I've studied SST and the others but I saw the equation that $SSR=(n-1)(β_1)^2(S_x)^2$.
I want to know how the equation is done. Thnks!!
$\endgroup$ 31 Answer
$\begingroup$In simple regression, we have
$$ \bar{y} = \hat{\beta_0} + \hat{\beta_1} \bar{x} $$
by normal equations. Thus,
$$ \begin{align} SSR &= \sum_{i=1}^n (\hat{y} - \bar{y})^2 \\ &= \sum_{i=1}^n \left( (\hat{\beta_0} + \hat{\beta_1}x_i) - (\hat{\beta_0} + \hat{\beta_1} \bar{x}) \right)^2 \\ &= \sum_{i=1}^n \left( (\hat{\beta_1})^2 (x_i - \bar{x})^2 \right) \\ &= (\hat{\beta_1})^2 \sum_{i=1}^n ( (x_i - \bar{x})^2 \\ &= (\hat{\beta}_1)^2S_{xx} \end{align} $$
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