Forecast Linear Augmented Projection (FLAP): A free lunch to reduce forecast error variance
Abstract
A novel forecast linear augmented projection (FLAP) method that provably reduces the forecast error variance of any unbiased multivariate forecast without introducing bias, is introduced. The method first constructs new series, called components, as linear combinations of the original series. Forecasts are then generated for both the original and new 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 forecast error variances are non-increasing with the number of components, and mild conditions are established for which the sum of the forecast error variances is strictly decreasing. It is also shown that the proposed method achieves maximum forecast variance reduction among linear projections. 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.