# How to show VIF

The variance of the $$j$$th element of the OLS estimator is given by

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\sigma^{2}\left(X_{j}^{T} M_{-j} X_{j}\right)^{-1}$$

where $$X_j$$ is the column of regressors associated to the $$j$$ variable, and $$M_{-j}$$ is the maker of residuals (the projection off) of the space generated by all columns of the matrix $$X$$ besides the $$j$$th one.

Show that the variance of $$\hat{\beta}$$ can also be written as

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\frac{\sigma^{2}}{(n-1) \operatorname{Var}\left(X_{j}\right)}\left(\frac{1}{1-R_{X_{j} \mid X_{-j}}^{2}}\right)$$

Recall that (see e.g. here), in general, $$R^2=1-\frac{\hat{u}'\hat{u}}{\tilde{y}'\tilde{y}}.$$ Here, $$\hat u$$ denotes the vector of residuals and $$\tilde y$$ the vector of demeaned observations on the dependent variable.
In matrix notation, with dependent variable $$X_j$$, the numerator is just $$X_{j}^{T} M_{-j} X_{j}$$ and the denominator is sum of the squares of the demeaned observations of the $$X_j$$, i.e., $$n-1$$ its sample variance (which I prefer to denote by $$s^2_{X_j}$$ to make clear it is not the population variance of $$X_j$$).
Thus, $$R_{X_{j} \mid X_{-j}}^{2}=1-\frac{X_{j}^{T} M_{-j} X_{j}}{(n-1)s^2_{X_j}}.$$ Hence, $$\frac{1}{(n-1)s^2_{X_j}}\frac{1}{1-R_{X_{j} \mid X_{-j}}^{2}}=(X_{j}^{T} M_{-j} X_{j})^{-1}$$