Inverse Moment Methods for Sufficient Forecasting using High-Dimensional Predictors

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.