Nonparametric Conditional Local Independence Testing

Conditional local independence is an 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. However, no nonparametric test of conditional local independence has been available. We propose such a nonparametric test based on double machine learning. The test is based on a functional target parameter defined as the expectation of a stochastic integral. Under the hypothesis of conditional local independence the stochastic integral is a zero-mean martingale, and the target parameter is constantly equal to zero. We introduce the test statistic as an estimator of the target parameter and show that by using sample splitting or cross-fitting, its distributional limit is a Gaussian martingale under the hypothesis. Its variance function can be estimated consistently, and we derive specific univariate test statistics and their asymptotic distributions. An example based on a marginalized Cox model with time-dependent covariates is used throughout to illustrate the theory, and simulations based on this example show how double machine learning as well as sample splitting are needed for the test to work. Moreover, the simulation study shows that when both of these techniques are used in combination, the test works well without restrictive parametric assumptions.