I was reading some lecture notes on simple linear regression where one section said that when the slope is 0 (hence, H0: beta = 0 is actually true), (SSY - SSE)/(DFY - DFE) estimates sigma squared. This doesn't make sense to me. If beta is actually 0, then shouldn't SSY and SSE be the same value?
SSY = ∑(Y-Ybar)^2 SSE = ∑(Y-Yhat)^2
However, if beta = 0, that means the best estimate (Yhat) is essentially Ybar, which in turn means
SSY = SSE = ∑(Y-Ybar)^2
In other words
(SSY - SSE)/(DFY - DFE) = 0/(DFY - DFE)
Is there something that I am missing? I hope someone here can elaborate on this.