Using the notation from Exercises 11-13, Cook's distance for observation \(i\) is defined as

\[ D_{i}:=\frac{\widehat{\boldsymbol{Y}}-\widehat{\boldsymbol{Y}}^{(i)^{2}}}{p S^{2}} \]

It measures the change in the fitted values when the \(i\)-th observation is removed, relative to the residual variance of the model (estimated via \(S^{2}\) ).
By using similar arguments as those in Exercise 13, show that \[ D_{i}=\frac{\mathbf{P}_{i i} E_{i}^{2}}{\left(1-\mathbf{P}_{i}\right)^{2} p S^{2}} \]
It follows that there is no need to "omit and refit" the linear model in order to compute Cook's distance for the \(i\)-th response.