Proper scoring rules are used to assess the out-of-sample accuracy of probabilistic forecasts, with different scoring rules rewarding distinct aspects of forecast performance. Herein, we re-investigate the practice of using proper scoring rules to produce probabilistic forecasts that are ‘optimal’ according to a given score and assess when their out-of-sample accuracy is superior to alternative forecasts, according to that score. Particular attention is paid to relative predictive performance under misspecification of the predictive model. Using numerical illustrations, we document several novel findings within this paradigm that highlight the important interplay between the true data generating process, the assumed predictive model and the scoring rule. Notably, we show that only when a predictive model is sufficiently compatible with the true process to allow a particular score criterion to reward what it is designed to reward, will this approach to forecasting reap benefits. Subject to this compatibility, however, the superiority of the optimal forecast will be greater, the greater is the degree of misspecification. We explore these issues under a range of different scenarios and using both artificially simulated and empirical data.

适当的评分规则用于评估概率预测的样本外准确性，不同的评分规则奖励预测性能的不同方面。在此，我们重新研究了使用适当的评分规则根据给定分数生成“最佳”概率预测的做法，并根据该分数评估其样本外准确性何时优于替代预测。特别注意在预测模型指定错误的情况下的相对预测性能。使用数值插图，我们记录了该范式中的几个新发现，突出了真实数据生成过程、假设的预测模型和评分规则之间的重要相互作用。尤其，我们表明，只有当预测模型与真实过程充分兼容以允许特定的评分标准奖励它旨在奖励的内容时，这种预测方法才会获得收益。然而，以这种兼容性为条件，最优预测的优势将越大，错误指定的程度就越大。我们在一系列不同的场景下并使用人工模拟和经验数据来探讨这些问题。