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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 $BR_i$?

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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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To restate the requirements you've listed (for linear models), an instrument must be

  1. exogenous in the sense that it is independent of the error term $u_i$,
  2. correlated with the endogenous explanatory variable $STR$ after conditioning on the other covariates,
  3. and uncorrelated with the dependent term except through the explanatory variable (again, conditioning on the other variables).

Condition (1) is "instrument exogeneity" and condition (2) is "instrument" relevance. The last condition here (3) is in essence just a restatement of (1).

The idea is that the instrumental variable cannot suffer from the same problem that the endogenous variable suffers from. Is needs to be uncorrelated with that error. In a laboratory, random assignment solves this problem. In the typical social sciences setting, an instrument must be found that approximates random assignment. This condition is similar to condition (3) because, notice that if $BR$ affects $STR$, then $BR$ affects $T$ (through $STR$). However, the trouble arises when $BR$ affects $T$ through the error $u$.

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