Nonparametric Conditional Local Independence Testing

TL;DR

This paper introduces a model-free, nonparametric testing framework for conditional local independence among continuous-time stochastic processes, utilizing a new Local Covariance Measure and double machine learning techniques.

Contribution

It develops the first nonparametric test for conditional local independence using the Local Covariance Measure and double machine learning, applicable to continuous-time processes.

Findings

The proposed test controls level and power uniformly under certain conditions.

Simulation studies demonstrate the test's effectiveness without restrictive parametric assumptions.

Application to a Cox model example illustrates practical utility.

Abstract

Conditional local independence is an asymmetric independence relation among continuous time stochastic processes. It describes whether the evolution of one process is directly influenced by another process given the histories of additional processes, and it is important for the description and learning of causal relations among processes. We develop a model-free framework for testing the hypothesis that a counting process is conditionally locally independent of another process. To this end, we introduce a new functional parameter called the Local Covariance Measure (LCM), which quantifies deviations from the hypothesis. Following the principles of double machine learning, we propose an estimator of the LCM and a test of the hypothesis using nonparametric estimators and sample splitting or cross-fitting. We call this test the (cross-fitted) Local Covariance Test ((X)-LCT), and we show that its level and power can be controlled uniformly, provided that the nonparametric estimators are consistent with modest rates. We illustrate the theory by an example based on a marginalized Cox model with time-dependent covariates, and we show in simulations that when double machine learning is used in combination with cross-fitting, then the test works well without restrictive parametric assumptions.