The book I read (Davidson,MacKinnon - Econometric Theory and Methods) describes the definition of "optimal instrument variables" as the following:
Usually, and this is seen very often in other textbooks which cope with regression models, we want to estimate ${\mathbb E}\left(y | X\right)$. Another way of saying the same thing but in a more general sense is that we want to estimate ${\mathbb E}\left(y | \Omega\right)$. $\Omega$ is a information set containing all variables which determine the outcome of $y$.
In the context of instrument variable estimation we consider the usual model $y = X\beta + u$ with homoscedastic $u$'s. Now it is assumed that at least one explanatory variable of X is endogenous. Hence we want to find a matrix $W$ of instruments which minimize the asmyptotic covariance matrix of $\beta_{IV}$.
Here I get lost. It is assumed that in general $X$ can be assumed to satisfy the relation $X = \bar X + V$ where $E(V|\Omega) = 0$ and $\bar X = E(X|\Omega)$. Now $\bar X$ is supposed to be the optimal matrix of instruments but I don't get what $\bar X$ is supposed to be. Yes, it is the the expected outcome of $X$ given all the information which is in $\Omega$, but if one variable in $X$ is assumed to be endogenous and I replace this variable with a linear combination of the remaining variables which are assumed to be exogenous and that $\bar X$, I ask myself why $\bar X$ is a viable instrument. Why is this $\bar X$ not correlated with $u$ no more?
I'm not looking for a technical solution nor for a mathematical proof of some sort just. I would like to understand the idea behind this approach.