Another Look at Dependence: The Most Predictable Aspects of Time Series

Serial dependence and predictability are two sides of the same coin. The literature has considered alternative measures of these two fundamental concepts. In this article, we aim to distill the most predictable aspect of a univariate time series, that is, the one for which predictability is optimized. Our target measure is the mutual information between the past and future of a random process, a broad measure of predictability that takes into account all future forecast horizons, rather than focusing on the one-step-ahead prediction error mean square error. The most predictable aspect is defined as the measurable transformation of the series that maximizes the mutual information between past and future. This transformation arises from the linear combination of a set of basis functions localized at the quantiles of the unconditional distribution of the process. The mutual information is estimated as a function of the sample partial autocorrelations, using a semiparametric method that estimates an infinite sum by a regularized finite sum. The second most predictable aspect can also be defined, subject to suitable orthogonality restrictions. Finally, we illustrate the use of the most predictable aspect for testing the null hypothesis of no predictability and for point and interval prediction of the original time series.

序列相关性与可预测性实为一体两面。现有文献已针对这两个核心概念提出了多种替代测度方案。本文旨在从单变量时间序列中提炼出最具可预测性的成分，即可预测性达到最优的序列成分。我们选取的测度指标为随机过程的过去与未来之间的互信息（mutual information），这是一种覆盖所有未来预测时域的广义可预测性测度，而非仅聚焦于单步预测误差均方误差。最具可预测性的成分被定义为可最大化该序列过去与未来互信息的可测变换。该变换由一组基函数（basis functions）的线性组合构成，这些基函数定位于随机过程无条件分布的分位点处。我们采用半参数方法，以正则化有限和近似无穷级数，将互信息表示为样本偏自相关系数的函数并完成估计。在适当的正交性约束下，亦可定义第二大可预测性成分。最后，本文演示了如何利用最具可预测性的成分开展无可预测性原假设检验，以及对原始时间序列进行点预测与区间预测。