How is it possible to create an estimator with the method of moments for the correlation coefficient $\rho$ of a sample $(X,Y)$?
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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 $$ \hat{\rho} =\frac{\frac 1 n\sum x_iy_i - \frac 1 n\sum x_i \frac 1 n \sum y_i}{( \frac 1 n \sum(x_i-\bar x_n)^2\frac 1 n\sum(y_i-\bar y _n)^2)^{1/2}} = \frac{\sum x_i y_i -n\bar x_n\bar y_n}{S_XS_Y}. $$
