  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.
