If $\text E[\varepsilon|X] = 0$, why is $\text{cov}(\varepsilon , X) = 0$? If $\text E[\varepsilon|X] = 0$, why is $\text{cov}(\varepsilon , X) = 0$? Can someone please explain it algebraically and intuitively?
This is related to OLS and is based on the first assumption of CLRM.

Observe $$\operatorname{Cov}[\mathbf x, \varepsilon]=\operatorname{Cov}[\mathbf x, \operatorname E[ \varepsilon|\mathbf x]]=\operatorname{Cov}[\mathbf x, 0]=\mathbf 0.$$
