All simulators
Partialling Out (FWL)
A confounded scatter beside its added-variable plot: the residual-on-residual slope equals the multiple-regression coefficient.
Day 28: Frisch–Waugh–Lovell & Partialling OutRAW DATA
AFTER PARTIALLING OUT x₂
darker = larger x₂naive slopeFWL slope = multiple-regression β̂₁
naive slope
2.084
OVB ≈ β₂γ/(γ²+1) = 1.00
FWL slope
1.000
truth β₁ = 1
share of x₁ variance left
0.41
1 − R²₁ (sets the SE)
Left: raw y against x₁; the naive slope absorbs the effect of the confounder x₂. Right: the added-variable plot, residuals of y on x₂ against residuals of x₁ on x₂. Its slope is exactly the multiple-regression coefficient (true value 1).