Constrained Factor Models for High-Dimensional Matrix-Variate Time Series

Abstract

High-dimensional matrix-variate time series data are becoming widely available in many scientific fields, such as economics, biology, and meteorology. To achieve significant dimension reduction while preserving the intrinsic matrix structure and temporal dynamics in such data, Wang, Liu, and Chen proposed a matrix factor model, that is, shown to be able to provide effective analysis. In this article, we establish a general framework for incorporating domain and prior knowledge in the matrix factor model through linear constraints. The proposed framework is shown to be useful in achieving parsimonious parameterization, facilitating interpretation of the latent matrix factor, and identifying specific factors of interest. Fully utilizing the prior-knowledge-induced constraints results in more efficient and accurate modeling, inference, dimension reduction as well as a clear and better interpretation of the results. Constrained, multi-term, and partially constrained factor models for matrix-variate time series are developed, with efficient estimation procedures and their asymptotic properties. We show that the convergence rates of the constrained factor loading matrices are much faster than those of the conventional matrix factor analysis under many situations. Simulation studies are carried out to demonstrate finite-sample performance of the proposed method and its associated asymptotic properties. We illustrate the proposed model with three applications, where the constrained matrix-factor models outperform their unconstrained counterparts in the power of variance explanation under the out-of-sample 10-fold cross-validation setting. Supplementary materials for this article are available online.

Keywords:

• Constrained eigen-analysis
• Convergence in L2-norm
• Dimension reduction
• Factor model
• Matrix-variate time series

Additional information

Funding

EYC supported in part by NSF Grants DMS-1803241. RST supported in part by the Booth School of Business, University of Chicago. RC supported in part by NSF Grants DMS-1503409, DMS-1737857, DMS-1803241, and IIS-1741390.