  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.
