Let be two variables $y$ and $x$, the latter being expected to be a cause of the latter. If we suppose linearity, we can set up a model:
$$y=\beta_0+\beta_1x+u$$
Where $\beta_0$ and $\beta_1$ are coefficients to be determined, and $u$ is a random noise ($\mathbb{E}[u]=0$) representing all other effects on $y$ not mediated through $x$.
If we make an so-called "exogeneity" hypothesis, we can interpret $\beta_1$ as the effect of $x$ on $y$. (Intuitively, exogeneity means that we can only estimate the effect of $x$ on $y$ if there is no spurious causal pathways, like a third variable $z$ causing both $x$ and $y$, messing with the estimation.) However, there are two versions of exogeneity, weak and strong, and I cannot make my mind around the intuitive difference between the two.
Weak exogeneity is $\mathbb{E}[ux]=0$ ("null correlation") ;
Strong or strict exogeneity is $\mathbb{E}[u|x]=0$ ("zero conditionnal mean").
I know that $\mathbb{E}[u|x]=0$ implies $\mathbb{E}[ux]=0$, since: $$\mathbb{E}_{x, u}[u\cdot x]=\mathbb{E}_x[\mathbb{E}_u[u \cdot x|x]]=\mathbb{E}_x[x\cdot\mathbb{E}_u[u|x]]=\mathbb{E}_x[x\cdot 0]=0$$
... but what does it intuitively mean?
Also, if I have a third variable $z$, what do these conditions become? I usually read something like $\mathbb{E}[u|x,z]=0$ for strong exogeneity. Is this stronger than $\mathbb{E}[u|x]=0$ and $\mathbb{E}[u|z]=0$ holding simultaneously or is that equivalent? And week exogeneity? Is it only $\mathbb{E}[ux]=0$ and $\mathbb{E}[uz]=0$ ? Do hypotheses like $\mathbb{E}[uz|x]=0$ make sense? And if it does, what does it intuitively mean? (For context, I am only interested in correctly estimating $\beta_1$ and I actually don't care about $z$ except for its confounding effects on $x$ and $y$.)
