Discussion
For a standard multiple regression model \(\{Y, X\beta, \sigma^{2}I\}\), which of the following statements is not true ?
Answer: D
The statement that is not true is \(\hat{Y} \sim N_{n}(\beta, \sigma^{2}H)\).
With \(H = X(X'X)^{-1}X'\) the hat matrix, \(\hat{Y} = HY\). Since \(HX = X\), \(E(\hat{Y}) = HX\beta = X\beta\) (A). Since H is symmetric and idempotent, \(Cov(\hat{Y}) = H\sigma^{2}IH' = \sigma^{2}H\) (B). And \(Cov(HY, (I - H)Y) = \sigma^{2}H(I - H) = 0\) (C).
Under normality \(\hat{Y}\) is normal with mean \(X\beta\), not \(\beta\). The vector \(\beta\) has \(p\) components while \(\hat{Y}\) has \(n\), so the stated mean is not even of the right size.
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