Given a linear model $\mathbf{y} = \beta \mathbf{X} + \epsilon$, it is well known that the estimate for $\beta$ that gives the minimum residual sum of squares (RSS) is given by $\hat{\beta} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}$.

Of course, since $\hat{\beta}$ is just an estimate, then we want to know how far it deviates from the true values $\beta$.

In the derivation I am reading, the variance of the estimate is given by:

$$ V(\hat{\beta}) = V((\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{Y})= (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\sigma^2 \mathbf{I}\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1} = \sigma^2 (\mathbf{X}^T\mathbf{X})^{-1}$$

Please help me understand what happened in this derivation.

(Or you can present a simpler derivation)



I can present a deriviation with more steps:

$Var (\hat \beta)= Var ((X'X)^{-1}X'y) \quad | $ replace $y = Xβ + u$

$= Var ((X'X)^{-1}(X'X)β + (X'X)^{-1}X'u )\quad | (X'X)^{-1}(X'X)=I$

$= Var (β + (X'X)^{-1}X'u )\quad | Var(b+Y)=Var(Y)$

$= Var ( (X'X)^{-1}X'u )\quad | \texttt{factoring out} \ (X'X)^{-1}X'$

$=(X'X)^{-1}X'Var(u)X(X'X)^{-1} \quad | Var(u)=\sigma^2I \ \texttt{(homoscedasticity)}$

$=(X'X)^{-1}X'\sigma^2IX(X'X)^{-1} \quad $

$=\sigma^2(X'X)^{-1}\underbrace{X'X(X'X)^{-1}}_{I} $



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