In simple linear regression, I know the confidence interval can be calculated from:
$CI = t\sqrt{\frac{\sigma^2}{n}+\frac{(x-\bar{x})^2\sigma^2}{\sum{(x-\bar{x})^2}}}$
I know this equation comes from the relation $Y=\bar{Y} +\beta_1(x-\bar{x})$
Is there a way to either use Matrix Algebra to calculate this from the Standard Error matrix?
$\sigma^2(X'X)^{-1}$ where $\sigma^2 = \frac{SS_e}{\nu}$ and degrees of freedom $\nu = n-2$
I'm looking for a way to extend the confidence interval to multiple linear regression, including polynomial x.