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Suppose we have the linear regression model given by $y=X\beta+\epsilon$, but we have a violation of assumptions where $X$, the regressor matrix, and $\epsilon$ are correlated. Also, suppose there is an observable collection of variables $Z$, such that $Z$ is uncorrelated with $\epsilon$, and $Z$ is correlated with $X$.

To derive the IV estimator $\beta_{IV}$ that is consistent for the true value of $\beta$, I am told that the first step is to premultiply the model $y=X\beta+\epsilon$ by $Z^T$, which ultimately yields $\beta_{IV}=(Z^TX)^{-1}Z^Ty$. My question is, what exactly is the intuition for premultiplying both sides by the instrument, $Z$?

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  • $\begingroup$ Try to consider the special case of a single regressor $X$ and $Z$ is a dummy---you just get a (sub-)sample mean for $X$ and $Y$. In general, when instrumenting you are taking conditional averages of $y$ and $X$ conditional on $Z$. Since $Z$ is exogenous, conditioning on $Z$ washes out endogeneity. $X$ is randomly assigned---conditional on $Z$. The IV estimates are noisy because, informally, passing to conditional averages mean you now have a "smaller" sample---again consider the dummy IV case. $\endgroup$ – Michael Apr 13 '19 at 16:10
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Intuitively, $Z^T\epsilon$ is expected to be $0$ (as the sample size increases) since they are uncorrelated. So, the equation becomes $$Z^Ty\approx Z^TX\beta+\underbrace{Z^T\epsilon}_{0}$$ which yields the solution you have by multipying each side by $(Z^TX)^{-1}$

For OLS (i.e. when $X$ and $\epsilon$ are uncorrelated), you multiply each side by $X^T$ instead of $Z^T$, and you'll have the typical solution $(X^TX)^{-1}X^Ty$.

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