# What is the variance of residual of regression (matrix form)?

I am confused at the variance of residual in regression (matrix form)

Given $$Y = X*\beta \hat{}+ \epsilon$$

Y is a nx1 predicted vector

X is a nxp matrix

$$\beta\hat{}$$ is a px1 coefficient vector = $$(X^TX)^{-1}X^TY$$

$$\epsilon$$ is a nx1 residual vector

Then $$var(\epsilon)=var(Y-X\beta\hat{}$$) = var($$Y$$ - $$X(X^TX)^{-1}X^TY$$) (1)

Let projection matrix $$P$$ = $$X(X^TX)^{-1}X^T$$ . It is idempotent $$(P^n=P)$$

(1) becomes $$var(Y - PY)$$ (2)

Method 1: (2) = $$var ((I-P)Y) = (I-P)Var(Y)(I-P)^T = \sigma^2(I-P)$$

Method 2: (2) = $$var(Y)+var(PY) = \sigma^2I + P\sigma^2P^T = \sigma^2(I+P)$$

I expect that both methods should result in the same formula but they did not. I am not sure what is missed here

The second formula you have assumed independence (or at least zero covariance) of $$Y$$ and $$PY$$, that does not hold.
$$var (Y - PY) = var (Y) + var(PY) + cov(Y, -PY) + cov(-PY, Y)$$