Matrix Quantile Factor Model

doi:10.1080/07350015.2026.2719893

Matrix Quantile Factor Model

2026-09-22

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DOI：10.1080/07350015.2026.2719893

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This article introduces a matrix quantile factor model for matrix-valued data with low-rank structure. We estimate the row and column factor spaces via minimizing the empirical check loss function with orthogonal rotation constraints. We show that the estimates converge at rate in the average Frobenius norm, where , and 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 model into a large vector model. To establish the central limit theorem, we introduce a tricky augmented Lagrangian function, which is equivalent to the original constrained empirical check loss minimization problem. Via the equivalence, we prove that the Hessian matrix of the augmented Lagrangian function is locally positive definite, resulting in a locally convex penalized loss function around the true factors and their loadings. This easily leads to a feasible second-order expansion of the score function and readily established central limit theorems of the smoothed estimates of the loadings. We provide three consistent criteria to determine the pair of row and column factor numbers. Extensive simulation studies and an empirical study justify our theory.

Keywords:

Check loss function

Column factor space

Row factor space

Two-way factor model

Matrix Quantile Factor Model

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