$y=X\beta+u$ where $u \sim N(0,\Sigma)$ and $\Sigma$ is symmetric & idempotent.
$X: n*k$, $y:n*1$, $\beta=k*1$, $u:n*1$ vector.
Suppose you apply LS(least square) to the model.
Calculate the covariance matrix of $\hat{\beta}$.
Since $\hat{\beta}=(X'X)^{-1}X'y$,
$E(\hat{\beta})=\beta$ so $\hat{\beta}$ is an unbiased estimator,
$Cov(\hat{\beta})=E(\hat{\beta}-E(\hat{\beta}))(\hat{\beta}-E(\hat{\beta}))'=E[(X'X)^{-1}X'uu'X(X'X)^{-1}]=\Sigma (X'X)^{-1}$
Is it right?
Best Answer
you may use this property: $Cov\left( Ay \right)=ACov\left( y \right)A'$ where $A={{\left( {X}'X \right)}^{-1}}{X}',Cov\left( y \right)=\text{ }\!\!\Sigma\!\!\text{ }$
so ${{\left( {X}'X \right)}^{-1}}{X}'\text{ }\!\!\Sigma\!\!\text{ }X{{\left( {X}'X \right)}^{-1}}$.