Zero leverage

The idea here is simple: if you add an observation at exactly x = x̄, it cannot move the OLS slope — no matter how extreme its y-value is.

The OLS slope formula makes this transparent:

β̂ = Σᵢ (xᵢ − x̄)(yᵢ − ȳ) / Σᵢ (xᵢ − x̄)²

A point at x = x̄ contributes (xᵢ − x̄) = 0 to both the numerator and the denominator. Its y-value is completely irrelevant to β̂. This is the geometric content of zero leverage: in the hat matrix H = X(XᵀX)⁻¹Xᵀ, the diagonal entry hᵢᵢ measures how much observation i can pull the fitted value at its own location. At x = x̄, hᵢᵢ = 1/n → 0 as n → ∞.

Zero-leverage points can shift the intercept — the fitted line pivots around x̄, so changing the predicted value at x̄ shifts the whole line up or down. But the slope is immune.

Setup: 100 clean observations from y = x + ε, ε ~ N(0, 0.3²) — a very small error variance so the true line is unmistakable. Then one contaminating point is placed at (x̄, 15): sitting 15 units above where the regression line predicts at that x-value.

The result: the slope of the combined fit is identical to the clean fit to machine precision (difference = 0). The intercept shifts slightly upward — the line tilts to pass closer to y = 15 at x = x̄ — but the angle is frozen.