Background
Let me start with the fact that I read, this post, this post and this post. These questions deal with omitting exogenous variables from the main equation in the first stage. My question is however slightly different.
This question deals with the requirements of the first stage of an instrumental variable approach. Chapter 15.2 of Wooldridge (2012), Introductory Econometrics: A Modern Approach, mentions that: "Equation (15.26) [the first stage] is an example of a reduced form equation, which means that we have written an endogenous variable in terms of exogenous variables." This sentence appears to insinuate that the only variables that need to be in the first stage, are the exogenous variables from the main stage.
Situation
My conundrum is as follows: I use all exogenous variables of the main stage in the first stage, just like I am supposed to do (I am using the notation from Chapter 15.2, Wooldridge (2012), Introductory Econometrics: A Modern Approach, for my example). However, looking at my first stage, I am of the opinion that not all variables that explain my instrumented variable are in the first stage.
Main stage:
First stage:
As an example, let's say that in the US, the amount of money that your parents have is obviously correlated with education (because they might pay for it). I though this might be a problem for the estimation of the fitted values for instrumental variable first stage. I then however read the sentence:
Question
Does this sentence essentially say that omitting this variable (the amount of money your parents have) does not matter by construction? Does this mean I do not have to worry about omitted variables in my first-stage? If so, can someone explain me the rational behind it, because for me it is not immediately clear from the text.
EDIT: As a thought experiment: Assuming that omitted variables in the first stage are indeed not an issue. What would be the consequence of adding an additional variable to the first stage?
