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What is the point of using the identity matrix as weighting matrix in GMM?

GMM is the minimizer of the distance $g_n(\delta)'\hat{W}g_n({\delta})$, where $g_n = \frac{1}{n}\sum_ix_i\epsilon_i$. If we set $\hat{W}=I$, we would get a distance equal to $g_n(\delta)'g_n({\delta})$, i.e. the sum of squared coordinates of $g_n$.

The result of the minimization is still a GMM estimator but it is clearly not efficient (we should have set $\hat{W}=S^{-1}$, where $S = \frac{1}{n}\sum_ix_i'\epsilon_i'\epsilon_ix_i$).

So why should we proceed in this direction? Is it something common in practice as a first step towards the best GMM or are there other reasons?

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Yes, getting a first step estimator is the canonical use. Of course, the error terms in $$S = \frac{1}{n}\sum_i\epsilon_i^2x_ix_i'$$ are not observable, so that you need to replace them with something feasible. As the efficient GMM estimator depends on $\hat S$, you first need some feasible preliminary estimator such as the one using $I$ as the weighting matrix.

There may be some further interesting considerations in a multiple equation setup, in which misspecification in one equation can "pollute" the entire system. You can avoid that risk through a less efficient, but more robust block-diagonal weighting matrix, of which $I$ would be an example.

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