I am getting confused trying to intuitively understand the difference between the strict exogeneity assumption and the weak exogeneity assumption (in the case of multivariate linear regression model). I already did my research and found lots of questions like this, but I could not find any satisfactory answer to my specific problem.
I will try to better explain the point I can't understand. Let's consider a model like $Y_i = \beta_0 + \beta_1X_{1i} + \beta_2X_{2i} + u_i$, for each $i=1,...n$, where $n$ is the sample size.
Is assuming weak exogeneity the same as assuming $E(u_i | X_{1i}=x_{1i}) = 0$ and $E(u_i | X_{2i} = x_{2i})$ ?. If I am getting this right, we are stating that the error term $u_i$ does not contain any "systematic" information about the two regressors with regards to observation $i$, hence on the average we are predicting the dependent variable $Y_i$ the right way for each $i$.
From what I understood, assuming strict exogeneity means assuming the same conditions as before but also with respect to different observations: $E(u_i | X_{1j}=x_{1j}) = 0$ and $E(u_i | X_{2j} = x_{2j})$, where $j=1,...,n$. At this point the error term $u_i$ does not contain any "systematic" information about the two regressors with regards to any other generic observation $j$. Didn't we reach the same conclusion as before? To me it seems that now we are still predicting the dependent variable $Y_i$ the right way in the average for each observation $i$. What am I getting wrong?
Also,isn't random sampling enough to guarantee that different observations will not influence each other?
