27.9 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 $Z$