One of the main objectives of econometrics is to predict future values of important economics-related quantities, such as unemployment level, stock prices, currency exchange rates, etc. — and especially to predict how different possible economy-boosting measures will affect these quantities. To perform this prediction, we design a model of such effect and train it on the available data. Usually, the daily (or weekly) data are used for this training. However, economics-related quantities fluctuate all the time. So, for each moment of time (e.g., for each day) instead of a single value of the corresponding quantity, we have smallest and largest daily values — i.e., the interval range of daily values. At first glance, it may seem that using both endpoints of this interval for training will lead to more accurate predictions, but in reality, predictions become less accurate. Predictions become more accurate if we only use midpoints of the corresponding intervals. Several recent papers showed that even more accurate predictions are possible if we allow general convex combinations of the intervals’ endpoints — and select the corresponding coefficients so as to best fit the data. In this paper, we provide a theoretical explanation for these results.

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为什么凸组合的使用对区间数据效果很好：理论解释

计量经济学的主要目标之一是预测重要经济相关量的未来值，例如失业率、股票价格、货币汇率等，尤其是预测不同的可能促进经济的措施将如何影响这些量。为了执行这个预测，我们设计了一个这种效果的模型，并在可用数据上对其进行训练。通常，每日（或每周）数据用于此培训。然而，与经济相关的数量一直在波动。因此，对于每个时刻（例如，对于每一天）而不是对应数量的单个值，我们有最小和最大的每日值——即每日值的区间范围。乍一看，似乎使用这个区间的两个端点进行训练会导致更准确的预测，但实际上，预测变得不那么准确。如果我们只使用相应区间的中点，预测会变得更加准确。最近的几篇论文表明，如果我们允许区间端点的一般凸组合，并选择相应的系数以最好地拟合数据，则更准确的预测是可能的。在本文中，我们为这些结果提供了理论解释。