Standard Error for Regression Coefficient

Definition

Assume that $\mathbf{y}\sim N_n(\mathbf{X}\mathbf{\beta },{\mathbf{I}}_n{\sigma }^2)$ The standard error for regression coefficient ${\beta }_j$ is calculated as $SE({\hat{\beta }}_j)=\sqrt{c_{j+1}}s$ where $c_{j+1}=\mathrm{diag}[({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{]}_{j+1}$, $s^2=\frac{{\sum }_{i=1}^n(Y_i-\hat{Y}{)}^2}{n-p}$, and $p-1$ is the number of explanatory variables.