Method of moments for correlation coefficient How is it possible to create an estimator with the method of moments for the correlation coefficient $\rho$ of a sample $(X,Y)$?

Note that 
$$
\rho = \frac{cov(X,Y)}{\sigma_x\sigma_y}= \frac{EXY - EXEY}{(E(X-EX)^2E(Y-EY)^2)^{1/2}},
$$
so just plug-in the sample moments instead of the population moment
$$
\hat{\rho} =\frac{\frac 1 n\sum x_iy_i - \frac 1 n\sum x_i \frac 1 n \sum y_i}{( \frac 1 n  \sum(x_i-\bar x_n)^2\frac 1 n\sum(y_i-\bar y _n)^2)^{1/2}} = \frac{\sum x_i y_i -n\bar x_n\bar y_n}{S_XS_Y}.
$$
