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How can I show that the instrumental variables (IV) estimator is consistent from this equation using the two stage least squares method?

Where does this equation come from?

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  • $\begingroup$ Two hints: a property of matrices is that A(B+ C) = AB + AC. Also what is the long range behavior of $(X^\prime Z (Z^\prime Z)^{=1}Z^\prime X)^{-1}\epsilon$? $\endgroup$ – AdamO Oct 8 '18 at 15:11
  • $\begingroup$ Are you sure the behavior of that matrix is to be examined? $\endgroup$ – Christoph Hanck Oct 9 '18 at 5:29
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Since this might be self-study, a couple of hints:

$y=X\beta+\epsilon$ has been substituted in the definition of the IV or 2SLS estimator. The tools you'll need are $n/n=1$ (so suitably expand in numerators and denominators), $$Z'Z/n\to_p\Sigma_{zz}\quad\text{and}\quad Z'X/n\to_p\Sigma_{xz}$$ equal to some finite matrix by the law of large numbers and $$Z'\epsilon/n\to_p E(Z'\epsilon)=0$$ by the assumption that the instruments are not correlated with the error term.

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