A Projective Approach to Conditional Independence Test for Dependent Processes

Conditional independence is a fundamental concept in many scientific fields. In this article, we propose a projective approach to measuring and testing departure from conditional independence for dependent processes. Through projecting high-dimensional dependent processes on to low-dimensional subspaces, our proposed projective approach is insensitive to the dimensions of the processes. We show that, under the common <i>β</i>-mixing conditions, our proposed projective test statistic is <i>n</i>-consistent if these processes are conditionally independent and root-<i>n</i>-consistent otherwise. We suggest a bootstrap procedure to approximate the asymptotic null distribution of the test statistic. The consistency of this bootstrap procedure is also rigorously established. The finite-sample performance of our proposed projective test is demonstrated through simulations against various alternatives and an economic application to test for Granger causality.

条件独立（Conditional Independence）是诸多科学领域中的核心概念。本文提出一种用于度量与检验相依过程（dependent processes）的条件独立偏离程度的投影方法。通过将高维相依过程投影至低维子空间，所提投影方法对过程维度不敏感。研究表明，在常见的β混合（β-mixing）条件下，当相依过程满足条件独立时，所提投影检验统计量具有n阶相合性（n-consistent）；反之则具有根号n阶相合性（root-n-consistent）。本文提出一种自助法（Bootstrap）程序以近似检验统计量的渐近原假设分布，且该自助法程序的相合性亦得到了严格证明。本文通过针对各类备择假设的模拟实验，以及一项用于检验格兰杰因果关系（Granger Causality）的经济学应用，验证了所提投影检验的有限样本性能。