Joint Confidence Region for Regression Coefficient

Definition

Assume that $\mathbf{y}\sim N_n(\mathbf{X}\mathbf{\beta },{\mathbf{I}}_n{\sigma }^2)$ The $100(1-\alpha )\%$ join confidence region for $\mathbf{\beta }$ is defined as $\{\mathbf{\beta }|(\mathbf{\beta }-\hat{\mathbf{\beta }}{)}^{⊺}{\mathbf{X}}^{⊺}\mathbf{X}(\mathbf{\beta }-\hat{\mathbf{\beta }})\le p{s}^2{F}_{\alpha }(p,n-p)\}$ where $s^2=\frac{{\sum }_{i=1}^n(Y_i-\hat{Y}{)}^2}{n-p}$ and $p-1$ is the number of explanatory variables.

The joint confidence region is ellipsoidal shape whose center is $\hat{\mathbf{\beta }}$ by the inequality.