We propose a flexible dual functional factor model for modelling high-dimensional functional time series. In this model, a high-dimensional fully functional factor parametrisation is imposed on the observed functional processes, whereas a low-dimensional version (via series approximation) is assumed for the latent functional factors. We extend the classic principal component analysis technique for the estimation of a low-rank structure to the estimation of a large covariance matrix of random functions that satisfies a notion of (approximate) functional "low-rank plus sparse" structure; and generalise the matrix shrinkage method to functional shrinkage in order to estimate the sparse structure of functional idiosyncratic components. Under appropriate regularity conditions, we derive the large sample theory of the developed estimators, including the consistency of the estimated factors and functional factor loadings and the convergence rates of the estimated matrices of covariance functions measured by various (functional) matrix norms. Consistent selection of the number of factors and a data-driven rule to choose the shrinkage parameter are discussed. Simulation and empirical studies are provided to demonstrate the finite-sample performance of the developed model and estimation methodology.

翻译：我们提出了一种灵活的双重函数因子模型，用于建模高维函数型时间序列。该模型对观测到的函数过程施加了高维全函数因子参数化形式，同时假设潜在函数因子具有低维版本（通过级数逼近实现）。我们将经典的主成分分析技术从低秩结构估计扩展到满足（近似）函数型"低秩加稀疏"结构的大维随机函数协方差矩阵估计，并将矩阵收缩方法推广至函数型收缩，以估计函数型异质成分的稀疏结构。在适当的正则条件下，我们推导了所提出估计量的大样本理论，包括因子和函数因子载荷估计的一致性，以及通过多种（函数型）矩阵范数度量的协方差函数矩阵估计的收敛速率。我们还讨论了因子数的一致选择及收缩参数的数据驱动选取准则。通过模拟和实证研究展示了所提出模型及估计方法在有限样本下的表现。