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Multicollinearity & VIF

Two predictors x₁ and x₂ are simulated with a controllable correlation r, then y = 1 + 2x₁ − 1.5x₂ + noise. The left panel shows one sample of the predictors; the right panel refits the regression on 150 independent resamples (n = 30 each) at the current r and plots the resulting (b₁, b₂) estimates — watch the cloud stretch into a narrow ridge as the predictors become collinear, even though it stays centered on the true values (red cross).

Predictors (single sample, n = 60)

Estimated (b₁, b₂) across 150 resamples

VIF =  |  SD(b̂₁) =  |  SD(b̂₂) =  |  true (b₁, b₂) = (2, −1.5)