For an IV to be valid, it must be:
Correlated with the endogenous variable in the model
Uncorrelated with the dependent variable in the model
What does the random assignment of an IV mean? How does one assess whether an IV is actually randomly assigned or not?
For example, given the model:
$T_i = \beta_0 + \beta_1STR_i + u_i$
where $T_i$ is average test score in school district i and $STR_i$ is student-teacher ratio in school district i.
Suppose $E(u_i|STR_i)\neq0$. Let an IV be the birth rate in school district i, $BR_i$.
What argument can be given for the random assignment of $BR_i$?