Research

2026

1. Multilevel Main Effects Matrix Factor Model

   Kaixin Liu, Clifford Lam, and Zetai Cen

   arXiv preprint arXiv:2610.18335, 2026

   Abs HTML PDF

   We propose a multilevel main effects matrix factor model (MMEFM) to extract meaningful row and column effects from groups of matrix data. As a generalization of a multilevel matrix factor model, we provide rigorous definitions of MMEFM and identifications, together with iterative algorithms for the estimators of all global and local components in the main effects and common component, including global/local main effects and core ranks, with rates of convergence spelt out. An important feature of the model is that all main effects can be nonstationary and still be consistently estimated at each time point. At the same time, our estimated global common factors are alignment independent, and hence are truly global, and can be presented and given interpretations more easily. We propose a Bootstrap procedure for detecting the existence and identifying the groups of data that are driven by the global common factors. The power of the test, together with the accuracy of various estimators, are demonstrated in extensive simulation settings. Estimation performance is also compared to other state-of-the-art methods for matrix time series by simulations and a set of U.S. equity data. All numerical works are done using the R package MMEFM, which is available at CRAN.

2. Main Effect Factor Models in High-Dimensional Matrix Time Series: Identification and Sparsity

   Zetai Cen^*, Kaixin Liu^*, and Clifford Lam

   arXiv preprint arXiv:2508.12510, 2026

   * Equal contribution.

   Abs HTML PDF

   We propose a general identification framework for main effect factor models for matrix-valued time series. The classical sum-to-zero restriction on the row and column main effects is replaced by a broad class of shift-varying functions, which includes weighted averages, quantiles, and reference-unit anchors as special cases. We prove that the shift-varying property is necessary and sufficient for parameter identification, and we derive closed-form estimators under minimal assumptions. Asymptotic convergence rates of all estimated components are derived under weak, heterogeneous factor strengths. Building on this flexible identification, we address sparsity in the main effects by selecting a minimum-based shift-varying function that sets the smallest main effect to zero, and we introduce a doubly adaptive fused Lasso estimator that consistently recovers the true sparse and dense blocks. A modified Mallows’s statistic is developed for tuning parameter selection, and the entire procedure is computationally practical via an equivalent generalized lasso formulation. Simulation experiments are performed under a variety of settings, showing our proposed methods work well. Two real applications on employment growth and economic indices are demonstrated to illustrate the empirical value of our approach.