How to calculate a variance-covariance matrix of coefficients for multivariate (multiple) linear regression?
Something like (equation below), but for the multivariate case.
Being more specific I'm interested in equations for diagonal terms.
The question is different from:
How to derive variance-covariance matrix of coefficients in linear regression
Because of the MULTIVARIATE regression case.
The general equation is the same for both the univariate and multivariate cases,
$$ V[\hat{\beta}] = V[(X^{T}X)^{-1}X^{T}Y]\\ = (X^{T}X)^{-1}X^{T}V[Y]X(X^{T}X)^{-1}\\ = \sigma^2(X^{T}X)^{-1} $$
This is unfortunately challenging to calculate. Examining $X^{T}X$ we see (assuming an intercept is included in the model),
$$X^{T}X = \begin{bmatrix} n & \sum_{i=1}^{n}x_{i1} & \sum_{i=1}^{n}x_{i2} & \ldots& \sum_{i=1}^{n}x_{ik}\\ \sum_{i=1}^{n} x_{i1} & \sum_{i=1}^{n} x_{i1}^2 & \sum_{i=1}^{n} x_{i1}x_{i2} & \ldots & \sum_{i=1}^{n}x_{i1}x_{ik} \\ \vdots & \vdots & \vdots & & \vdots\\ \sum_{i=1}^{n}x_{ik} & \sum_{i=1}^{n} x_{ik}x_{i1} & \sum_{i=1}^{n} x_{ik}x_{i2} & \ldots & \sum_{i=1}^{n} x_{ik}^2 \end{bmatrix}$$
which is symmetric, but does not have a clean expression for the inverse. Therefore the simple formulas for the univariate case are not available for the multivariate case.
