Counterexample for zero conditional mean

Question

This has to do with linear regression. I know the zero conditional mean condition, where $E[u\mid X] = 0$. I know that $E[u] = 0$ does not imply $E[u \mid X] = 0$. However, suppose I assume $E[u]= 0$ and $E[uX] = 0$. Does these two conditions imply $E[u \mid X] = 0$? My hunch is no, but I'm struggling to find a counter example. Can I get a hint?

Answer 1

No. Consider $X$ being $N(0,1)$ and $u = X^2 - 1$. Then $E[u|X] = X^2 - 1$.

But the "linear" projection from $u$ on $X$ is indeed always zero.

The other exception is when $X$ and $u$ are jointly normal, in which case, the condition expectation coincides with the linear projection.