We introduce various methods that combine forecasts using constrained optimization with penalty. A non-negativity constraint is imposed on the weights, and several penalties are considered, taking the form of a divergence from a reference combination scheme. In contrast with most of the existing approaches, our framework performs forecast selection and combination in one step, allowing for potentially sparse combining schemes. Moreover, by exploiting the analogy between forecasts combination and portfolio optimization, we provide the analytical expression of the optimal penalty strength when penalizing with the L2-divergence from the equally-weighted scheme. An extensive simulation study and two empirical applications allow us to investigate the impact of the divergence function, the reference scheme, and the non-negativity constraint on the predictive performance. Our results suggest that the proposed models outperform those considered in previous studies.

我们介绍了各种将使用约束优化与惩罚相结合的预测方法。对权重施加了非负约束，并考虑了几种惩罚，采取与参考组合方案发散的形式。与大多数现有方法相比，我们的框架一步执行预测选择和组合，允许潜在地稀疏组合方案。此外，通过利用预测组合和投资组合优化之间的类比，我们提供了在用来自等权重方案的 L2 散度进行惩罚时的最佳惩罚强度的分析表达式。广泛的模拟研究和两个经验应用使我们能够研究散度函数、参考方案和非负约束对预测性能的影响。我们的结果表明，所提出的模型优于先前研究中考虑的模型。