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