coefficient of determination: absence of cross products

Question

With regard to the coefficient of determination, why is the total variation equal to the sum of the explained variation and the unexplained variation and there are no cross-products?

Answer 1

\begin{align} \sum(Y_i - \bar{Y})^2 &= \sum[(Y_i - \hat{Y}_i)+( \hat{Y}_i-\bar{Y})]^2\\ &=\sum(Y_i - \hat{Y}_i)^2 + \sum(\hat{Y}_i - \bar{Y})(Y_i - \hat{Y}_i) + \sum(\hat{Y}_i - \bar{Y})^2\\ & =\sum(Y_i - \hat{Y})^2+\sum(\hat{Y}_i - \bar{Y})^2 \end{align} because $$ \sum(\hat{Y}_i - \bar{Y})(Y_i - \hat{Y}_i) = \sum(\hat{Y}_i - \bar{Y})e_i = \sum \hat{Y}_ie_i - \sum\bar{Y}e_i = 0 + 0=0. $$

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The last two assertions follow from the OLS construction $$ s(\beta) = \sum \left(Y_i - \beta_0- \sum \beta_jX_j\right)^2\\ s_{\beta_0}'(\hat{\beta}) =-2\sum \left(Y_i - \hat{\beta_0}- \sum \hat{\beta}_jX_j\right)=-2\sum e_i =0. $$ For the first assertion, $\sum \hat{Y}_ie_i $ try to prove the orthogonality of $\hat{Y}$ and $e$ vectors in the simple regression model, then it is straightforward to extend it to $p$ number of $\beta$.