Thread: proof the mean of the residuals must always be zero

1. 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

2. 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

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

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

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

$=0-0=0$

3. 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?

4. what's the diff?
divide by n

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mean of residuals is always zero

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