Conditional independence testing with a single realization of a multivariate nonstationary nonlinear time series

That is, testing whether two random vectors X and Y are independent given a third random vector Z . For example, there are conditional independence tests based on conditional densities [SW08], characteristic functions [SW07], empirical likelihood ratios [SW14], discretization [Mar05; Hua10], permutation [Dor+14; Sen+17], kernels [Fuk+07; Zha+11; SP11], copulas [BRT12], and conditional mutual information [Run18b]. Also, there are many conditional independence tests based on regressing X on Z and Y on Z followed by testing for independence between the residuals [Pat+09; Pet+14; Ram14; FFX20; ZZG17; Zha+19]. Unfortunately, conditional independence tests oftentimes struggle to control the Type-I error in finite samples, as shown by Shah and Peters [SP20]. In fact, Shah and Peters [SP20] prove that conditional independence testing is fundamentally impossible without making further assumptions. This issue has sparked significant interest in conditional independence testing over the last several years. We begin by providing an overview of recent advances in conditional independence testing. Afterwards, we discuss how our work addresses limitations in the existing literature. Finally, we motivate our work by reviewing key applications of conditional independence tests for time series in areas such as variable selection and causal discovery.