Title: Inverse Moment Methods for Sufficient Forecasting using High-Dimensional Predictors,
Abstract: We consider forecasting a single time series using high-dimensional
predictors in the presence of a possible nonlinear forecast function. The
sufficient forecasting (Fan et al., 2016) used sliced inverse regression to
estimate lower-dimensional sufficient indices for nonparametric forecasting
using factor models. However, Fan et al. (2016) is fundamentally limited to the
inverse first-moment method, by assuming the restricted fixed number of
factors, linearity condition for factors, and monotone effect of factors on the
response. In this work, we study the inverse second-moment method using
directional regression and the inverse third-moment method to extend the
methodology and applicability of the sufficient forecasting. As the number of
factors diverges with the dimension of predictors, the proposed method relaxes
the distributional assumption of the predictor and enhances the capability of
capturing the non-monotone effect of factors on the response. We not only
provide a high-dimensional analysis of inverse moment methods such as
exhaustiveness and rate of convergence, but also prove their model selection
consistency. The power of our proposed methods is demonstrated in both
simulation studies and an empirical study of forecasting monthly macroeconomic
data from Q1 1959 to Q1 2016. During our theoretical development, we prove an
invariance result for inverse moment methods, which make a separate
contribution to the sufficient dimension reduction.