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Thread: Sst=ssr+sse

  1. #1
    Member
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    Acolman, Mexico
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    Sst=ssr+sse

    Hello,
    I am stuck proving the following:

    For simple linear regression prove that
    SST=SSR+SSE
    where
    $\displaystyle SSE = \sum(Y_i-\hat{Y}_i)^2$
    $\displaystyle SSR = \sum(\hat{Y}_i-\bar{Y})^2$
    $\displaystyle SST = \sum(Y_i-\bar{Y})^2$

    Here's what I have so far,

    $\displaystyle (Y_i-\bar{Y})^2=(Y_i-\hat{Y}_i+\hat{Y}_i-\bar{Y})^2=((Y_i-\hat{Y}_i)+(\hat{Y}_i-\bar{Y}))^2=$
    $\displaystyle (Y_i-\hat{Y}_i)^2-2(Y_i-\hat{Y}_i)(\hat{Y}_i-\bar{Y})+(\hat{Y}_i-\bar{Y})^2
    $

    So,$\displaystyle
    SST=\sum(Y_i-\bar{Y})^2=\sum{(Y_i-\hat{Y}_i)^2}-2\sum{(Y_i-\hat{Y}_i)(\hat{Y}_i-\bar{Y})}+\sum{(\hat{Y}_i-\bar{Y})^2}=$
    $\displaystyle SSE-2\sum{(Y_i-\hat{Y}_i)(\hat{Y}_i-\bar{Y})}+SSR$

    Thanks in advance.
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  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
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    5
    start with SSTotal and add and subtract Y hat's
    Then do the square and the intermediate term drops out.
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