zero conditional mean assumption coupled with random sampling assumption (deriving unbiasedness)

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?

• By conditioning on it, you can extract it from the expectation operator, as if it were constant (fixed). The idea is that strict exogeneity yields the same OLS properties as if we just treated X as nonrandom.
– VCG
Sep 20 '16 at 23:31