{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:52:19Z","timestamp":1791323539176,"version":"4.3.1"},"reference-count":22,"publisher":"Institute of Mathematical Statistics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"We propose a new method for identifying and estimating the CP-factor models for matrix time series. Unlike the generalized eigenanalysis-based method (J. R. Stat. Soc. Ser. B. Stat. Methodol. 85 (2023) 127\u2013148) for which the convergence rates of the associated estimators may suffer from small eigengaps as the asymptotic theory is based on some matrix perturbation analysis, the proposed new method enjoys faster convergence rates which are free from any eigengaps. It achieves this by turning the problem into a joint diagonalization of several matrices whose elements are determined by a basis of a linear system, and by choosing the basis carefully to avoid near colinearity (see Proposition 5 and Section 4.3). Furthermore, unlike the generalized eigenanalysis-based method which requires the two factor loading matrices to be full-ranked, the proposed new method can handle rank-deficient factor loading matrices. Illustration with both simulated and real matrix time-series data shows the advantages of the proposed new method.<\/jats:p>","DOI":"10.1214\/25-aos2608","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:50:21Z","timestamp":1782395421000},"page":"1372-1397","source":"Crossref","is-referenced-by-count":1,"title":["Identification and estimation for matrix time-series CP-factor models"],"prefix":"10.1214","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7933-4449","authenticated-orcid":false,"given":"Jinyuan","family":"Chang","sequence":"first","affiliation":[{"name":"Joint Laboratory of Data Science and Business Intelligence, Southwestern University of Finance and Economics"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yue","family":"Du","sequence":"additional","affiliation":[{"name":"Joint Laboratory of Data Science and Business Intelligence, Southwestern University of Finance and Economics"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guanglin","family":"Huang","sequence":"additional","affiliation":[{"name":"Joint Laboratory of Data Science and Business Intelligence, Southwestern University of Finance and Economics"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2065-8486","authenticated-orcid":false,"given":"Qiwei","family":"Yao","sequence":"additional","affiliation":[{"name":"Department of Statistics, London School of Economics and Political Science"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"16","doi-asserted-by":"publisher","unstructured":"Kolda, T. G.<\/small> and Bader, B. W.<\/small> (2009). Tensor decompositions and applications. SIAM Rev.<\/i> 51<\/b> 455\u2013500.","DOI":"10.1137\/07070111X"},{"key":"18","doi-asserted-by":"publisher","unstructured":"LAM, C.<\/small>, YAO, Q.<\/small> and BATHIA, N.<\/small> (2011). Estimation of latent factors for high-dimensional time series. Biometrika<\/i> 98<\/b> 901\u2013918.","DOI":"10.1093\/biomet\/asr048"},{"key":"10","doi-asserted-by":"publisher","unstructured":"DE LATHAUWER, L.<\/small> (2006). A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM J. Matrix Anal. Appl.<\/i> 28<\/b> 642\u2013666.","DOI":"10.1137\/040608830"},{"key":"8","doi-asserted-by":"publisher","unstructured":"CHEN, E. Y.<\/small>, TSAY, R. S.<\/small> and CHEN, R.<\/small> (2020). Constrained factor models for high-dimensional matrix-variate time series. J. Amer. Statist. Assoc.<\/i> 115<\/b> 775\u2013793.","DOI":"10.1080\/01621459.2019.1584899"},{"key":"9","doi-asserted-by":"publisher","unstructured":"CHEN, R.<\/small>, XIAO, H.<\/small> and YANG, D.<\/small> (2021). Autoregressive models for matrix-valued time series. J. Econometrics<\/i> 222<\/b> 539\u2013560.","DOI":"10.1016\/j.jeconom.2020.07.015"},{"key":"21","doi-asserted-by":"publisher","unstructured":"WANG, D.<\/small>, LIU, X.<\/small> and CHEN, R.<\/small> (2019). Factor models for matrix-valued high-dimensional time series. J. Econometrics<\/i> 208<\/b> 231\u2013248.","DOI":"10.1016\/j.jeconom.2018.09.013"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Lam, C.<\/small> and Yao, Q.<\/small> (2012). Factor modeling for high-dimensional time series: Inference for the number of factors. Ann. Statist.<\/i> 40<\/b> 694\u2013726.","DOI":"10.1214\/12-AOS970"},{"key":"1","doi-asserted-by":"publisher","unstructured":"AFSARI, B.