2
$\begingroup$

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) :

image

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!

$\endgroup$
3
  • $\begingroup$ Read here: stats.stackexchange.com/questions/215230/… $\endgroup$
    – markowitz
    Mar 28, 2021 at 11:04
  • $\begingroup$ @markowitz does this mean that $x$ is treated as a deterministic variable while in reality it is random?Anyway I struggle to answer my questions with your link, even if I see it is really related to my post $\endgroup$
    – Tortar
    Mar 28, 2021 at 12:00
  • $\begingroup$ "does this mean that $x$ is treated as a deterministic variable while in reality it is random?" sometimes it is so. "Anyway I struggle to answer my questions with your link, even if I see it is really related to my post", Indeed this was a related discussion only. I tried to summarize the point below. $\endgroup$
    – markowitz
    Mar 28, 2021 at 13:40

1 Answer 1

4
$\begingroup$

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

$\endgroup$
4
  • $\begingroup$ Thank you a lot ! Can I ask you if there's a book (not too advanced possibly) which contains more information about this than Wooldridge's one or should I abandon the hope ? $\endgroup$
    – Tortar
    Mar 28, 2021 at 14:11
  • 1
    $\begingroup$ Unfortunately it seems me that today do not exist an generalistic econometrics book that address properly all problems above. However, at introduction level Stock and Watson seems me a good book: pearson.com/uk/educators/higher-education-educators/program/…. About causality I suggest Angrist and Pischke (2009): press.princeton.edu/books/paperback/9780691120355/…; $\endgroup$
    – markowitz
    Mar 28, 2021 at 15:56
  • 1
    $\begingroup$ about forecasting, Elliot and Timmermann (2016): press.princeton.edu/books/hardcover/9780691140131/… Said that. If you are a student stay focused on the material that your professor suggest. $\endgroup$
    – markowitz
    Mar 28, 2021 at 15:56
  • $\begingroup$ I value your advice to stay focused but I like to try to understand things more than it is required in a course (unfortunately for my mental health :P). So I really thank you for the suggestions. $\endgroup$
    – Tortar
    Mar 28, 2021 at 19:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.