Statistical inference for large-dimensional tensor factor model by iterative projections

Tensor Factor Models (TFM) are appealing dimension reduction tools for high-order large-dimensional tensor time series, and have wide applications in economics, finance and medical imaging. In this paper, we propose a projection estimator for the Tucker-decomposition based TFM, and provide its least-square interpretation which parallels to the least-square interpretation of the Principal Component Analysis (PCA) for the vector factor model. The projection technique simultaneously reduces the dimensionality of the signal component and the magnitudes of the idiosyncratic component tensor, thus leading to an increase of the signal-to-noise ratio. We derive a convergence rate of the projection estimator of the loadings and the common factor tensor which are faster than that of the naive PCA-based estimator. Our results are obtained under mild conditions which allow the idiosyncratic components to be weakly cross- and auto- correlated. We also provide a novel iterative procedure based on the eigenvalue-ratio principle to determine the factor numbers. Extensive numerical studies are conducted to investigate the empirical performance of the proposed projection estimators relative to the state-of-the-art ones.