  Functional time series have become an integral part of both functional data
and time series analysis. Important contributions to methodology, theory and
application for the prediction of future trajectories and the estimation of
functional time series parameters have been made in the recent past. This paper
continues this line of research by proposing a first principled approach to
estimate invertible functional time series by fitting functional moving average
processes. The idea is to estimate the coefficient operators in a functional
linear filter. To do this a functional Innovations Algorithm is utilized as a
starting point to estimate the corresponding moving average operators via
suitable projections into principal directions. In order to establish
consistency of the proposed estimators, asymptotic theory is developed for
increasing subspaces of these principal directions. For practical purposes,
several strategies to select the number of principal directions to include in
the estimation procedure as well as the choice of order of the functional
moving average process are discussed. Their empirical performance is evaluated
through simulations and an application to vehicle traffic data.
