Matrix-valued factor model with time-varying main effects

Abstract

We introduce the matrix-valued time-varying Main Effects Factor Model (MEFM). MEFM is a generalization to the traditional matrix-valued factor model (FM). We give rigorous definitions of MEFM and its identifications, and propose estimators for the time-varying grand mean, row and column main effects, and the row and column factor loading matrices for the common component. Rates of convergence for different estimators are spelt out, with asymptotic normality shown. The core rank estimator for the common component is also proposed, with consistency of the estimators presented. As time series, the row and column main effects $\left\{{\alpha }_t\right\}$ and $\left\{{\beta }_t\right\}$ can be non-stationary without affecting the estimation accuracy of our estimators. The number of main effects factors contributing to row or column main effects is also consistently estimated by our proposed estimators. We propose a test for testing if FM is sufficient against the alternative that MEFM is necessary, and demonstrate the power of such a test in various simulation settings. We also demonstrate numerically the accuracy of our estimators in extended simulation experiments. A set of NYC Taxi traffic data is analyzed and our test suggests that MEFM is indeed necessary for analyzing the data against a traditional FM.