Least Squares Estimator with Linear Restrictions

Definition

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ and linear restrictions $A\mathbf{\beta }=\mathbf{c}$ By Method of Lagrange Multipliers, the least squares estimator with the linear restrictions is estimated by ${\hat{\mathbf{\beta }}}_H=\underset{\mathbf{\beta }}{\mathrm{argmin}}{\mathbf{ϵ}}^{⊺}\mathbf{ϵ}+\mathbf{\lambda }(A\mathbf{\beta }-\mathbf{c})$ The analytic solution of ${\hat{\mathbf{\beta }}}_H$ is obtained as ${\hat{\mathbf{\beta }}}_H=\hat{\mathbf{\beta }}+({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{A}^{⊺}[A({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{A}^{⊺}{]}^{-1}(A\hat{\mathbf{\beta }}-\mathbf{c})$

Facts

$RS{S}_H=||\mathbf{y}-\mathbf{X}{\hat{\mathbf{\beta }}}_H|{|}^2=||\mathbf{y}-\mathbf{X}\hat{\mathbf{\beta }}|{|}^2+||X(\hat{\mathbf{\beta }}-{\hat{\mathbf{\beta }}}_H)|{|}^2$