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For a variable $x = (x_1,\cdots,x_n)$ and its sample matrix $X,$ we know the partial correlation of $x_i,x_j$ is the correlation of their residuals respect to the left variables. And the precision matrix $(X^TX)^{-1}$ is just the partial covariance matrix of $x.$ Then the diagonal elements $(X^TX)_{ii}^{-1}$ is the $x_i$'s squared residual $RSS_i.$

On the other hand, we know the variance of linear regression estimation $Var(\hat{\beta}_i) = s^2(X^TX)^{-1}_{ii}.$ Therefore, the variance should be $$Var(\hat{\beta}_i) = s^2 RSS_i.$$

However in the discussion of VIF in wiki is:

https://en.wikipedia.org/wiki/Variance_inflation_factor

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I don't where I am wrong. Actually I also don't understand the last step: $RSS_j$ should be

$$RSS_j = (X_j - X_{-j}\hat{\beta}_{*j})^T(X_j - X_{-j}\hat{\beta}_{*j}).$$

How to get

$$RSS_j = X^T_jX_j - (X_{-j}\hat{\beta}_{*j})^T(X_{-j}\hat{\beta}_{*j})?$$

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