# Why is $\sigma^2 = Var(\epsilon)$ when computing the standard error of a simple linear regression slope parameter

Assume the true underlying linear approximation of a set of data is equal to $$Y=2+3X +\epsilon$$ where $$\epsilon$$ represents the irreducible error that is inherent in a linear approximation. I then perform a linear regression and arrive at my $$\hat{\beta}_0$$ and $$\hat{\beta}_1$$ parameters. In order to determine the standard error of $$\hat{\beta}_0$$ and $$\hat{\beta}_1$$, I use the following formulas:

$$SE(\hat{\beta}_0)^2= \sigma^2 \big{[} \frac{1}{n} + \frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^2} \big{]}$$

$$SE(\hat{\beta}_1)^2= \frac{\sigma^2}{\sum_{i=1}^n(x_i-\bar{x})^2}$$

where $$\sigma^2 = Var(\epsilon)$$

Why does $$\sigma^2 = Var(\epsilon)$$ and not $$\sigma^2 = Var(x)$$ (the variance of my sample population)?

Where does $$\epsilon$$ work its way in to the derivation of the standard errors of these two parameters? $$\epsilon$$ has nothing to do with the least squares approach for calculating $$\hat{\beta}_0$$ and $$\hat{\beta}_1$$ in the first place. Also, how do we determine $$\epsilon$$? Doesn't $$\epsilon$$ require knowing that the "true" linear approximation of the data is $$Y=2+3X +\epsilon$$? We obviously would not know this in real life.

EDIT: one last question: what does it mean to assume that the errors $$\epsilon_i$$ for each observation are uncorrelated with common variance $$\sigma^2$$?

• Ponder the different meanings of "$Var$" you are using: in $\sigma^2=\operatorname{Var}(\epsilon)$ the variance is a property of a random variable whereas in $\sigma^2=\operatorname{Var}(x)$ the variance measures the spread of a set of regressor values. – whuber Oct 5 '18 at 13:11