Definition of "optimal" instruments

Question

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.

Answer 1

I haven't read the particular page in the book, but here is my take on that.

$\bar X = E(X|\Omega)$ $\rightarrow$ that's your 'ideal' first stage regression here. $\bar X$ is the set of all variables that are informative after controlling for everything else ($\Omega$) that determines Y.

$E(V|\Omega) = 0$ $\rightarrow$ that's the validity assumption. The unobserved factors of $\bar X$ are not correlated with the determinants of Y.

I provide you an example, Card (1992) instruments "years of education" with "travel distance to school" to determine the return on wages. He assumes that "intelligence" or "ability" is an unobserved factor that has not been controlled for. "Education" is therefore assumed endogenous as it relates to "ability".

What is the set of optimal instruments? It is the set that fulfills the definition by Davidson and MacKinnon.

Although Card uses only one instrument, the intuition can be exemplified here. In the first stage regression, $\bar X = E(X|\Omega)$, you would regress education on "distance to school" AND anything else that determines wages, e.g. age, grades, experience etc. If you can safely assume that after doing so, $E(V|\Omega) = 0$, then you have found your proper instrument.

That the first assumption includes the whole information set $\Omega$ is necessary. It is wickedly conceivable that "distance to school" is related with "age" which is assumed to be related with "wage" but also "ability", as for example, older people live farther away from cities where schools were built but may be less/more able as they are a different generation. Failing to control for that, would invalidate your second assumption, $E(V|\Omega) = 0$.

So the ideal set of instruments, needs to be correlated with your regression after 'partialling out' anything else that is known determine Y.