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While conducting estimation via the Generalised Method of Moments, or GMM, I understand that we need to minimise the following expression:

$Q_n(\theta)=g_n(\theta)'W_ng_n(\theta)$

Where $g_n(\theta)$ is a vector of sample moments, $W_n$ is the weighting matrix, and of course, $\theta$ is our parameter of interest. I have also been told that the optimal weighting matrix is the inverse of the covariance of the sample moments. I do not understand the reason for this.

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Each estimated moment is a random variable with unequal variances, and, usually, non-zero cross-moment covariances. Using the inverse of the covariance matrix re-weights the moments, so that you effectively pay more attention to moments that are more informative or perhaps better measured, in the sense of having smaller variance. This gives you the most efficient GMM estimate and is analogous to same benefit with GLS.

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