# Solved – Variance of new \$y\$ – \$hat{y}\$

Suppose we have a linear regression $$Y = Xbeta + e$$, and we have new data come in. $$Y_{new} = X_{new}beta + e_{new}$$. And we know that $$Y_{new}$$ and $$hat{Y_{new}}$$ are independent. So $$Var(Y_{new} – hat{Y_{new}})$$ = $$Var(Y_{new}) + Var(hat{Y_{new}})$$ = $$(I + H)sigma^2$$, where $$H$$ is the projection matrix. But according to the formula $$Var(Ay) = AVar(y)A^{T}$$, so we can get $$Var(Y_{new} – hat{Y_{new}})$$ = $$(I-H)sigma^2(I-H)^{T}$$ = $$(I – H)sigma^2$$.
My question is that are they he same? Or I missed something? Thank you!!

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The problem here is that $$hat{Y}_{new} neq H Y_{new}$$, so your working falls apart. The hat matrix comes from the data that was used to fit the model. Using the model estimated from the initial data, you have:
begin{equation} begin{aligned} hat{Y}_{new} = X_{new} hat{beta} = X_{new} (X^text{T} X) X^text{T} Y neq X (X^text{T} X) X^text{T} Y_{new}. end{aligned} end{equation}