{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T15:23:26Z","timestamp":1648740206058},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":6922,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:p>Let {<jats:bold>X<\/jats:bold>(<jats:italic>t<\/jats:italic>)} be a multivariate Gaussian stationary process with the spectral density matrix <jats:italic>f<\/jats:italic><jats:sub>0<\/jats:sub>(\u03c9), where \u03b8 is an unknown parameter vector. Using a quasi-maximum likelihood estimator <jats:private-char description=\"\u03b8\u0302\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600004928_char1\"><jats:alt-text>\u03b8\u0302<\/jats:alt-text><\/jats:inline-graphic><\/jats:private-char> of \u03b8, we estimate the spectral density matrix <jats:italic>f<\/jats:italic><jats:sub>0<\/jats:sub>(\u03c9) by <jats:italic>f<\/jats:italic><jats:private-char description=\"\u03b8\u0302\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600004928_char2\"><jats:alt-text>\u03b8\u0302<\/jats:alt-text><\/jats:inline-graphic><\/jats:private-char>(\u03c9). Then we derive asymptotic expansions of the distributions of functions of <jats:italic>f<\/jats:italic><jats:private-char description=\"\u03b8\u0302\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600004928_char2\"><jats:alt-text>\u03b8\u0302<\/jats:alt-text><\/jats:inline-graphic><\/jats:private-char>(\u03c9). Also asymptotic expansions for the distributions of functions of the eigenvalues of <jats:italic>f<\/jats:italic><jats:private-char description=\"\u03b8\u0302\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600004928_char2\"><jats:alt-text>\u03b8\u0302<\/jats:alt-text><\/jats:inline-graphic><\/jats:private-char>(\u03c9) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of <jats:private-char description=\"\u03b8\u0302\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600004928_char1\"><jats:alt-text>\u03b8\u0302<\/jats:alt-text><\/jats:inline-graphic><\/jats:private-char>, the estimated coherency, and contribution ratio in the principal component analysis based on <jats:private-char description=\"\u03b8\u0302\"><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600004928_char1\"><jats:alt-text>\u03b8\u0302<\/jats:alt-text><\/jats:inline-graphic><\/jats:private-char> in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.<\/jats:p>","DOI":"10.1017\/s0266466600004928","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T15:48:17Z","timestamp":1236700097000},"page":"75-96","source":"Crossref","is-referenced-by-count":2,"title":["Asymptotic Expansions of the Distributions of Statistics Related to the Spectral Density Matrix in Multivariate Time Series and Their Applications"],"prefix":"10.1017","volume":"6","author":[{"given":"Masanobu","family":"Taniguchi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Koichi","family":"Maekawa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600004928_ref003","volume-title":"Time series: data analysis and theory","author":"Brillinger","year":"1975"},{"key":"S0266466600004928_ref018","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600011002"},{"key":"S0266466600004928_ref017","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176346066"},{"key":"S0266466600004928_ref010","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316429"},{"key":"S0266466600004928_ref011","volume-title":"The economic analysis of time series","author":"Harvey","year":"1981"},{"key":"S0266466600004928_ref019","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(86)90055-2"},{"key":"S0266466600004928_ref001","unstructured":"1. Amos D.E. & Koopmans L.H. . Tables of the distribution of the coefficient of coherence of stationary bivariate Gaussian process. Sandia Corp. Monograph SCR-483. Sandia Corp. (The table is reprinted in Koopmans [12].)"},{"key":"S0266466600004928_ref002","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/56.2.375"},{"key":"S0266466600004928_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(82)90081-1"},{"key":"S0266466600004928_ref005","doi-asserted-by":"publisher","DOI":"10.4159\/harvard.9780674334076"},{"key":"S0266466600004928_ref006","volume-title":"Multivariate Analysis IV","author":"Fujikoshi","year":"1977"},{"key":"S0266466600004928_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(79)90082-4"},{"key":"S0266466600004928_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(78)90020-9"},{"key":"S0266466600004928_ref008","volume-title":"Introduction to statistical time series","author":"Fuller","year":"1976"},{"key":"S0266466600004928_ref009","volume-title":"Spectral analysis of time series","author":"Granger","year":"1964"},{"key":"S0266466600004928_ref012","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176345696"},{"key":"S0266466600004928_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1889(88)90040-1"},{"key":"S0266466600004928_ref013","volume-title":"The spectral analysis of time series","author":"Koopmans","year":"1974"},{"key":"S0266466600004928_ref016","volume-title":"Multivariate Analysis-Ill","author":"Priestley","year":"1973"},{"key":"S0266466600004928_ref020","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(87)90083-2"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600004928","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:49:05Z","timestamp":1558216145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600004928\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":20,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0266466600004928"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600004928","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}