3.11 Centering and Scaling the Explanatory Variables: 3.11.1 Centering, 3.11.2 Scaling

If $\tilde{Y}_{i} = Y_{i}-\bar{Y}$ and $\tilde{\mathbf{Y}} = \left(\tilde{Y}_{1}, \ldots , \tilde{Y}_{n}\right)^{T}$, prove from (3.54) that RSS $= \tilde{\mathbf{Y}}^{T}\left(\mathbf{I}_{n}-\tilde{\mathbf{P}}\right) \tilde{\mathbf{Y}}$.

Suppose that we consider fitting a model in which the $Y$-data are centered and scaled as well as the $x$-data. This means that we use $Y_{i}^{*} = \left(Y_{i}-\bar{Y}\right) / s_{y}$ instead of $Y_{i}$, where $s_{y}^{2} = \sum_{i}\left(Y_{i}-\bar{Y}\right)^{2}$. Using (3.54), obtain an expression for RSS from this model.