In the paper, the autoregressive moving average model for matrix time series (MARMA) is investigated. The properties of the MARMA model are investigated by using the conditional least square estimation, the conditional maximum likelihood estimation, the projection theorem in Hilbert space and the decomposition technique of time series, which include necessary and sufficient conditions for stationarity and invertibility, model parameter estimation, model testing and model forecasting.

References

• Chen, R., Xiao, H., & Yang, D. (2021). Autoregressive models for matrix-valued time series. Journal of Econometrics, 222(1), 539–560. https://doi.org/10.1016/j.jeconom.2020.07.015  
• Getmanov, V. G., Chinkin, V. E., Sidorov, R. V., Gvishiani, A. D., Dobrovolsky, M. N., Soloviev, A. A., Dmitrieva, A. N., Kovylyaeva, A. A., & Yashin, I. I. (2021). Methods for recognition of local anisotropy in muon fluxes in the URAGAN hodoscope matrix data time series. Physics of Atomic Nuclei, 84(6), 1080–1086. https://doi.org/10.1134/S106377882113010X  
• Graham, A. (2018). Kronecker products and matrix calculus with applications. Dover Publications Inc.  
• Karl, W., & Simar, L. (2015). Applied multivariate statistical analysis (4th ed.). Springer-Verlag.  
• Samadi, S. (2014). Matrix time series analysis [PhD Dissertation]. The University of Georgia.  
• Walden, A., & Serroukh, A. (2002). Wavelet analysis of matrix-valued time series. Proceedings: Mathematical, Physical and Engineering Sciences, 458(2017), 157–179. 
• Wang, D., Liu, X., & Chen, R. (2019). Factor models for matrix-valued high-dimensional time series. Journal of Econometrics, 208(1), 231–248. https://doi.org/10.1016/j.jeconom.2018.09.013
• Wang, H., & West, M. (2009). Bayesian analysis of matrix normal graphical models. Biometrika, 96(4), 821–834. https://doi.org/10.1093/biomet/asp049 
• Wu, S. J., & Hua, N. (2022). Autoregressive model of time series with matrix cross-section data (in Chinese). Acta Mathematica Sinica, 65(6), 1093–1104.  
• Zhou, L. H., Du, G. W., Tao, D. P., Chen, H. M., Cheng, J., & Gong, L. B. (2018). Clustering multivariate time series data via multi-nonnegative matrix factorization in multi-relational networks. IEEE Access, 2018(6), 74747–74761. https://doi.org/10.1109/Access.6287639 doi: 10.1109/ACCESS.2018.2882798. 

To cite this article: Shujin Wu & Ping Bi (04 Oct 2023): Autoregressive moving average model for matrix time series, Statistical Theory and Related Fields, DOI: 10.1080/24754269.2023.2262360 To link to this article: https://doi.org/10.1080/24754269.2023.2262360