  We address the problem of forecasting high-dimensional functional time series
through a two-fold dimension reduction procedure. The difficulty of forecasting
high-dimensional functional time series lies in the curse of dimensionality. In
this paper, we propose a novel method to solve this problem. Dynamic functional
principal component analysis is first applied to reduce each functional time
series to a vector. We then use the factor model as a further dimension
reduction technique so that only a small number of latent factors are
preserved. Classic time series models can be used to forecast the factors and
conditional forecasts of the functions can be constructed. Asymptotic
properties of the approximated functions are established, including both
estimation error and forecast error. The proposed method is easy to implement
especially when the dimension of the functional time series is large. We show
the superiority of our approach by both simulation studies and an application
to Japanese age-specific mortality rates.
