Identification and estimation for matrix time series CP-factor models

The demand of modeling and forecasting high-dimensional matrix time series brings the opportunities with challenges. Let Y t " p y i,j,tq be a p ˆ q matrix recorded at time t, where y i,j,trepresents the value of, for example, the j -th variable on the i -th individual at time t . A popular approach to model Y t in the existing literature is via the so-called Tucker decomposition, namely the matrix Tucker-factor model. See, for example, Wang et al. (2019), Chen et al. (2020) and Chen and Chen (2022). It represents a high-dimensional matrix time series as a linear combination of a lower-dimensional matrix process. The Tucker decomposition can be viewed as a natural extension of the factor model for vector time series considered in Lam and Yao (2012) and Chang et al. (2015). Similarly we can only identify the factor loading spaces (the linear spaces spanned by the columns of the factor loading matrices) in the matrix Tucker-factor model while the factor loading matrices themselves are not uniquely defined. Parallel to the Tucker decomposition for matrix time series Y t, Chang et al. (2023) and Han et al. (2024) consider to model Y t via the so-called canonical polyadic (CP) decomposition, namely the matrix CP-factor model. It provides a more comprehensive dimensionality reduction as the dynamic structure of a matrix time series is driven by a vector process rather than a matrix process. Furthermore the factor loading matrices in the matrix CP-factor model can be identified uniquely upto the column reflection and permutation indeterminacy under some regularity conditions. The CP-factor model for matrix time series specified in Chang et al. (2023) admits the form Y t " AX tB J ` ε t, t ě 1, (1) where X t " diag p x tq with x t " p x t, 1, .. .