Sparsity of the Main Effect Matrix Factor Model

We introduce sparsity detection and estimation in main effect matrix factor models for matrix-valued time series. A carefully chosen set of identification conditions for the common component and the potentially nonstationary main effects is proposed to strengthen the interpretations of sparse main effects, while estimators of all model components are presented. Sparse estimation of the latent main effects is proposed using a doubly adaptive fused lasso estimation to allow for sparse sub-block detection, with theoretical guarantees and rates of convergence spelt out for the final estimators. Sparse block consistency for the main effects is also proved as a result. A realized Mallow's $C_p$ is developed for tuning parameter selection, with practical implementation described. Simulation experiments are performed under a variety of settings, showing our proposed estimators work well. A set of NYC taxi traffic data is analyzed, clearly showing the effects of Covid-19 lockdown, with prolonged sparse main effects detected.