin. As a by-product, we generalize a theorem by Fox and Taqqu (1987, <jats:italic>Probability Theory and Related Fields<\/jats:italic> \n74, 213\u2013240) concerning the asymptotic behavior of Toeplitz \nmatrices.<\/jats:p>\n\t  <jats:p>Lieberman, Rousseau, and Zucker (2003, \n<jats:italic>Annals of Statistics<\/jats:italic> 31, 586\u2013612) establish a valid \nEdgeworth expansion for the maximum likelihood estimator for stationary \nlong-memory Gaussian models. For a significant class of models, their \nexpansion is shown to have an error of \n<jats:italic>o<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>\u22121<\/jats:sup>). The results given here improve \nupon those of Lieberman et al. in that the results provide an Edgeworth \nexpansion for an asymptotically efficient estimator, as Lieberman et al. \ndo, but the error of the expansion is shown to be \n<jats:italic>o<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>\u2212(<jats:italic>s<\/jats:italic>\u22122)\/2<\/jats:sup>), not \n<jats:italic>o<\/jats:italic>(<jats:italic>n<\/jats:italic><jats:sup>\u22121<\/jats:sup>), for a broad range of \nmodels.<\/jats:p>","DOI":"10.1017\/s0266466605050383","type":"journal-article","created":{"date-parts":[[2005,7,26]],"date-time":"2005-07-26T17:17:06Z","timestamp":1122398226000},"page":"710-734","source":"Crossref","is-referenced-by-count":16,"title":["VALID EDGEWORTH EXPANSIONS FOR THE WHITTLE MAXIMUM LIKELIHOOD \nESTIMATOR FOR STATIONARY LONG-MEMORY GAUSSIAN TIME SERIES"],"prefix":"10.1017","volume":"21","author":[{"given":"Donald W.K.","family":"Andrews","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Offer","family":"Lieberman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2005,7,19]]},"reference":[{"key":"S0266466605050383_ref014","doi-asserted-by":"crossref","unstructured":"Lahiri, S.N. 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(1988)Asymptotic expansions of the distributions of some test statisticsfor Gaussian ARMA processes.Journal of Multivariate Analysis 27,494\u2013511.","DOI":"10.1016\/0047-259X(88)90144-3"},{"key":"S0266466605050383_ref005","doi-asserted-by":"crossref","unstructured":"Durbin, J. (1980)Approximations for densities of sufficient statistics.Biometrika 67,311\u2013333.","DOI":"10.1093\/biomet\/67.2.311"},{"key":"S0266466605050383_ref010","doi-asserted-by":"crossref","unstructured":"Granger, C.W.J. & R. Joyeux (1980)An introduction to long-memory time series and fractionaldifferencing.Journal of Time Series Analysis 1,15\u201330.","DOI":"10.1111\/j.1467-9892.1980.tb00297.x"},{"key":"S0266466605050383_ref019","unstructured":"Searle, S.R. (1971)Linear Models.Wiley."},{"key":"S0266466605050383_ref016","doi-asserted-by":"crossref","unstructured":"Lieberman, O. , J. Rousseau , & D.M. 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Taqqu (1986)Large sample properties of parameter estimates for stronglydependent stationary Gaussian time series.Annals of Statistics 14,517\u2013532.","DOI":"10.1214\/aos\/1176349936"},{"key":"S0266466605050383_ref011","unstructured":"Grenander, U. & G. Szeg\u00f6 (1956)Toeplitz Forms and Their Applications.University of California Press.2nd ed., 1984, Chelsea."},{"key":"S0266466605050383_ref004","doi-asserted-by":"crossref","unstructured":"Dahlhaus, R. (1989)Efficient parameter estimation for self-similar processes.Annals of Statistics 17,1749\u20131766.","DOI":"10.1214\/aos\/1176347393"},{"key":"S0266466605050383_ref012","doi-asserted-by":"crossref","unstructured":"Hosking, J.R.M. (1981)Fractional differencing.Biometrika 68,165\u2013176.","DOI":"10.1093\/biomet\/68.1.165"},{"key":"S0266466605050383_ref009","doi-asserted-by":"crossref","unstructured":"G\u00f6tze, F. & C. Hipp (1994)Asymptotic distribution of statistics in time series.Annals of Statistics 22,2062\u20132088.","DOI":"10.1214\/aos\/1176325772"},{"key":"S0266466605050383_ref008","doi-asserted-by":"crossref","unstructured":"G\u00f6tze, F. & C. Hipp (1983)Asymptotic expansions for sums of weakly dependent randomvectors.Zeitschrift f\u00fcr Wahrscheinlichskeitstheorie und VerwandteGebiete 64,211\u2013239.","DOI":"10.1007\/BF01844607"},{"key":"S0266466605050383_ref002","unstructured":"Beran, J. (1994)Statistics for Long-Memory Processes.Chapman and Hall."},{"key":"S0266466605050383_ref025","doi-asserted-by":"crossref","unstructured":"Taniguchi, M. & Y. Kakizawa (2000)Asymptotic Theory of Statistical Inference for Time Series.Springer.","DOI":"10.1007\/978-1-4612-1162-4"},{"key":"S0266466605050383_ref003","doi-asserted-by":"crossref","unstructured":"Bhattacharya, R.N. & J.K. Ghosh (1978)On the validity of the formal Edgeworth expansion.Annals of Statistics 6,434\u2013451.","DOI":"10.1214\/aos\/1176344134"},{"key":"S0266466605050383_ref018","doi-asserted-by":"crossref","unstructured":"Robinson, P.M. (1995)Log periodogram regression of time series with long rangedependence.Annals of Statistics 23,1048\u20131072.","DOI":"10.1214\/aos\/1176324636"},{"key":"S0266466605050383_ref021","doi-asserted-by":"crossref","unstructured":"Taniguchi, M. (1984)Validity of Edgeworth expansions for statistics of timeseries.Journal of Time Series Analysis 5,37\u201351.","DOI":"10.1111\/j.1467-9892.1984.tb00377.x"},{"key":"S0266466605050383_ref007","doi-asserted-by":"crossref","unstructured":"Fox, R. & M.S. Taqqu (1987)Central limit theorems for quadratic forms in random variableshaving long-range dependence.Probability Theory and Related Fields 74,213\u2013240.","DOI":"10.1007\/BF00569990"},{"key":"S0266466605050383_ref013","doi-asserted-by":"crossref","unstructured":"Hurst, H.E. (1951)Long-term storage capacity of reservoirs.Transactions of the American Society of Civil Engineers 116,770\u2013808.","DOI":"10.1061\/TACEAT.0006518"},{"key":"S0266466605050383_ref001","unstructured":"Andrews, D.W.K. , O. Lieberman , & V. Marmer (2005)Higher-order improvements of the parametric bootstrap forlong-memory Gaussian processes.Journal of Econometrics,forthcoming.Cowles Foundation Discussion paper 1378, Yale University.Available at http:\/\/cowles.econ.yale.edu."},{"key":"S0266466605050383_ref022","doi-asserted-by":"crossref","unstructured":"Taniguchi, M. (1986)Third order asymptotic properties of maximum likelihood estimatorsfor Gaussian ARMA processes.Journal of Multivariate Analysis 18,1\u201331.","DOI":"10.1016\/0047-259X(86)90055-2"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466605050383","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,21]],"date-time":"2025-06-21T03:58:40Z","timestamp":1750478320000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466605050383\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,7,19]]},"references-count":25,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2005,12]]}},"alternative-id":["S0266466605050383"],"URL":"https:\/\/doi.org\/10.1017\/s0266466605050383","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,7,19]]}}}