 Generalized covariance measure statistic

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]