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.