Summary 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 nonmonotone 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 of Fan et al. (2017). The proposed methods are demonstrated both in 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.

中文翻译：

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使用高维预测器进行充分预测的逆矩法

总结 我们考虑在存在可能的非线性预测函数的情况下使用大量预测变量来预测单个时间序列。假设预测变量通过潜在因素影响响应，我们建议首先进行因素分析，然后对估计的因素进行足够的降维，以得出降维数据用于后续预测。在充分降维阶段，利用方向回归和逆三次矩法，所提方法可以捕捉因素对响应的非单调效应。我们还允许不同数量的因子，并且只对因子的分布施加一般规律性条件，避免后者对因子的不希望的时间可逆性。这些使得所提出的方法从根本上比 Fan 等人的充分预测方法更适用。（2017）。所提出的方法在模拟研究和预测 1959 年至 2016 年每月宏观经济数据的实证研究中得到证明。此外，我们的理论有助于充分降维的文献，因为它包括不变性结果，执行充分降维的路径在不假设稀疏的高维设置下，以及相应的顺序确定过程。