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Usually the exogeneity assumption is states, given the vector E[$\epsilon$|x]=0.

what this implies then is E[$\epsilon_i$|x$_i$]=0 for all i. The individual notation part is what is confusing me.

What is the intuition for the individual notation? is $\epsilon_i$ a random variable for each i, or is it just one random variable $\epsilon$,who's realizations vary with the population? are we taking the expectation over all values for individual i, or are we taking the expectation of the error across individuals?

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2 Answers 2

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$\epsilon$ is a vector random variable, with one component for each observation; each $\epsilon_i$ is a univariate random variable.

I encourage you to use this to try to answer your other questions, and then post a comment in a few hours or days if you’re still confused.

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In econometrics, the first definition refers to strict exogeneity.
The second definition is for contemporaneous exogeneity.

The first implies the second but the second doesn't imply the first (contemporaneous is a weaker condition than strict, as the latter represents past, present and future independence).

You can find some useful details here:

https://stats.stackexchange.com/a/247418/292958
https://www.statisticshowto.com/exogeneity/

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