Covariance Function Estimation for High-Dimensional Functional Time Series with Dual Factor Structures

We propose a flexible dual functional factor model for modelling high-dimensional functional time series. In this model, a high-dimensional fully functional factor structure 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 estimating 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 generalize the matrix shrinkage method to functional shrinkage in order to estimate the sparse structure of functional idiosyncratic components. The developed methodology can be used to estimate both the functional contemporaneous covariance and lag-h autocovariance matrices. Under appropriate regularity conditions, we derive the large sample theory of the resulting estimators, including the consistency of the estimated factors and functional factor loadings and convergence rates of the estimated matrices of covariance and autocovariance 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.