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Let's consider the linear model with endogenous errors: $$ y_t = \beta x_t + e_t $$ with $E[x_t e_t] \neq 0$.

How do we find in practice an instrumental variable (IV)? i.e. a variable $z_t$ that verifies: \begin{align} E[z_t x_t] \neq 0 \\ E[z_t e_t] = 0 \end{align}

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In practice, we have measures of $x$, $y$ and some $z$. The measure of $z$ is an instrument, i.e., satisfies the conditions you specified, if it affects $x$ and affects $y$ only through its effect on $y$. The only way to find such a valid measure $z$ is to understand the context one is studying; in particular, one needs a good, contextual understanding of the relationship between $x$ and $y$, and why $z$ would affect $y$ only through its effect on $x$. There is no purely theoretical way to find a good instrument. One has to consider the "true world" outside of mathematics. I suggest you read some applications that use the instrumental variables strategy, for example in epidemiology or economics.

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  • $\begingroup$ An arbitrary example that I found interesting: time of hospital admission affects the likelihood of admission to different hospital departments / assignment to specific doctors and through this pathway also influences health outcomes. But it (presumably) has no other effect on health outcomes except through this mechanism, so it represents an IV. Another example would be that commuting distance to a teaching facility affects the likelihood of participating in a course, but presumably has no other effect on teaching outcomes. Thus also an IV. $\endgroup$
    – Eike P.
    May 13, 2022 at 21:44
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Finding an instrument is literally "art".

In a sentence, we just consider a variable satisfyimg the two IV condition, and no theory is applied.

If you can convince people, then that is enough.

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