Factor modelling for high-dimensional functional time series

Many economic and scientific problems involve the analysis of high-dimensional functional time series, where the number of functional variables $p$ diverges as the number of serially dependent observations $n$ increases. In this paper, we present a novel functional factor model for high-dimensional functional time series that maintains and makes use of the functional and dynamic structure to achieve great dimension reduction and find the latent factor structure. To estimate the number of functional factors and the factor loadings, we propose a fully functional estimation procedure based on an eigenanalysis for a nonnegative definite and symmetric matrix. Our proposal involves a weight matrix to improve the estimation efficiency and tackle the issue of heterogeneity, the rationale of which is illustrated by formulating the estimation from a novel regression perspective. Asymptotic properties of the proposed method are studied when $p$ diverges at some polynomial rate as $n$ increases. To provide a parsimonious model and enhance interpretability for near-zero factor loadings, we impose sparsity assumptions on the factor loading space and then develop a regularized estimation procedure with theoretical guarantees when $p$ grows exponentially fast relative to $n.$ Finally, we demonstrate that our proposed estimators significantly outperform the competing methods through both simulations and applications to a U.K. temperature data set and a Japanese mortality data set.