DFBETAS

Definition

The DFBEATS consider the influence of the $i$-th observation on ${\hat{\mathbf{\beta }}}_j$

Single Observation

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $DFBETA{S}_{j,i}=\frac{{\hat{\mathbf{\beta }}}_j-{\hat{\mathbf{\beta }}}_{j(i)}}{s_{(i)}\sqrt{({\mathbf{X}}^{⊺}\mathbf{X}{)}_{jj}^{-1}}}=\frac{c_{ji}}{\sqrt{{\sum }_{l=1}^n{c}_{jl}^2}}\frac{e_i}{s_{(i)}(1-h_{ii})}$ where $s_{(i)}$ is the externally studentized residual and $c_{ji}$ is the element of the matrix $C:=({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{\mathbf{X}}^{⊺}$

Set of Observations

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $DFBETA{S}_{j,K}=\frac{{\hat{\mathbf{\beta }}}_j-{\hat{\mathbf{\beta }}}_{j(K)}}{s_{(K)}\sqrt{({\mathbf{X}}^{⊺}\mathbf{X}{)}_{jj}^{-1}}}=\frac{{\mathbf{c}}_{jk}^{⊺}(I-{\mathbf{H}}_K{)}^{-1}{\mathbf{e}}_K}{s_{(K)}\sqrt{{\sum }_{l=1}^n{c}_{jl}^2}}$ where $\mathbf{H}$ is the matrix of leverages and $c_{ji}$ is the element of the matrix $C:=({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{\mathbf{X}}^{⊺}$