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Demeaning the moment conditions

Because $E[g(w_i,\theta_0)]=0$ at $\theta_0$ there is a tendency to use $\frac{1}{N} \sum_{i=1}^N \left(g(w_i,\hat{\theta}) \, g(w_i,\hat{\theta} )'\right)^{-1}$ (without de-meaning the moments). In theory this would work fine, but in practice \textbf{it is nearly always a bad idea}.\\

Not sure if this is an issue or a question, maybe I just misunderstood :)

Here, you say it's almost always a bad idea not to demean. However, below (linear IV estimator), you give $\hat{S}$ without demeaning because $E(z_i \epsilon_i) = 0$. So, perhaps it makes sense either to explain when it is OK not to demean, or else we should demean $\hat{S}$ below too?

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