24.4 Regression Kink Design

• If the slope of the treatment intensity changes at the cutoff (instead of the level of treatment assignment), we can have regression kink design

• Example: unemployment benefits

Sharp Kink RD parameter

\[ \alpha_{KRD} = \frac{\lim_{x \downarrow c} \frac{d}{dx}E[Y_i |X_i = x]- \lim_{x \uparrow c} \frac{d}{dx}E[Y_i |X_i = x]}{\lim_{x \downarrow c} \frac{d}{dx}b(x) - \lim_{x \uparrow c} \frac{d}{dx}b(x)} \]

where \(b(x)\) is a known function inducing “kink”

Fuzzy Kink RD parameter

\[ \alpha_{KRD} = \frac{\lim_{x \downarrow c} \frac{d}{dx}E[Y_i |X_i = x]- \lim_{x \uparrow c} \frac{d}{dx}E[Y_i |X_i = x]}{\lim_{x \downarrow c} \frac{d}{dx}E[D_i |X_i = x]- \lim_{x \uparrow c} \frac{d}{dx}E[D_i |X_i = x]} \]