# instrumental variables property

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?

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

$$y=X\beta+\epsilon$$ 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.