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Math Help - SSE simple linear regression

  1. #1
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    SSE simple linear regression

    I am given SSE= \sum_{i=1}^n(y_i-\hat{y}_i)^2 and am asked to show that it =\sum_{i=1}^n(y_i-\bar{y})^2-\hat{\beta}_1\sum_{i=1}^n(x_i-\bar{x})(y_i-\bar{y})

    where \hat{y}_i=\hat{\beta}_0+\hat{\beta}_1x_i and \hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}

    So here's what I got:


    \sum_{i=1}^n(y_i-\hat{y}_i)^2

    =\sum_{i=1}^n(y_i-\hat{\beta}_0-\hat{\beta}_1x_i)^2

    =\sum_{i=1}^n(y_i-\bar{y}+\hat{\beta}_1\bar{x}-\hat{\beta}_1x_i)^2

    =\sum_{i=1}^n(y_i-\bar{y}-\hat{\beta}_1(x_i-\bar{x}))^2

    =\sum_{i=1}^n[(y_i-\bar{y})^2-2\hat{\beta}_1(y_i-\bar{y})(x_i-\bar{x})+\hat{\beta}_1^2(x_i-\bar{x})^2]

    =\sum_{i=1}^n(y_i-\bar{y})^2-2\hat{\beta}_1\sum_{i=1}^n(y_i-\bar{y})(x_i-\bar{x})+\hat{\beta}_1^2\sum_{i=1}^n(x_i-\bar{x})^2

    and i've got nothing from here.... anyone? thanks
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  2. #2
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    You are just one step short. For simplicity convert everything into sums of squares notation, with

    S_{xx} = \sum (x_i - \bar{x})^2, S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) and note that \hat{\beta_1} = \frac{S_{xy}}{S_{xx}} (which I'm assuming you can take as given).
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  3. #3
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    Quote Originally Posted by theodds View Post
    You are just one step short. For simplicity convert everything into sums of squares notation, with

    S_{xx} = \sum (x_i - \bar{x})^2, S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) and note that \hat{\beta_1} = \frac{S_{xy}}{S_{xx}} (which I'm assuming you can take as given).
    =\sum_{i=1}^n(y_i-\bar{y})^2-2\hat{\beta}_1\sum_{i=1}^n(y_i-\bar{y})(x_i-\bar{x})+\hat{\beta}_1^2\sum_{i=1}^n(x_i-\bar{x})^2

    =S_{yy}-2\hat{\beta}_1S_{xy}+\hat{\beta}_1\frac{S_{xy}}{S_  {xx}}\cdot S_{xx}

    =S_{yy}-\hat{\beta}_1S_{xy}


    Good call
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