#StackBounty: #regression #econometrics #intuition #instrumental-variables #endogeneity Question about Instrumental variables, endogene…

Bounty: 50

I have seen his notation to describe the Instrumental Variable framework, and I wish to make sure I understand it. Y is the dependent variable, x is treatment, and z is the instrument:

$y = f(x,epsilon)$

$x = g(z,eta)$

and the endogeneity structure is defined as: $cov(epsilon,eta)neq0$, $cov(z,epsilon)=0$, $cov(z,eta)=0$

I want to make sure I understand what this is saying.

  1. First, is any variable z that can fit this an instrument?

  2. If I am say approximating these functions with linear equations, that $x = pi z + eta$, is this saying we can partition the entire variation of x as the variation explained by z and then all the remaining variation $eta$, and the endogeneity can be expressed as $cov(epsilon,eta)neq0$? I am confused because usually this is simply expressed as $cov(x,epsilon)neq0$, and I am not familiar with writing this all in terms of errors. is this the same since I can just plug in the model of x as $cov(pi z + eta,epsilon) = cov(eta,epsilon)$ given the exogeneity of z?

  3. Is this equivalent as saying there exists some subset of variables, $rin epsilon$ and $r in eta$, i.e. omitted variables that determine x and determine y?

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