# Deriving the relation $\sum (y_i - \bar{y})^2 - \sum (y_i - \hat{y}_i)^2 = \sum (\bar{y} - \hat{y}_i)^2$

$y_i$ denotes the observed values of $y$, $\hat{y}_i$ are the projected values from the explanatory variables, and $\bar{y}$ is the mean. This relation is supposed to hold up when one of the explanatory variables is the constant. So far, I have gotten $$\sum (y_i - \bar{y})^2 - \sum (y_i - \hat{y}_i)^2 = \sum(-y_i \bar{y} + 2 y_i \hat{y}_i - \hat{y}_i^2)$$ It seems that if I can somehow show that $\sum y_i \bar{y} = \sum y_i \hat{y}_i$ then the derivation would be complete. Can someone show me how to proceed from here on?

• It is qualified by stating that this is true only when $\hat{\beta}_0 \neq 0$ so the constant explanatory variable contributes to the regression. I agree that this is not true in the general case. – Astaboom May 6 '16 at 8:53

## 1 Answer

Somewhat ugly answer: \begin{align*} \sum_i \left[ \left( y_i - \bar{y}\right)^2 - \left( y_i - \hat{y}_i\right)^2 - \left(\bar{y} - \hat{y}_i \right)^2 \right] &= \sum_i \left( y_i^2 - 2\bar{y}y_i + \bar{y}^2 - y_i^2 + 2y_i \hat{y}_i - \hat{y}_i^2 - \bar{y}^2 + 2\bar{y}\hat{y}_i - \hat{y}_i^2\right) \\ &= \sum_i \left( - 2\bar{y}y_i + 2y_i \hat{y}_i + 2\bar{y}\hat{y}_i- 2\hat{y}_i^2 \right)\\ &= -2n\bar{y}^2 + 2n\bar{y}^2 + 2\sum_i \hat{y}_i\left(y_i - \hat{y}_i \right) \\ &= 2 \sum_i \hat{y}_i\left( y_i - \hat{y_i}\right) \end{align*} Which equals zero because it's basically the orthogonality condition that's used to estimate your $\hat{b}$ vector in ordinary least squares.

Sketch of linear algebra to show this explicitly (it's so much easier and succinct to use matrix notation...)

\begin{align*} \sum_i \hat{y}_i\left( y_i - \hat{y_i}\right) &=(Xb)'(y - Xb)\\ &= b'X'y-b'X'Xb \\ &= y'X(X'X)^{-1}X'y - y'X(X'X)^{-1}X'X(X'X)^{-1}X'y\\ &= y'X(X'X)^{-1}X'y - y'X(X'X)^{-1}X'y\\ &= 0 \end{align*}

• Thanks for the input. I would have never thought of flipping the LHS to the RHS and equate the whole thing to 0. Nice. Though, just one minor point: when you wrote $\sum (-2 \bar{y} y_i + 2 \bar{y} \hat{y}_i) = -2\bar{y}^2 + 2 \bar{y}$, you actually meant $-2n\bar{y}^2 + 2 n \bar{y}$, right? After all, I do see that $\sum y_i = \sum \hat{y}_i = n \bar{y}$. – Astaboom May 7 '16 at 4:56
• And going by the matrix form does make everything simpler by a mile. It's uncanny how simple it becomes. – Astaboom May 7 '16 at 4:59
• @Astaboom good catch, yeah there should be an $n$ there (i just fixed it). – Matthew Gunn May 7 '16 at 13:54