Why the Use of Convex Combinations Works Well for Interval Data: A Theoretical Explanation

Abstract

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.