Generalized covariance measure statistic ($T$)

To test $X\perp \perp Y\mid Z$, regress $X$ and $Y$ on $Z$ and average the products of their residuals. Dividing the scaled average by its empirical second moment produces the generalized covariance measure statistic. Small mean-square nuisance-regression errors and a product-rate condition make it asymptotically pivotal.

If the numerator of the generalized covariance measure statistic obeys a central limit theorem with variance $\mathbb{E}({\epsilon }^2{\xi }^2)$ and its empirical residual-product second moment converges in probability to the same positive quantity, the Slutsky theorem makes the studentized statistic converge in distribution to $N(0,1)$.