I'm having trouble understanding the math behind $Var(u|x)=\sigma^2$. The conditional expectation of the variance is defined as
$$Var(u|x)=E(u^2|x)-[E(u|x)]^2$$
zero conditional mean assumption:
$$E(u|x)=E(u)=0$$
using this assumption yields:
$$Var(u|x)=E(u^2|x)$$
$$\sigma^2=E(u^2|x)$$
Until here I understand.
In my textbook the author writes then:
$\sigma^2$ is also the unconditional expectation of $u^2$ and therefore $\sigma^2=E(u^2)=Var(u)$, because $E(u)=0$. In other words, $\sigma^2$ is the unconditional variance of u.
Why is $\sigma^2$ the unconditional expectation of $u^2$? What is the unconditional expectation/variance? I really don't understand the last part at all. The textbook just doesn't explain it very well.
The assumption seems to be that you are dealing with a random variable $u$ for which $E(u) = 0$. The variance is defined as
$$\sigma ^2 = E((u - E(u))^2)$$
If $E(u) = 0$ this reduces to $\sigma ^2 = E(u^2)$.
The word "unconditional" is just for emphasis since in the absence of explicit conditioning variance is unconditional.