TITLE: Prove $SST=SSE+SSR$
QUESTION [17 upvotes]: Prove $$SST=SSE+SSR$$
I start with $$SST= \Sigma (y_i-\bar{y})^2=...=SSE+SSR+ \Sigma 2( y_i-y_i^*)(y_i^*-\bar{y} )$$ and I don't know how to prove that  $\Sigma 2( y_i-y_i^*)(y_i^*-\bar{y} )=0$

a note on notation: the residuals $e_i$ is $e_i=y_i-y_i^*$. A more common notation is $\hat{y}$.

REPLY [3 votes]: When an intercept is included in linear regression(sum of residuals is zero), $SST=SSE+SSR$.
prove
$$
\begin{eqnarray*}
SST&=&\sum_{i=1}^n (y_i-\bar y)^2\\&=&\sum_{i=1}^n (y_i-\hat y_i+\hat y_i-\bar y)^2\\&=&\sum_{i=1}^n (y_i-\hat y_i)^2+2\sum_{i=1}^n(y_i-\hat y_i)(\hat y_i-\bar y)+\sum_{i=1}^n(\hat y_i-\bar y)^2\\&=&SSE+SSR+2\sum_{i=1}^n(y_i-\hat y_i)(\hat y_i-\bar y)
\end{eqnarray*}
$$
Just need to prove last part is equal to 0: 
$$\begin{eqnarray*}
\sum_{i=1}^n(y_i-\hat y_i)(\hat y_i-\bar y)&=&\sum_{i=1}^n(y_i-\beta_0-\beta_1x_i)(\beta_0+\beta_1x_i-\bar y)\\&=&(\beta_0-\bar y)\sum_{i=1}^n(y_i-\beta_0-\beta_1x_i)+\beta_1\sum_{i=1}^n(y_i-\beta_0-\beta_1x_i)x_i
\end{eqnarray*}
$$
In Least squares regression, the sum of the squares of the errors is minimized.
$$
SSE=\displaystyle\sum\limits_{i=1}^n \left(e_i \right)^2= \sum_{i=1}^n\left(y_i - \hat{y_i} \right)^2= \sum_{i=1}^n\left(y_i -\beta_0- \beta_1x_i\right)^2
$$
Take the partial derivative of SSE with respect to $\beta_0$ and setting it to zero.
$$
\frac{\partial{SSE}}{\partial{\beta_0}} = \sum_{i=1}^n 2\left(y_i - \beta_0 - \beta_1x_i\right)^1 (-1) = 0
$$
So
$$
\sum_{i=1}^n \left(y_i - \beta_0 - \beta_1x_i\right)^1 (-1) = 0
$$
Take the partial derivative of SSE with respect to $\beta_1$ and setting it to zero.
$$
\frac{\partial{SSE}}{\partial{\beta_1}} = \sum_{i=1}^n 2\left(y_i - \beta_0 - \beta_1x_i\right)^1 (-x_i) = 0
$$
So
$$
\sum_{i=1}^n \left(y_i - \beta_0 - \beta_1x_i\right)^1 x_i = 0
$$
Hence,
$$
\sum_{i=1}^n(y_i-\hat y_i)(\hat y_i-\bar y)=(\beta_0-\bar y)\sum_{i=1}^n(y_i-\beta_0-\beta_1x_i)+\beta_1\sum_{i=1}^n(y_i-\beta_0-\beta_1x_i)x_i=0
$$
$$SST=SSE+SSR+2\sum_{i=1}^n(y_i-\hat y_i)(\hat y_i-\bar y)=SSE+SSR$$
similar question: 
https://stats.stackexchange.com/a/401299/243636