I don't get part of the explanation of deriving unbiasedness of OLS in my textbook. I understand that to derive unbiasedness we have to use conditional expectation (conditioning on $x$) so that the error term goes to zero $E(u|x)=0$ and we can proof unbiasedness of OLS. The author of my textbook writes:

"In addition to restricting the relationship between u and x in the population, the zero conditional mean assumption - coupled with the random sampling assumption - allows for technical simplification. We can derive the statistical properties of the OLS estimators as conditional on the values of $x_i$ in our sample. Technically, in statistical derivations, conditioning on the sample values of the independent variable is the same as treating the $x_i$ as fixed in repeated samples."

My question is now: Why is conditioning on sample values of $x_i$ the same as treating $x_i$ fixed in repeated samples? In my opinion unbiasedness of OLS can be derived just by using the conditional expectation properties. I hope someone may explain me intuitively how this works with the zero conditional mean assumption coupled with the random sampling assumption. And why exactly the random sampling assumption makes such a difference/simplification.

I don't get part of the explanation of deriving unbiasedness of OLS in my textbook.

I understand that to derive unbiasedness we have to use conditional expectation (conditioning on $x$) so that the error term goes to zero $E(u|x)=0$ and we can prove unbiasedness of OLS.

The author of my textbook writes:

In addition to restricting the relationship between u and x in the population, the zero conditional mean assumption - coupled with the random sampling assumption - allows for technical simplification. We can derive the statistical properties of the OLS estimators as conditional on the values of $x_i$ in our sample. Technically, in statistical derivations, conditioning on the sample values of the independent variable is the same as treating the $x_i$ as fixed in repeated samples.

My question is now: Why is conditioning on sample values of $x_i$ the same as treating $x_i$ fixed in repeated samples? In my opinion unbiasedness of OLS can be derived just by using the conditional expectation properties. I hope someone may explain me intuitively how this works with the zero conditional mean assumption coupled with the random sampling assumption. And why exactly the random sampling assumption makes such a difference/simplification?

I don't get part of the explanation of deriving unbiasedness of OLS in my textbook. I understand that to derive unbiasedness we have to use conditional expectation (conditioning on x) so that the error term goes to zero (E(u|x)=0) and we can proof unbiasedness of OLS. The author of my textbook writes:

"In addition to restricting the relationship between u and x in the population, the zero conditional mean assumption - coupled with the random sampling assumption - allows for technical simplification. We can derive the statistical properties of the OLS estimators as conditional on the values of x_i in our sample. Technically, in statistical derivations, conditioning on the sample values of the independent variable is the same as treating the x_i as fixed in repeated samples."

My question is now: Why is conditioning on sample values of x_i the same as treating x_i fixed in repeated samples? In my opinion unbiasedness of OLS can be derived just by using the conditional expectation properties. I hope someone may explain me intuitively how this works with the zero conditional mean assumption coupled with the random sampling assumption. And why exactly the random sampling assumption makes such a difference/simplification.

I don't get part of the explanation of deriving unbiasedness of OLS in my textbook. I understand that to derive unbiasedness we have to use conditional expectation (conditioning on $x$) so that the error term goes to zero $E(u|x)=0$ and we can proof unbiasedness of OLS. The author of my textbook writes:

"In addition to restricting the relationship between u and x in the population, the zero conditional mean assumption - coupled with the random sampling assumption - allows for technical simplification. We can derive the statistical properties of the OLS estimators as conditional on the values of $x_i$ in our sample. Technically, in statistical derivations, conditioning on the sample values of the independent variable is the same as treating the $x_i$ as fixed in repeated samples."

My question is now: Why is conditioning on sample values of $x_i$ the same as treating $x_i$ fixed in repeated samples? In my opinion unbiasedness of OLS can be derived just by using the conditional expectation properties. I hope someone may explain me intuitively how this works with the zero conditional mean assumption coupled with the random sampling assumption. And why exactly the random sampling assumption makes such a difference/simplification.