# Definition of "optimal" instruments

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.

• Druss, could you provide page number references to the Davidson and MacKinnon book? Dec 5, 2012 at 17:58
• Hey fg nu, the page number is 318. The equation number is 8.18. I guess ive already developed a good idea. Since 2SLS estimates $E(X|W)$, which is not equal to $E(X|Ω)$ but a approximation, this is nothing else what 2SLS actually does. Dec 7, 2012 at 16:12
• Druss, I haven't had a chance to take a look at this yet. I intend to do so soon. Dec 7, 2012 at 16:14
• Not to worry :-) You already helped me alot! Dec 7, 2012 at 16:17

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.

• This gives the overall intuition of IV regressions but the current explanation falls a bit short regarding the intuition behind the optimal matrix of instruments $W$ which minimizes the asymptotic covariance matrix. It's not really a problem given that the OP's main confusion seems to lie with IV in general but for future readers this should be kept in mind. In this sense the Card example doesn't apply because he only has one instrument whereas Davidson and MacKinnon talk about the case of many instruments.
– Andy
Jun 7, 2015 at 11:56
• I didn't want to go into details. The op specifically asks the intuition behind why is X a viable instrument and not correlated with u. It is not correlated with u by assumption after partialling out any other effect of V. Jun 7, 2015 at 12:08
• That's why you got a +1 from me ;) I just wanted to highlight to future readers that the paragraph in the book is about something slightly different in case people have a similar question.
– Andy
Jun 7, 2015 at 12:11
• Thanks ;) In fact, I am trying to answer a related question posted a few days ago and sidelined into this one. Jun 7, 2015 at 12:15