  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. 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 analysed and our test suggests that MEFM is
indeed necessary for analysing the data against a traditional FM.
