## 24.3 Fuzzy RD Design

When you have cutoff that does not perfectly determine treatment, but creates a discontinuity in the likelihood of receiving the treatment, you need another instrument

For those that are close to the cutoff, we create an instrument for $$D_i$$

$Z_i= \begin{cases} 1 & \text{if } X_i \ge c \\ 0 & \text{if } X_c < c \end{cases}$

Then, we can estimate the effect of the treatment for compliers only (i.e., those treatment $$D_i$$ depends on $$Z_i$$)

The LATE parameter

$\lim_{c - \epsilon \le X \le c + \epsilon, \epsilon \to 0}( \frac{E(Y |Z = 1) - E(Y |Z=0)}{E(D|Z = 1) - E(D|Z = 0)})$

equivalently, the canonical parameter:

$\frac{lim_{x \downarrow c}E(Y|X = x) - \lim_{x \uparrow c} E(Y|X = x)}{\lim_{x \downarrow c } E(D |X = x) - \lim_{x \uparrow c}E(D |X=x)}$

Two equivalent ways to estimate

1. First

1. Sharp RDD for $$Y$$

2. Sharp RDD for $$D$$

3. Take the estimate from step 1 divide by that of step 2

2. Second: Subset those observations that are close to $$c$$ and run instrumental variable $$Z$$