Clarification on the assumptions $E[u|x]=0$ and the $x_i$ being fixed in repeated samples in Wooldridge Introductory Econometrics

Question

The author is writing on the assumption $E[u|x]=0$.

The part of the text which is not clear to me is this (the red lines emphasize where the critical portions are located) :

In the first piece I don't get why the zero conditional mean assumption can make it possible to "derive the properties of the OLS estimators as conditional on the values of the $x_i$"; it seems to me that it's always possible, it is just that with this assumption we can simplify the expression much more and derive unbiasedness for $\hat\beta_0$ and $\hat\beta_1$ , is this the meaning of what's written?

In the last part, instead, the author says that "nothing is lost in derivations by treating the $x_i$ as nonrandom", but why is this the case? Assuming the fixed-in-repeated-samples is not how the datas are generated, so why assuming this in the first place?

Then he says that "the danger is that the fixed-in repeated-samples assumption always implies that $u_i$ and $x_i$ are independent" ; does this mean that assuming the fixed-in repeated-samples is stronger than the zero conditional mean assumption + the random sampling assumption? So the author is simplifying all the analysis using this one? It appeared to me, from what is written at the beginning, that the zero conditional mean assumption + the random sampling assumption implied the fixed in repeated samples one.

Note : the random sampling assumption was defined as "We have a random sample $\{(x_i,y_i) : i = 1,2,\dots,n\}$ following the population model".

Thanks in advance!

Answer 1

Your question do not admit short and exhaustive answer. The assumption you interested in $E[u|x]=0$, usually named exogeneity assumption, is crucial in Econometrics. Unfortunately exogeneity concept (and so related assumptions) is bad treated, many problems come from that.

I wrote a lot about them, I suggest you some related discussion that can help:

The origin of the problems: How would econometricians answer the objections and recommendations raised by Chen and Pearl (2013)?

a possible way for solutions: Under which assumptions a regression can be interpreted causally?

note that "fixed-in repeated-samples assumption" is a common way for conflate causal and statistical concepts. Unfortunately the book you refers on do not help.

some more specific points about exogeneity:

endogenous regressor and correlation

Random Sampling: Weak and Strong Exogenity (and links therein)

Regression's population parameters

Homoscedasticity and independence of errors

What are the differences between stochastic and fixed regressors in linear regression model?

Endogeneity in forecasting