Rajen Shah, University of Cambridge

Abstract:

It is a common saying that testing for conditional independence, i.e., testing whether X is independent of Y, given Z, is a hard statistical problem if Z is a continuous random variable. We provide a formalisation of this result and show that a test with correct size does not have power against any alternative.
Given the non-existence of a uniformly valid conditional independence test, we argue that tests must be designed so their suitability for a particular problem setting may be judged easily. To address this need, we propose to nonlinearly regress X on Z, and Y on Z and then compute a test statistic based on the sample covariance between the residuals, which we call the generalised covariance measure (GCM). We prove that validity of this form of test relies almost entirely on the weak requirement that the regression procedures are able to estimate the conditional means X given Z, and Y given Z, at a slow rate. While our general procedure can be tailored to the setting at hand by combining it with any regression technique, we develop the theoretical guarantees for kernel ridge regression. A simulation study shows that the test based on GCM is competitive with state of the art conditional independence tests.