Optimal and robust combination of forecasts via constrained optimization and shrinkage

Roccazzella, Francesco [UCL] Gambetti, Paolo [UCL] Vrins, Frédéric [UCL]

Combining forecasts formed by various models can substantially improve the prediction performances compared to those obtained from the individual models. Standard combination approaches consist in a forecast selection step followed by a weighting scheme. It is not clear, however, which models to include, and how to combine them. This is a central question, having a substantial impact on the quality of the aggregate forecast. We propose a robust method that mitigates estimation uncertainty and implicitly features forecast selection. Our approach relies on constrained optimization with penalty (COP). We take advantage of the equivalence existing between COP and constrained optimization with shrinkage of the prediction errors’ covariance matrix (COS) to determine the optimal L2 penalty, thereby making the economy of an expensive (and potentially harmful) cross-validation stage. Our method is tested empirically in a simulation exercise and on two applications in economics. The proposed combination schemes outperform the simple average forecast, trimmed simple average forecast and perform at least as well as the best individual model(s) in the considered cases.