{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:34:19Z","timestamp":1759638859318,"version":"3.40.5"},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T00:00:00Z","timestamp":1657756800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2023,10]]},"abstract":"<jats:p>Functional coefficient (FC) regressions allow for systematic flexibility in the responsiveness of a dependent variable to movements in the regressors, making them attractive in applications where marginal effects may depend on covariates. Such models are commonly estimated by local kernel regression methods. This paper explores situations where responsiveness to covariates is locally flat or fixed. The paper develops new asymptotics that take account of shape characteristics of the function in the locality of the point of estimation. Both stationary and integrated regressor cases are examined. The limit theory of FC kernel regression is shown to depend intimately on functional shape in ways that affect rates of convergence, optimal bandwidth selection, estimation, and inference. In FC cointegrating regression, flat behavior materially changes the limit distribution by introducing the shape characteristics of the function into the limiting distribution through variance as well as centering. In the boundary case where the number of zero derivatives tends to infinity, near parametric rates of convergence apply in stationary and nonstationary cases. Implications for inference are discussed and a feasible pre-test inference procedure is proposed that takes unknown potential flatness into consideration and provides a practical approach to inference.<\/jats:p>","DOI":"10.1017\/s0266466622000287","type":"journal-article","created":{"date-parts":[[2022,7,14]],"date-time":"2022-07-14T09:55:00Z","timestamp":1657792500000},"page":"900-949","source":"Crossref","is-referenced-by-count":2,"title":["LIMIT THEORY FOR LOCALLY FLAT FUNCTIONAL COEFFICIENT REGRESSION"],"prefix":"10.1017","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2341-0451","authenticated-orcid":false,"given":"Peter C. B.","family":"Phillips","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ying","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2022,7,14]]},"reference":[{"key":"S0266466622000287_r17","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.jeconom.2011.05.013","article-title":"Measuring correlations of integrated but not cointegrated variables: A semiparametric approach","volume":"164","author":"Sun","year":"2011","journal-title":"Journal of Econometrics"},{"key":"S0266466622000287_r21","unstructured":"Wang, Y. and Phillips, P.C.B. (2022) Limit theory of local $p$ -th order polynomial estimator in functional coefficient cointegrating regression. Working Paper, Renmin University of China."},{"key":"S0266466622000287_r13","unstructured":"Phillips, P.C.B. and Magdalinos, T. (2009) Econometric inference in the vicinity of unity. CoFie Working Paper 7, Singapore Management University."},{"key":"S0266466622000287_r9","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1017\/S0266466615000109","article-title":"Uniform consistency of nonstationary kernel-weighted sample covariances for nonparametric regression","volume":"32","author":"Li","year":"2016","journal-title":"Econometric Theory"},{"key":"S0266466622000287_r11","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1111\/1467-9469.00056","article-title":"Local polynomial estimation of regression functions for mixing processes","volume":"24","author":"Masry","year":"1997","journal-title":"Scandinavian Journal of Statistics"},{"key":"S0266466622000287_r19","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1080\/07474938.2019.1624402","article-title":"Adaptive estimation of heteroskedastic functional-coefficient regressions with an application to fiscal policy evaluation on asset markets","volume":"39","author":"Tu","year":"2020","journal-title":"Econometric Reviews"},{"key":"S0266466622000287_r7","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1017\/S0266466608080365","article-title":"Regression asymptotics using martingale convergence methods","volume":"24","author":"Ibragimov","year":"2008","journal-title":"Econometric Theory"},{"key":"S0266466622000287_r15","doi-asserted-by":"crossref","unstructured":"Phillips, P. C. B. and Wang, Y. (2021) When bias contributes to variance: True limit theory in functional coefficient cointegrating regression. Journal of Econometrics, published online 20 October 2021. doi: 10.1016\/j.jeconom.2021.09.007.","DOI":"10.1016\/j.jeconom.2021.09.007"},{"volume-title":"Nonlinear Time Series: Nonparametric and Parametric