# Intuition for Instrumental Variable estimator in Linear Regression Model

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, $$Z$$?

• 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. Commented Apr 13, 2019 at 16:10

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 $$(X^TX)^{-1}X^Ty$$.