Generalized covariance measure statistic ($T$)

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]

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]