# #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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