A novel forecast linear augmented projection (FLAP) method is introduced, which reduces the forecast error variance of any unbiased multivariate forecast without introducing bias. The method first constructs new component series which are linear combinations of the original series. Forecasts are then generated for both the original and component series. Finally, the full vector of forecasts is projected onto a linear subspace where the constraints implied by the combination weights hold. It is proven that the trace of the forecast error variance is non-increasing with the number of components, and mild conditions are established for which it is strictly decreasing. It is also shown that the proposed method achieves maximum forecast error variance reduction among linear projection methods. The theoretical results are validated through simulations and two empirical applications based on Australian tourism and FRED-MD data. Notably, using FLAP with Principal Component Analysis (PCA) to construct the new series leads to substantial forecast error variance reduction.

翻译：本文介绍了一种新颖的预测线性增强投影（FLAP）方法，该方法能在不引入偏差的前提下，降低任何无偏多元预测的预测误差方差。该方法首先构建新的成分序列，这些序列是原始序列的线性组合。随后，分别为原始序列和成分序列生成预测。最后，将完整的预测向量投影到一个线性子空间上，该子空间满足由组合权重所隐含的约束条件。研究证明，预测误差方差的迹随成分数量的增加而单调非增，并建立了其严格递减的温和条件。同时，研究表明所提方法在线性投影方法中实现了最大的预测误差方差缩减。理论结果通过仿真以及基于澳大利亚旅游数据和FRED-MD数据的两个实证应用得到了验证。值得注意的是，使用FLAP结合主成分分析（PCA）来构建新序列，能显著降低预测误差方差。