  We consider forecasting a single time series using a large number of
predictors in the presence of a possible nonlinear forecast function. Assuming
that the predictors affect the response through the latent factors, we propose
to first conduct factor analysis and then apply sufficient dimension reduction
on the estimated factors, to derive the reduced data for subsequent
forecasting. Using directional regression and the inverse third-moment method
in the stage of sufficient dimension reduction, the proposed methods can
capture the non-monotone effect of factors on the response. We also allow a
diverging number of factors and only impose general regularity conditions on
the distribution of factors, avoiding the undesired time reversibility of the
factors by the latter. These make the proposed methods fundamentally more
applicable than the sufficient forecasting method in Fan et al. (2017). The
proposed methods are demonstrated in both simulation studies and an empirical
study of forecasting monthly macroeconomic data from 1959 to 2016. Also, our
theory contributes to the literature of sufficient dimension reduction, as it
includes an invariance result, a path to perform sufficient dimension reduction
under the high-dimensional setting without assuming sparsity, and the
corresponding order-determination procedure.
