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!