Studentization of the generalized covariance measure statistic

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]