Show that for a model fit by least squares, the total sum of squares is equal to the model sum of squares plus the residual sum of squares.

That is, show that

\[ \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}=\sum_{i=1}^{n}\left(\widehat{y}_{i}-\bar{y}\right)^{2}+\sum_{i=1}^{n}\left(y_{i}-\widehat{y}_{i}\right)^{2} \]

is an algebraic identity if the residuals sum to 0 ; that is,

\[ \sum_{i=1}^{n}\left(y_{i}-\widehat{y}_{i}\right)=0 \]