A Projective Approach to Conditional Independence Test for Dependent Processes

Abstract

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 β-mixing conditions, our proposed projective test statistic is n-consistent if these processes are conditionally independent and root-n-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.

Keywords:

• Conditional independence
• Empirical process
• Granger causality
• Nonlinear dependence
• Projection correlation

Supplementary Materials

The supplement contains detailed proofs of Theorem 1 through Theorem 5.

Additional information

Funding

Zhou’s work is supported by National Natural Science Foundation of China (12001405) and China Postdoctoral Science Foundation (2019M661603). Zhang’s work is supported by National Natural Science Foundation of China (11801349). Zhu’s work is supported by National Natural Science Foundation of China (11731011, 11931014) and Beijing Natural Science Foundation (Z190002).