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In GMM, the efficient weight matrix minimizes the asymptotic variance of the GMM estimator by setting:

$$ W_T^{opt} = S_T^{-1}$$

where $S_T$ is an estimator of the asymptotic variance of the moments, $S = Var(\sqrt{T}g_T(\beta))$.

In my textbook, it says:

How to estimate $S_T$ depends on whether the moments are:

  1. IID over time
  2. independent over time, but heteroskedastic
  3. autocorrelated and heteroskedastic over time

My question is: how do we test in which of the three cases we are?

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