Methods","year":"2008","author":"Fan","key":"S0266466622000287_r3"},{"key":"S0266466622000287_r12","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.jeconom.2016.09.013","article-title":"Estimating smooth structural change in cointegration models","volume":"196","author":"Phillips","year":"2017","journal-title":"Journal of Econometrics"},{"key":"S0266466622000287_r20","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1016\/j.jeconom.2020.12.010","article-title":"Spurious functional-coefficient regression models and robust inference with marginal integration","volume":"229","author":"Tu","year":"2022","journal-title":"Journal of Econometrics"},{"key":"S0266466622000287_r22","unstructured":"Wang, Y. , Phillips, P.C.B. , and Tu, Y. (2019) Limit theory and inference in non-cointegrated functional coefficient regression. Working Paper, University of Auckland."},{"key":"S0266466622000287_r23","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.econlet.2016.10.023","article-title":"Improving inflation prediction with the quantity theory","volume":"149","author":"Wang","year":"2016","journal-title":"Economics Letters"},{"key":"S0266466622000287_r24","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.jeconom.2009.01.008","article-title":"Functional-coefficient cointegration models","volume":"152","author":"Xiao","year":"2009","journal-title":"Journal of Econometrics"},{"key":"S0266466622000287_r4","doi-asserted-by":"crossref","unstructured":"Gao, J. and Phillips, P.C.B. (2013) Functional coefficient nonstationary regression with non-and semi-parametric cointegration. Working Paper.","DOI":"10.2139\/ssrn.2327604"},{"key":"S0266466622000287_r2","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/j.jeconom.2008.10.003","article-title":"Functional-coefficient models for nonstationary time series data","volume":"148","author":"Cai","year":"2009","journal-title":"Journal of Econometrics"},{"key":"S0266466622000287_r1","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1080\/01621459.2000.10474284","article-title":"Functional-coefficient regression models for nonlinear time series","volume":"95","author":"Cai","year":"2000","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466622000287_r6","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1093\/biomet\/78.2.263","article-title":"On optimal data-based bandwidth selection in kernel density estimation","volume":"78","author":"Hall","year":"1991","journal-title":"Biometrika"},{"key":"S0266466622000287_r8","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1111\/j.1368-423X.2005.00160.x","article-title":"Functional-coefficient models under unit root behaviour","volume":"8","author":"Juhl","year":"2005","journal-title":"The Econometrics Journal"},{"key":"S0266466622000287_r14","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1214\/aos\/1176348666","article-title":"Asymptotics for linear processes","volume":"20","author":"Phillips","year":"1992","journal-title":"Annals of Statistics"},{"key":"S0266466622000287_r5","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1007\/BF00363516","article-title":"Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation","volume":"74","author":"Hall","year":"1987","journal-title":"Probability Theory and Related Fields"},{"key":"S0266466622000287_r18","doi-asserted-by":"crossref","first-page":"1401","DOI":"10.1111\/obes.12309","article-title":"Functional coefficient cointegration models subject to time\u2013varying volatility with an application to the purchasing power parity","volume":"81","author":"Tu","year":"2019","journal-title":"Oxford Bulletin of Economics and Statistics"},{"volume-title":"Nonparametric Econometrics: Theory and Practice","year":"2007","author":"Li","key":"S0266466622000287_r10"},{"key":"S0266466622000287_r16","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1111\/j.1467-9892.1983.tb00368.x","article-title":"Nonparametric estimators for time series","volume":"4","author":"Robinson","year":"1983","journal-title":"Journal of Time Series Analysis"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466622000287","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,28]],"date-time":"2023-09-28T09:10:19Z","timestamp":1695892219000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466622000287\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,14]]},"references-count":24,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["S0266466622000287"],"URL":"https:\/\/doi.org\/10.1017\/s0266466622000287","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"type":"print","value":"0266-4666"},{"type":"electronic","value":"1469-4360"}],"subject":[],"published":{"date-parts":[[2022,7,14]]}}}