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For observations $(X_i,Y_i,Z_i)$, estimate the conditional means $f(z)=\mathbb{E}[X\mid Z=z]$ and $g(z)=\mathbb{E}[Y\mid Z=z]$, and form residual products $R_i=(X_i-\hat{f}(Z_i))(Y_i-\hat{g}(Z_i))$. The univariate generalized covariance measure statistic isThe denominator is an estimated standard deviation, not the second moment itself, and must be positive. Under conditional independence, accurate nuisance regressions, suitable moment assumptions and a product-rate condition give a standard normal limit. The test detects residual covariance; conditional dependence with zero residual covariance can escape it. Its validity depends on regression quality rather than allowing arbitrary distributions with no restrictions.^[ref]

Jonas Peters studies causal inference and the foundations of conditional independence testing. With Rajen D. Shah, he proved that a test with uniform finite-sample validity over an unrestricted continuous-data null class cannot obtain nontrivial power against alternatives in their setting.^[ref]

Their constructive response is the generalized covariance measure statistic: regress each of two variables on the conditioning variables and examine the covariance of their residuals. Suitable regression-error rates and moment conditions yield asymptotic validity. Such a test targets conditional residual association; arbitrary forms of conditional dependence need not produce a nonzero residual covariance.^[ref]

Rajen D. Shah develops statistical inference for high-dimensional data. His author publication list verifies work on regression, model checking and testing conditional independence.^[ref]

With Jonas Peters, he introduced the generalized covariance measure statistic, based on the covariance of residuals after each response is regressed on the same conditioning variables. The resulting hypothesis test combines regression estimates with a calibrated covariance statistic. Their accompanying impossibility result explains why unrestricted distribution-free conditional independence testing needs additional structure; the regression-based method instead states approximation and moment assumptions explicitly.^[ref]

Suppose the scaled mean residual product $\sqrt{n}\overline{R}$ in the generalized covariance measure statistic converges in distribution to $N(0,v)$, with $v>0$, and its empirical variance $s_R^2=n^{-1}{\sum }_i{R}_i^2-{\overline{R}}^2$ converges in probability to $v$. Then the Slutsky theorem gives $\sqrt{n}\overline{R}/s_R\Rightarrow N(0,1)$. It is the square root of the variance estimate that belongs in the denominator. Under the null and suitable regression conditions, $v=\mathbb{E}[{\epsilon }^2{\xi }^2]$ for the two true regression residuals. This is a calibration argument; it does not remove the conditions needed for the numerator limit or variance consistency.^[ref]