We investigate the performance and sampling variability of estimated forecast combinations, with particular attention given to the combination of forecast distributions. Unknown parameters in the forecast combination are optimized according to criterion functions based on proper scoring rules, which are chosen to reward the form of forecast accuracy that matters for the problem at hand, and forecast performance is measured using the out-of-sample expectation of said scoring rule. Our results provide novel insights into the behavior of estimated forecast combinations. Firstly, we show that, asymptotically, the sampling variability in the performance of standard forecast combinations is determined solely by estimation of the constituent models, with estimation of the combination weights contributing no sampling variability whatsoever, at first order. Secondly, we show that, if computationally feasible, forecast combinations produced in a single step -- in which the constituent model and combination function parameters are estimated jointly -- have superior predictive accuracy and lower sampling variability than standard forecast combinations -- where constituent model and combination function parameters are estimated in two steps. These theoretical insights are demonstrated numerically, both in simulation settings and in an extensive empirical illustration using a time series of S&P500 returns.

翻译：我们调查了估计预测组合的性能和抽样变异性,特别注意预测分布的组合。预测组合中的未知参数根据基于适当的评分规则的标准功能优化,选择这些参数是为了奖励对目前问题很重要的预测准确性形式,预测性能则使用上述评分规则的超模预期值来衡量。我们的结果为估计预测组合的行为提供了新颖的洞察力。首先,我们表明,标准预测组合的性能的抽样变异性仅通过对构成模型的估计来确定,而组合权重的估算在第一个顺序上则没有任何抽样变异性。第二,我们表明,如果从计算上看可行,单步(即组成模型和合并功能参数是共同估计的)产生的预测组合比标准预测组合的预测性更准确,抽样变异性也更低。在两个步骤中,对组成模型和组合功能参数进行了估计。这些理论洞察通过数字方式展示,既在模拟环境中,也在使用S&P500返回的时间序列进行广泛的实验性说明。