Evaluating the Calibration of Multi-Step-Ahead Density Forecasts Using Raw Moments

The evaluation of multi-step-ahead density forecasts is complicated by the serial correlation of the corresponding probability integral transforms. In the literature, three testing approaches can be found that take this problem into account. However, these approaches rely on data-dependent critical values, ignore important information and, therefore lack power, or suffer from size distortions even asymptotically. This article proposes a new testing approach based on raw moments. It is extremely easy to implement, uses standard critical values, can include all moments regarded as important, and has correct asymptotic size. It is found to have good size and power properties in finite samples if it is based on the (standardized) probability integral transforms.

多步提前密度预测（multi-step-ahead density forecasts）的评估会因对应概率积分变换（probability integral transforms）的序列相关而变得复杂。现有文献中已提出三种可应对该问题的检验方法，但此类方法均存在局限：要么依赖数据依赖临界值（data-dependent critical values），要么忽略关键信息以致检验效力不足，甚至在渐近情形下仍存在规模扭曲问题。本文提出一种基于原始矩（raw moments）的全新检验方法，该方法极易实现，可直接使用标准临界值，能够纳入所有被视作重要的矩信息，且具备正确的渐近规模属性。研究表明，若基于（标准化）概率积分变换构建该检验方法，其在有限样本下同样拥有优良的规模与检验功效特性。