<\/small> (2008). Sensitivity analysis for the problem of matrix joint diagonalization. SIAM J. Matrix Anal. Appl.<\/i> 30<\/b> 1148\u20131171.","DOI":"10.1137\/060655997"},{"key":"2","doi-asserted-by":"crossref","unstructured":"CHANG, J.<\/small>, DU, Y.<\/small>, HUANG, G.<\/small> and YAO, Q.<\/small> (2026). Supplement to \u201cIdentification and Estimation for Matrix Time-Series CP-factor Models.\u201d https:\/\/doi.org\/10.1214\/25-AOS2608SUPP<\/a>","DOI":"10.1214\/25-AOS2608SUPP"},{"key":"3","doi-asserted-by":"publisher","unstructured":"CHANG, J.<\/small>, GUO, B.<\/small> and YAO, Q.<\/small> (2015). High dimensional stochastic regression with latent factors, endogeneity and nonlinearity. J. Econometrics<\/i> 189<\/b> 297\u2013312.","DOI":"10.1016\/j.jeconom.2015.03.024"},{"key":"4","doi-asserted-by":"publisher","unstructured":"CHANG, J.<\/small>, GUO, B.<\/small> and YAO, Q.<\/small> (2018). Principal component analysis for second-order stationary vector time series. Ann. Statist.<\/i> 46<\/b> 2094\u20132124.","DOI":"10.1214\/17-AOS1613"},{"key":"5","unstructured":"CHANG, J.<\/small>, HE, J.<\/small>, LIN, C.<\/small> and YAO, Q.<\/small> (2024). HDTSA: An R package for high-dimensional time series analysis. Available at arXiv:2412.17341<\/a>."},{"key":"11","doi-asserted-by":"crossref","unstructured":"FAMA, E. F.<\/small> and MACBETH, J. D.<\/small> (1973). Risk, return, and equilibrium: Empirical tests. J. Polit. Econ.<\/i> 81<\/b> 607\u2013636.","DOI":"10.1086\/260061"},{"key":"12","doi-asserted-by":"publisher","unstructured":"HAN, Y.<\/small>, CHEN, R.<\/small>, YANG, D.<\/small> and ZHANG, C.-H.<\/small> (2024a). Tensor factor model estimation by iterative projection. Ann. Statist.<\/i> 52<\/b> 2641\u20132667.","DOI":"10.1214\/24-aos2412"},{"key":"14","doi-asserted-by":"publisher","unstructured":"HAN, Y.<\/small>, YANG, D.<\/small>, ZHANG, C.-H.<\/small> and CHEN, R.<\/small> (2024b). CP factor model for dynamic tensors. J. R. Stat. Soc. Ser. B. Stat. Methodol.<\/i> 86<\/b> 1383\u20131413.","DOI":"10.1093\/jrsssb\/qkae036"},{"key":"19","doi-asserted-by":"publisher","unstructured":"PHAM, D.-T.<\/small> and CARDOSO, J.-F.<\/small> (2001). Blind separation of instantaneous mixtures of nonstationary sources. IEEE Trans. Signal Process.<\/i> 49<\/b> 1837\u20131848.","DOI":"10.1109\/78.942614"},{"key":"20","doi-asserted-by":"crossref","unstructured":"VOLLGRAF, R.<\/small> and OBERMAYER, K.<\/small> (2006). Quadratic optimization for simultaneous matrix diagonalization. IEEE Trans. Signal Process.<\/i> 54<\/b> 3270\u20133278.","DOI":"10.1109\/TSP.2006.877673"},{"key":"22","unstructured":"ZIEHE, A.<\/small>, LASKOV, P.<\/small>, NOLTE, G.<\/small> and M\u00dcLLER, K.-R.<\/small> (2004). A fast algorithm for joint diagonalization with non-orthogonal transformations and its application to blind source separation. J. Mach. Learn. Res.<\/i> 5<\/b> 777\u2013800."},{"key":"13","doi-asserted-by":"publisher","unstructured":"HAN, Y.<\/small>, CHEN, R.<\/small> and ZHANG, C.-H.<\/small> (2022). Rank determination in tensor factor model. Electron. J. Stat.<\/i> 16<\/b> 1726\u20131803.","DOI":"10.1214\/22-ejs1991"},{"key":"15","doi-asserted-by":"publisher","unstructured":"HAN, Y.<\/small> and ZHANG, C.-H.<\/small> (2023). Tensor principal component analysis in high dimensional CP models. IEEE Trans. Inf. Theory<\/i> 69<\/b> 1147\u20131167.","DOI":"10.1109\/tit.2022.3203972"},{"key":"6","doi-asserted-by":"publisher","unstructured":"CHANG, J.<\/small>, HE, J.<\/small>, YANG, L.<\/small> and YAO, Q.<\/small> (2023). Modelling matrix time series via a tensor CP-decomposition. J. R. Stat. Soc. Ser. B. Stat. Methodol.<\/i> 85<\/b> 127\u2013148.","DOI":"10.1093\/jrsssb\/qkac011"},{"key":"7","doi-asserted-by":"crossref","unstructured":"CHEN, E. Y.<\/small> and CHEN, R.<\/small> (2022). Modeling dynamic transport network with matrix factor models: An application to international trade flow. J. Data Sci.<\/i> 21<\/b> 490\u2013507.","DOI":"10.6339\/22-JDS1065"}],"container-title":["The Annals of Statistics"],"original-title":[],"deposited":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:19:24Z","timestamp":1791321564000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-54\/issue-3\/Identification-and-estimation-for-matrix-time-series-CP-factor-models\/10.1214\/25-AOS2608.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,1]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,6,1]]}},"URL":"https:\/\/doi.org\/10.1214\/25-aos2608","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,1]]}}}