Multicollinearity and the VIF
When two regressors are correlated, OLS still hits the true value on average, but each coefficient is harder to pin down. With two regressors the auxiliary fit gives R²ⱼ = ρ², so the variance inflation factor is VIF = 1 / (1 − ρ²) and the standard error widens by √VIF versus no collinearity, holding σ² and SSTⱼ fixed.
Variance inflation factor1.33
VIF = 1 / (1 − ρ²) 1.33SE multiplier √VIF 1.15×Auxiliary R²ⱼ = ρ² 0.25
Correlation ρ = corr(x₁, x₂)0.50
The two curves share the same centre, so β̂₁ stays unbiased — collinearity does not move the estimate, it only spreads it out. The gold curve is 1.15× wider than the baseline, so the standard error of β̂₁ is 1.15× larger and its t-statistic is 0.87× smaller, holding σ² and SSTⱼ fixed. Note this inflates the variance of the individual coefficients; the overall fit and any joint prediction can still be precise.
Closed-form variance algebra (no simulation): VIF = 1 / (1 − ρ²), the (1 − R²ⱼ) factor in Var(β̂ⱼ) = σ² / [SSTⱼ(1 − R²ⱼ)]. A rule of thumb flags VIF > 10 (|ρ| > 0.95). In Stata, read it with
estat vif after regress.