Prove VIF of jth predictor is jth diagonal entry of inverse of correlation matrix

Question

I am trying to prove that the variance inflation factor for the jth predictor variable in a centered/scaled regression model is the jth diagonal entry of the inverse of the correlation matrix of the predictors. The first idea I had was to try $VIF_j=\frac{Var({\hat{\beta_j}})}{\sigma^2}$, where $\hat{\beta_j}$ is the estimator for the regression coefficient of the jth predictor. But I do not know how to get from here to the diagonal entry I want. Can anyone help?

Answer 1

Recall that in OLS, $$ \text{Var}(\hat{\beta}|X)=\sigma^2(X'X)^{-1}, $$ hence, $$ \frac{\text{Var}(\hat{\beta_j})}{\sigma^2} = \frac{\sigma^2(X'X)^{-1}_{jj}}{\sigma^2} = (X'X)^{-1}_{jj} $$