In my econometrics class regarding multiple linear regression, we learned that one of the Gauss-Markov assumptions is the zero conditional mean, expressed as $ E(y|\boldsymbol{x}) = 0$.
My question is: If we transform $\boldsymbol{x}$ into $g(\boldsymbol{x})$, does this zero conditional mean still hold for the new conditioned variable? In other words, does $E(y|g(\boldsymbol{x})) = 0$ for arbitrary $g$?
The law of iterated expectations is useful here.
In particular, the function $g$ can at best maintain your information set, but may decrease it (e.g., when $g$ is such that it drops some regressor from $x$). And given that the smaller information set dominates, we may write $$ E[y|g(x)]=E[E(y|x)|g(x)], $$ which, by assumption, equals $E[0|g(x)]=0$.
Comment: You may want to check your notes again in that $y$ is often the notation for the dependent variable in your regression model, where the assumption of a zero conditional mean is typically for the error term, i.e., something like $E(u|x)=0$.
Indeed, if we had $E(y|x)=0$ that would be tantamount to saying that there is no (mean) relationship between $y$ and $x$, which would imply that regressing $y$ on $x$ would not be interesting.
