# Interpretation of $det(X'X)$ in MLR

I would like to understand the interpretation of $$det(X'X)$$ in case of multiple regressors.

$$Var(x) = \sum_i^n(x_i-\bar{x})^2 = \frac{1}{n}\sum_i^nx_i^2 - \bar{x}^2 = \frac{1}{n}\sum_i^nx_i^2 - \frac{(\sum_i^n x_i)^2}{n^2}$$

SLR case: $$\hat{y_i} = \hat{\beta_0} + \hat{\beta_1}x_i$$

$$\beta = (X'X)^{-1}X'y$$

$$X'X =\begin{bmatrix} n & \sum_i^nx_i\\ \sum_i^nx_i & \sum_i^nx_i^2 \end{bmatrix}$$

$$(X'X)^{-1} =\frac{1}{n\sum_i^nx_i^2 - (\sum_i^nx_i)^2}\begin{bmatrix} \sum_i^nx_i^2 & -\sum_i^nx_i\\ -\sum_i^nx_i & n \end{bmatrix} = \frac{n}{n^2Var(x)}\begin{bmatrix} \frac{\sum_i^nx_i^2}{n} & -\bar{x}\\ -\bar{x} & 1 \end{bmatrix}$$

Thus, if I get it right, in SLR case $$det(X'X) = n^2Var(x)$$

Now, let's consider case with two regressors, but without intercept, then $$(X'X)$$ is still 2x2.

$$\tilde{y_i} = \tilde{\beta_1}x_i + \tilde{\beta_2}x_i$$

$$X'X =\begin{bmatrix} \sum_i^nx_{1i}^2 & \sum_i^nx_{1i}x_{2i}\\ \sum_i^nx_{1i}x_{2i}& \sum_i^nx_{2i}^2 \end{bmatrix}$$

$$(X'X)^{-1} =\frac{1}{\sum_i^nx_{1i}^2\sum_i^nx_{2i}^2 - \sum_i^nx_{1i}\sum_i^nx_{2i}}\begin{bmatrix} \sum_i^nx_{2i}^2 & -\sum_i^nx_{1i}x_{2i}\\ -\sum_i^nx_{1i}x_{2i} & \sum_i^nx_{1i}^2 \end{bmatrix}$$

Can someone please provide the interpretation of $$det(X'X)$$ in the case of two regressors? I think that it should be linked with variances of $$x_{1i}, x_{2i}$$ and their covariance, but I don't have a solid understaning. Also, it will be great to get insights about more general case of several regressors with intercept.

• There are various interpretations possible. The one you seem to be looking for might be closely related to the account I gave of the connection between the covariance matrix and the Normal equations of multiple regression at stats.stackexchange.com/a/108862/919. – whuber May 8 at 17:20