How to check exogeneity of residuals in linear regression model? Can you please give me some advice in testing exogeneity of residuals ? I check the internet and it says a lot of test or other ways to prove or disapprove other assumptions, but I couldn't find any for exogeneity.
By exogenity I mean that $E(\epsilon)=0$ and $E(\epsilon_i|X)=0$ where $\epsilon_i$ is residual for $i$
-th observation.
 A: Exogeneity cannot be tested in observational studies1 using statistical tests.  Exogeneity is important for causal inference, which should be distinguished from statistical estimation.
To fix ideas, let us assume that we have the following linear model:
$y_i=\beta_0 + B X_i + \delta D_i + \epsilon_i$
Let us assume that $i$ is the unit of analysis, $y_i$ is the observed outcome for unit $i$, $X_i$ is a vector of covariates, and $D_i$ is a binary causal variable. Further assume that we are interested in estimating the effect $\delta$ of $D$ on the outcome.  For example, $y$ could be income at the age of 30 and $D$ could be "college degree."  Then, $\delta$ gives the wage differential for college education.
To estimate this causal effect, we need $D$ to be exogenous, i.e., there is no unmeasured confounding between $D$ and $y$ after we control for $X$.  Exogeneity is established through theoretical or qualitative arguments in observational studies, not by statistical tests.  In randomized control trials, the treatment would be exogenous by design (if the trial was executed properly and subjects complied with the design, etc.).
If this content is new to you, I would suggest that you watch the following excellent lecture by the famous economist Alberto Abadie, starting at 8 minutes and 40 seconds.
https://www.aeaweb.org/conference/webcasts/2017/Cross-Section-Econometrics-View-Part-1
1 Studies in which a causal variable of interest was not randomized to the study units.
