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 \(Z\)