Helping Effects Against Curse of Dimensionality in Threshold Factor Models for Matrix Time Series

As is known, factor analysis is a popular method to reduce dimension for high-dimensional data. For matrix data, the dimension reduction can be more effectively achieved through both row and column directions. In this paper, we introduce a threshold factor models to analyze matrix-valued high-dimensional time series data. The factor loadings are allowed to switch between regimes, controlling by a threshold variable. The estimation methods for loading spaces, threshold value, and the number of factors are proposed. The asymptotic properties of these estimators are investigated. Not only the strengths of thresholding and factors, but also their interactions from different directions and different regimes play an important role on the estimation performance. When the thresholding and factors are all strong across regimes, the estimation is immune to the impact that the increase of dimension brings, which breaks the curse of dimensionality. When the thresholding in two directions and factors across regimes have different levels of strength, we show that estimators for loadings and threshold value experience 'helping' effects against the curse of dimensionality. We also discover that even when the numbers of factors are overestimated, the estimators are still consistent. The proposed methods are illustrated with both simulated and real examples.