Deriving the variance-covariance matrix for parameter vector of a linear regression model

Question

I came across this formula in the book Elements of Statistical Learning in Chapter 3.

$$\operatorname{Var}(\hat\beta) = (\mathbf X^T\mathbf X)^{-1}\hat\sigma^2$$

Here, $\hat\beta$ is the estimated parameter vector for a linear regression equation satisfying: $$\mathbf y= \mathbf X\mathbf \beta + \epsilon$$

$\epsilon$ in turn follows a normal distribution with mean 0 and variance $\sigma^2$. $\hat\beta$ follows the normal equation, which has been derived previously in the same chapter: $$ \hat\beta = (\mathbf X^T \mathbf X)^{-1}\mathbf X^T\mathbf y$$

What I understand as per the text is that the $\operatorname{Var}(\hat\beta)$ is the variance-covariance derivation, and for a matrix I can calculate the variance-matrix given values. But I'm getting stuck in this case as $\hat\beta$ is expressed as an equation, which when I use to expand $\hat\beta$ in my derivation, I end up with an equation that I cannot simplify further.

Answer 1

I found the solution to be very well written in the following question: Variance Estimate in linear regression

The key to solving this equation is $\operatorname{Var}(AW) = A\operatorname{Var}(W)A^T$

Marking this question as answered.