# What is the difference between strict / strong and weak exogeneity

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 $$y$$.)

• Nov 20, 2019 at 16:22