How Variance of error term is equal to the conditional variance of y given x in linear regression

Question

How the variance of the error term is equal to the conditional variance of y given x in linear regression?

Answer 1

I'm presuming your model assumption is something like $$y = \alpha + \beta x + \epsilon$$ where $\epsilon$ is noise and is independent of $x$.

If you are conditioning on $x$, then the only random quantity is $\epsilon$, so $y$ is just $\epsilon$ plus some non-random quantities. Shifting a random variable does not change its variance. $$\text{Var}(y \mid x) = \text{Var}(\alpha + \beta x + \epsilon \mid x) = \text{Var}(\epsilon \mid x) = \text{Var}(\epsilon).$$ The last step is due to independence of $\epsilon$ and $x$.