Hey im having trouble proving the mean of the residuals must always be zero.

Can someone help me with this?

Thanks

For a simple regression it is straight forward since you can express the slope intercept form of the model as:

Yhat = b0 + b1*X

Yhat = YBar + R*(Sy/Sx)*(X - XBar)

And, the Error (E) terms are expressed as:

E=(Y - Yhat)

SumE=Sum(Y - Yhat)

SumE=Sum(Y - YBar + R*(Sy/Sx)*(X - XBar) )

SumE=SumY - SumYBar + R*(Sy/Sx)*Sum(X - XBar)

**SumE=N*YBar - N*YBar + 0 = 0**,

because the summation of a constant N times is N times that constant (N*YBar) and Sum(X - XBar) = 0.

Thus, as indicated above, if the sum of the error terms are zero, so will the mean.