Matrix Quantile Factor Model

In this paper, we introduce a matrix quantile factor model for matrix sequence data analysis. For matrix-valued data with a low-rank structure, we estimate the row and column factor spaces via minimizing the empirical check loss function over all panels. We show that the estimates converge at rate $1/\min\{\sqrt{p_1p_2}, \sqrt{p_2T}, \sqrt{p_1T}\}$ in the sense of average Frobenius norm, where $p_1$, $p_2$ and $T$ are the row dimensionality, column dimensionality and length of the matrix sequence, respectively. This rate is faster than that of the quantile estimates via ``flattening" the matrix quantile factor model into a large vector quantile factor model, if the interactive low-rank structure is the underground truth. We provide three criteria to determine the pair of row and column factor numbers, which are proved to be consistent. Extensive simulation studies and an empirical study justify our theory.