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Tensor time series data appears naturally in a lot of fields, includingfinance and economics. As a major dimension reduction tool, similar to itsfactor model counterpart, the idiosyncratic components of a tensor time seriesfactor model can exhibit serial correlations, especially in financial andeconomic applications. This rules out a lot of state-of-the-art methods thatassume white idiosyncratic components, or even independent/Gaussian data. Whilethe traditional higher order orthogonal iteration (HOOI) is proved to beconvergent to a set of factor loading matrices, the closeness of them to thetrue underlying factor loading matrices are in general not established, or onlyunder some strict circumstances like having i.i.d. Gaussian noises (Zhang andXia, 2018). Under the presence of serial and cross-correlations in theidiosyncratic components and time series variables with only bounded fourthorder moments, we propose a pre-averaging method that accumulates informationfrom tensor fibres for better estimating all the factor loading spaces. Theestimated directions corresponding to the strongest factors are then used forprojecting the data for a potentially improved re-estimation of the factorloading spaces themselves, with theoretical guarantees and rate of convergencespelt out. We also propose a new rank estimation method which utilizescorrelation information from the projected data, in the same spirit as Fan, Guoand Zheng (2022) for factor models with independent data. Extensive simulationresults reveal competitive performance of our rank and factor loadingestimators relative to other state-of-the-art or traditional alternatives. Aset of matrix-valued portfolio return data is also analyzed.

Rank and Factor Loadings Estimation in Time Series Tensor Factor Model by Pre-averaging