I am revisiting the basic notions of linear regression and stumbled upon the following idea in Cameron and Trivedi's Microeconometrics book:
However, for the conditional mean to be linear in x, so that $E[y|x] = α+xγ$, requires the assumption that $E[u|x] = 0$, in addition to $E[u] = 0$ and $Cov[x,u] = 0$.
They way I thought about linear prediction is that it is the same as the conditional mean (or conditional expectation function) if that is actually linear (otherwise it is still the best linear prediction) . What I fail to understand is how exactly $E[u|x]=0$ guarantees that the conditional mean $E[y|x]$ is linear.
What if the underlying distribution is non-linear? The conditional mean should be non-linear as well. How does $E[u|x]=0$ change that? Is this somehow related to the idea of omitted variables in the actual estimation (e.g. via OLS)?