Solved – the logic behind method of moments

A sample consisting of $n$ realizations from identically and independently distributed random variables is ergodic. In a such a case, "sample moments" are consistent estimators of theoretical moments of the common distribution, if the theoretical moments exist and are finite.

This means that

$$hat mu_k(n) = mu_k(theta) + e_k(n), ;;; e_k(n) xrightarrow{p} 0 tag{1}$$

So by equating the theoretical moment with the corresponding sample moment we have

$$hat mu_k(n) = mu_k(theta) Rightarrow hat theta(n) = mu_k^{-1}(hat mu_k(n)) = mu_k^{-1}[mu_k(theta) + e_k(n)]$$

So ($mu_k$ does not depend on $n$)

$$text{plim} hat theta(n) = text{plim}big[mu_k^{-1}(mu_k(theta) + e_k)big] = mu_k^{-1}big(mu_k(theta) + text{plim}e_k(n)big)$$

$$=mu_k^{-1}big(mu_k(theta) + 0big) = mu_k^{-1}mu_k(theta) = theta$$

So we do that because we obtain consistent estimators for the unknown parameters.