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Math Help - proof the mean of the residuals must always be zero

  1. #1
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    proof the mean of the residuals must always be zero

    so I need to be able to prove that, given the residual is given by ei=yi-y(hat)i, the mean of the residuals, ie e-bar, is always equal to zero
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  2. #2
    MHF Contributor matheagle's Avatar
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    but that's false, if your model is Y=\beta X+\epsilon

    However it is true if the model does include the constant term, say Y=\beta_0+\beta_1 X+\epsilon

    In that second case, which using matrices is a lot easier we have... from http://en.wikipedia.org/wiki/Regression_analysis

    \sum(y_i-\hat y_i)=\sum(y_i-\hat \beta_0-\hat\beta_1 x_i)

    =\sum(y_i-\bar y+\hat \beta_1\bar x-\hat\beta_1 x_i)

    =\sum(y_i-\bar y)-\hat \beta_1\sum(x_i-\bar x)

    =0-0=0
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  3. #3
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    but isnt that just the proof that the sum of the residuals is equals to zero, not that the sum of the mean of the residuals is equal to zero?
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  4. #4
    MHF Contributor matheagle's Avatar
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    what's the diff?
    divide by n
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