arXiv:2504.21647v5 Announce Type: replace-cross Abstract: Identifying relationships among stochastic processes is a core objective in many fields, such as economics. While the standard toolkit for multivariate time series analysis has many advantages, it can be difficult to capture nonlinear dynamics using linear vector autoregressive models. This difficulty has motivated the development of methods for causal discovery and variable selection for nonlinear time series, which routinely employ tests for conditional independence. In this paper, we introduce the first framework for conditional independence testing that works with a single realization of a nonstationary nonlinear process. The proposed test is designed to have power against alternatives in which the expected conditional covariance is non-zero for at least some times. We also discuss an approach for gaining power against a broader range of alternatives. The key technical ingredients of our framework are time-varying nonlinear regression, estimation of local long-run covariance matrices of products of error processes, and a distribution-uniform strong Gaussian approximation.