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Apparently Wooldridge, Introductory Econometrics, 2002ed is the only book showing that two-stage least squares (2SLS) is asymptotically efficient. I cannot get a copy of the proof.

Is it correct to just use the generalized method of moments (GMM) proof substituting in the instrumental variables?

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Yes, the key is to use homoscedasticity so that the GMM weight matrix simplifies. Given that, a bit of cancellation will show that the variance of the optimal GMM estimator reduces to the 2SLS variance.

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