Econometrics – Definition of Validity of an Instrumental Variable

Question

What does "validity of an instrument" mean exactly?

In my econometrics course we have just defined instrument validity as $E[Z|u]=0$, where $Z$ is the instrumental variable and $u$ is the error term of a univariate regression model. Then, we also talked about strength of an instrument, but I am pretty sure I have understood correctly that it is a different requirement than validity.

In applications, I often find the definition of validity as ${\rm corr}(Z,X) \neq 0$, where $Z$ is the instrument and $X$ is the endogenous explanatory variable, plus the requirement that $E[Z|u]=0$ (as above), which is usually defined as exclusion restriction.

I am a bit confused and it is not so easy to find the kind of a primer on IV approaches I need. Is anyone able to unravel these issues?

Answer 1

Requirements for Z to be a valid instrument for X are:

• Relevance = Z needs to highly correlated with X
• Exogenous = Z is correlated with Y solely through its correlation with X; so Z is uncorrelated with the error in the outcome equation

The main idea behind IV is that when Z changes, it should also alter X, but not the troublesome part of X that is correlated with the error. To get the effect of X on Y we are only using part of the variation in X, the part that's driven by variation in Z.