Variance Inflation Factor and Multicollinearity Diagnostics

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Multicollinearity.

Your Task
In multiple linear regression $y = X\beta + \epsilon$, the variance of the $j$-th OLS regression coefficient is $\text{Var}(\hat{\beta}_j) = \frac{\sigma^2}{(n-1) s_j^2} \text{VIF}_j$, where $s_j^2$ is the sample variance of feature $X_j$.
(1) Prove that $\text{VIF}_j = \frac{1}{1 - R_j^2}$, where $R_j^2$ is the coefficient of determination from regressing feature $X_j$ on all remaining explanatory variables.
(2) Show that $\text{VIF}_j$ equals the $j$-th diagonal element of the inverse correlation matrix $(R_{XX})^{-1}$.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.