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Math Help - Sst=ssr+sse

  1. #1
    Member
    Joined
    Mar 2008
    From
    Acolman, Mexico
    Posts
    118

    Sst=ssr+sse

    Hello,
    I am stuck proving the following:

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

    Here's what I have so far,

    (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=
    (Y_i-\hat{Y}_i)^2-2(Y_i-\hat{Y}_i)(\hat{Y}_i-\bar{Y})+(\hat{Y}_i-\bar{Y})^2<br />

    So, <br />
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}=
    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
    Posts
    2,763
    Thanks
    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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