Zero Conditional Mean (ZCM), or Strict Exogeneity, is given by:
$E[u|X]=0$
Equivalently,
$E[u_t|X]=0, t=1,...,T$

Is it true that this implies:

• Zero Unconditional Mean: $E[u_t]=0, \forall t$

• Contemporaneous Exogeneity: $E[u_t|x_t]=0, \forall t $ (Where $x_t$ is a vector of explanatory variables)

• $E[x_su_t]=0, \forall t,s$

And are there any other things that ZCM imply that are particularly useful?