{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,24]],"date-time":"2026-07-24T06:06:49Z","timestamp":1784873209329,"version":"3.55.0"},"reference-count":40,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2017,7,24]],"date-time":"2017-07-24T00:00:00Z","timestamp":1500854400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["www.cambridge.org"],"crossmark-restriction":true},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2018,10]]},"abstract":"<jats:p>This article revisits the asymptotic inference for nonstationary AR(1) models of Phillips and Magdalinos (2007a) by incorporating a structural change in the AR parameter at an unknown time<jats:italic>k<\/jats:italic><jats:sub>0<\/jats:sub>. Consider the model<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline1\"\/><jats:tex-math>${y_t} = {\\beta _1}{y_{t - 1}}I\\{ t \\le {k_0}\\} + {\\beta _2}{y_{t - 1}}I\\{ t &gt; {k_0}\\} + {\\varepsilon _t},t = 1,2, \\ldots ,T$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where<jats:italic>I<\/jats:italic>{\u00b7} denotes the indicator function, one of<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline2\"\/><jats:tex-math>${\\beta _1}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline3\"\/><jats:tex-math>${\\beta _2}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>depends on the sample size<jats:italic>T<\/jats:italic>, and the other is equal to one. We examine four cases: Case (I):<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline4\"\/><jats:tex-math>${\\beta _1} = {\\beta _{1T}} = 1 - c\/{k_T}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline5\"\/><jats:tex-math>${\\beta _2} = 1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>; (II):<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline6\"\/><jats:tex-math>${\\beta _1} = 1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline7\"\/><jats:tex-math>${\\beta _2} = {\\beta _{2T}} = 1 - c\/{k_T}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>; (III):<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline8\"\/><jats:tex-math>${\\beta _1} = 1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline9\"\/><jats:tex-math>${\\beta _2} = {\\beta _{2T}} = 1 + c\/{k_T}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>; and case (IV):<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline10\"\/><jats:tex-math>${\\beta _1} = {\\beta _{1T}} = 1 + c\/{k_T}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline11\"\/><jats:tex-math>${\\beta _2} = 1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where<jats:italic>c<\/jats:italic>is a fixed positive constant, and<jats:italic>k<\/jats:italic><jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>is a sequence of positive constants increasing to \u221e such that<jats:italic>k<\/jats:italic><jats:sub><jats:italic>T<\/jats:italic><\/jats:sub>=<jats:italic>o<\/jats:italic>(<jats:italic>T<\/jats:italic>). We derive the limiting distributions of the<jats:italic>t<\/jats:italic>-ratios of<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline12\"\/><jats:tex-math>${\\beta _1}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and<jats:inline-formula><jats:alternatives><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0266466617000317_inline13\"\/><jats:tex-math>${\\beta _2}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>and the least squares estimator of the change point for the cases above under some mild conditions. Monte Carlo simulations are conducted to examine the finite-sample properties of the estimators. Our theoretical findings are supported by the Monte Carlo simulations.<\/jats:p>","DOI":"10.1017\/s0266466617000317","type":"journal-article","created":{"date-parts":[[2017,7,24]],"date-time":"2017-07-24T03:55:40Z","timestamp":1500868540000},"page":"985-1017","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":15,"title":["STRUCTURAL CHANGE IN NONSTATIONARY AR(1) MODELS"],"prefix":"10.1017","volume":"34","author":[{"given":"Tianxiao","family":"Pang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Terence","family":"Tai-Leung Chong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Danna","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yanling","family":"Liang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2017,7,24]]},"reference":[{"key":"S0266466617000317_ref6","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.2010.00714.x"},{"key":"S0266466617000317_ref8","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176350492"},{"key":"S0266466617000317_ref24","doi-asserted-by":"publisher","DOI":"10.2307\/1891113"},{"key":"S0266466617000317_ref37","doi-asserted-by":"publisher","DOI":"10.1111\/j.1368-423X.2004.00120.x"},{"key":"S0266466617000317_ref4","doi-asserted-by":"publisher","DOI":"10.1111\/j.1368-423X.2008.00237.x"},{"key":"S0266466617000317_ref36","doi-asserted-by":"publisher","DOI":"10.3982\/QE82"},{"key":"S0266466617000317_ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2012.01.024"},{"key":"S0266466617000317_ref23","doi-asserted-by":"publisher","DOI":"10.1086\/294632"},{"key":"S0266466617000317_ref17","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-012-9306-7"},{"key":"S0266466617000317_ref11","volume-title":"An Introduction to Probability Theory and its Applications","volume":"2","author":"Feller","year":"1971"},{"key":"S0266466617000317_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2012.05.024"},{"key":"S0266466617000317_ref19","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1176347498"},{"key":"S0266466617000317_ref40","first-page":"1","volume-title":"Cointegration, Causality and Forcasting: A Festschrift in Honour of Clive. 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