Conditional and unconditional expectation for the variance of error term in linear regression

Question

I'm working through the book Introductory Econometrics and stumbled across a statement regarding the variance of the error term, $u$, of a linear regression model, $y = \beta_0 + \beta_1 x + u$.

To give some context, two assumptions were introduced beforehand:

1. Zero conditional mean, i.e. $E(u|x) = 0$, and
2. Homoskedasticity, i.e. $Var(u|x) = \sigma^2$.

Then, the argument goes on as follows:

Because $Var(u|x) = E(u^2|x) - [E(u|x)]^2$ and $E(u|x) = 0$, $\sigma^2 = E(u^2|x)$, which means $\sigma^2$ is also the unconditional expectation of $u^2$.

While I understand the first part of the sentence, I have no idea where the bolded part comes from. It seems to say that because $E(u^2|x)=\sigma^2$ (i.e. the conditional expectation of $u^2$), it follows that $E(u^2) = \sigma^2$ (i.e. the unconditional expectation of $u^2$). I might be missing something very basic here, but I can't figure it out.

Answer 1

It follows from the law of iterated expectations: the expected value of the conditional expected value of $u$ given $X$ is the same as the expected value of $u$.

$$E[u^2] = E[E[u^2|X]] = E[\sigma^2] = \sigma^2$$