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I understand that when omitted-variable bias occurs the coefficient estimated for some regressors is the sum of the direct effect and indirect effect through the omitted-variable. What I fail to see is how this causes exogeneity to fail. Let's say the true relationship is given by $y= c+ ax+bz+error$ where the error term fulfills all the assumptions and $x$ and $z$ are correlated. Now if we run a regression with only a constant and $x$ we will get some result $y=f+gx+error$. Now the coefficient $g$ will be higher or lower than $a$ depending on the direction of the bias. However wouldn't the error term in this second equation still fulfill exogeneity?

edit: To clarify my question consider the following. Let's say $y= c+ ax+bz+error$ (where the error term fulfills all the assumptions) descibes reality. If we have $z=vx$ than $y=h+mx+error$ will also describe reality. When reading about OVB I have seen statemets like "if the true modell is $y= c+ ax+bz+error$ and x and z are correlated than regressing only on x will overestimate(or underestimate) the true coefficient of $x$". But this seems to be trying to establish causality? If we are only talking about descriptive models than there could be many different correct coefficients for $x$ depending on the variables included in the model couldn't there?

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Exogeneity won't hold if the correlation between z and x is nonzero and $b\not=0$ because then the error will be correlated to x in model 2 with the omitted z (which is now part of the error). If the true value of b is zero then you would also still have exogeneity. Deriving the estimator of a in both cases might help you see this.

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  • $\begingroup$ but if if have for example x=vz wouldn't the two models be identical? $\endgroup$
    – Jagol95
    Dec 27, 2019 at 21:07
  • $\begingroup$ No, they wouldn't be identical. Plug it in and see that it is not identical. $\endgroup$
    – LSC
    Dec 27, 2019 at 21:40
  • $\begingroup$ if I plug it in I get y=c+ax+bvx = c+(a+bv)x so if I regress y=f+gx I should get c for f and a+bv for g which would be the same as the first equation wouldn't it? $\endgroup$
    – Jagol95
    Dec 27, 2019 at 21:44
  • $\begingroup$ I can see that omitted variable bias is a problem for causal inferece but I fail to see how it's a problem problem for descriptive statements $\endgroup$
    – Jagol95
    Dec 27, 2019 at 23:17

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