0

I'm trying to figure out why when

$ {\hat{\beta}}_0 = 0 $

then

$ {\sum_{i=1}^n}{[(y_i-{\hat{y}}_i)({\hat{y}}_i-\bar{y})]} \neq 0 $

At least it is not necessarily true that it is 0. I tried reading other stackexchange posts on this, and I'm using the Woolridge textbook, however, no proofs were given for why this is the case. I understand the proof for why it is zero based on the OLS first order conditions, but not why it is not necessarily zero when $ {\hat{\beta}}_0 = 0 $. Thank you very much for your help!

Memiya
  • 153
  • Are you saying that having a fitted intercept value of $0$ necessarily guarantees that the given sum is nonzero? This is not the case if the model fits the data perfectly ($y_i = \hat{y}_i$); the sum is then $0$. – Craveable Banana Oct 05 '20 at 08:07
  • I apologise if my wording was imprecise. I meant that a zero \beta_0 meant that the sum is not necessarily 0. I was trying to find a case where the sum is not 0, and thereby show that R^2 can be negative, since sum of squared residuals is more than variance of fitted values. – Memiya Oct 05 '20 at 14:55

0 Answers0