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In econometrics, we typicall assume exogeneity as, starting with $y = x\beta + \epsilon$:

$E[\epsilon|x]=0$.

I always intuitively thought about this in an abstract sense, away from the individual observations per se, as just the underlying relationship between x and y in the population is such that y doesnt cause x and x isnt correlated with any other unobservable determinant of y. However, I think in the context of individual notation, this becomes confusing. For instance, if the above is actually saying:

$E[\epsilon_i|x_i]$=0 for all i, then what would the following imply:

say for observation j in the data set, $E[\epsilon_j|x_j]$ = f($x_j)$, f`(.)>0 - so higher levels of $x_j$ lead to higher draws of $\epsilon_j$ in expectation, but for all i =/= j, $E[\epsilon_i|x_i]$ = 0? Could this logically make sense? what does it imply about the exogeneity assumptions? how does it even intuitively arrive- would the idea be, if say I am observation i, If i were to receive higher levels of x, it is also because of some other unobservable effecting y?

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  • $\begingroup$ "$E[\epsilon_i|x_i]$ = 0" is not good notation that can be made precise (although it wouldn't be surprising if it shows up in empirical literature). The proper statement is "$E[\epsilon|x]=0$ and $(x_i, \epsilon_i)$ is i.i.d., with observed data being a transformation of it, $(x_i, y_i)$. $\endgroup$
    – Michael
    Commented Aug 17, 2020 at 0:11

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In linear regression, the assumption is that, for all individuals "i", $E[\epsilon_i|x]=0$. If there are a set of individuals that doesn't follow this assumption, it can mean there is a missing variable in the model or a misspecification.

For example, let's say you have two groups of individuals in your data ("men" and "women"), a set of variables X that affect the outcome of men and women, and a variable Z that affect the outcome of women but doesn't affect the outcome of men. The correct specification would be : $Y = \beta_0 + \beta_W*1_W + X\beta_X + Z*1_W*\beta_Z + \epsilon$ where $1_W$ is 1 for women and 0 for men.

If you don't use Z in the model it will be a part of the residuals, the expectation of the residuals will be be 0 for men but not for women...

more generally, the observations in your data can arise from differents population which share common characteristics but can also have specificities of their own. The purpose of the model is to find the variables that explain the outcome for all the populations and make the residual expectation a white noise.

hope this can help...

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  • $\begingroup$ Then why not state the assumption as $E(y|x) = x\beta$ rather than $E(\epsilon | x) = 0$? The former seems much more intuitive: It states that the conditional mean function is exactly within the family of functions you specify in your model. Further, truth of $E(y|x) = x\beta$ implies truth of $E(\epsilon | x) = 0$, so the assumption $E(\epsilon | x) = 0$ is superfluous in this context. $\endgroup$
    – BigBendRegion
    Commented Aug 17, 2020 at 10:29
  • $\begingroup$ I think this is equivalent no? the general specification of a model is $Y = E(y|x) + \epsilon$. When you perform a linear regression, you assume $ E(y|x)=x\beta $ for some variables X and you search these variables such as $E(\epsilon|x)=0$. Am I missing something? $\endgroup$
    – Florent
    Commented Aug 17, 2020 at 11:10
  • $\begingroup$ Yes, I just wonder why to use $E(\epsilon | x) = 0$. It just seems to obfuscate the issue. $\endgroup$
    – BigBendRegion
    Commented Aug 17, 2020 at 11:59
  • $\begingroup$ I'm not sure, since under the assumption $E(\epsilon|x)=0$ we have $cov(x,\epsilon)=E(yx)-E(E(y|x)x)$, I suppose to keep $\epsilon$ uncorrelated from X, it's better not to make assumption on $E(y|x)$. But I'm confused because for me, choosing the linear regression is making the assumption that $E(y|x)=x\beta$ $\endgroup$
    – Florent
    Commented Aug 17, 2020 at 17:52

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