27.3 Formal Definition
Let \(X_i\) be the running variable, \(c\) the cutoff, and \(D_i\) the treatment indicator:
\[ D_i = 1_{X_i > c} \]
or equivalently,
\[ D_i = \begin{cases} 1, & X_i > c \\ 0, & X_i < c \end{cases} \]
where:
\(D_i\): Treatment assignment
\(X_i\): Running variable (continuous)
\(c\): Cutoff value
27.3.1 Identification Assumptions
27.3.1.1 Continuity-Based Identification
RD estimates the [Local Average Treatment Effect] at the cutoff:
\[ \begin{aligned} \alpha_{SRDD} &= E[Y_{1i} - Y_{0i} | X_i = c] \\ &= E[Y_{1i}|X_i = c] - E[Y_{0i}|X_i = c] \\ &= \lim_{x \to c^+} E[Y_{1i}|X_i = x] - \lim_{x \to c^-} E[Y_{0i}|X_i = x] \end{aligned} \]
This relies on the assumption that, in the absence of treatment, the conditional expectation of potential outcomes is continuous at the threshold \(c\).
27.3.1.2 Local Randomization-Based Identification
Alternatively, identification can be achieved using local randomization within a small bandwidth \(W\) (i.e., a neighborhood around the cutoff). The [Local Average Treatment Effect] in this case is:
\[ \begin{aligned} \alpha_{LR} &= E[Y_{1i} - Y_{0i}|X_i \in W] \\ &= \frac{1}{N_1} \sum_{X_i \in W, D_i = 1} Y_i - \frac{1}{N_0} \sum_{X_i \in W, D_i = 0} Y_i \end{aligned} \]
Since RD estimates are local, they may not generalize to the entire population. However, for many applications, internal validity is of primary concern (rather than external validity).