Solved – How to determine if an instrumental variable is randomly assigned

For an IV to be valid, it must be:

  1. Randomly assigned

  2. Correlated with the endogenous variable in the model

  3. 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 $BR_i$?

To add to @jmbejara's answer, there is no formal statistical test for the validity of an instrument (beyond, obviously, that there shouldn't be any apparent correlation with the outcome except through the endogenous variable). Selection of a valid instrument depends on subject matter knowledge.

To your example, I can imagine scenarios in which the birth rate could affect test scores. Maybe parents who are busy with new children spend less time with their kids? And why would student-teacher ratio be affected by birth rate, unless there are lags involved? (Infants don't go to school). The point is that instrument validity involves qualitative reasoning.

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