27.9 Multi-Cutoff Regression Discontinuity Design

The Multi-Cutoff Regression Discontinuity Design extends the standard RD framework by allowing for multiple cutoff points across different groups or geographic regions. Instead of a single threshold $c$, different subgroups are assigned different cutoffs $C_i$. This framework allows for a heterogeneous treatment effect function:

$$\tau (x,c)= E[Y_{1i} - Y_{0i}|X_i = x, C_i = c].$$

Why Use Multi-Cutoff RD?

• Policy Variation: Policies often implement different cutoffs across regions or institutions (e.g., different states setting different minimum test scores for scholarship eligibility).
• Generalizability: Allows estimation of treatment effects across multiple populations instead of relying on a single threshold.
• Improved Precision: Leveraging multiple thresholds can enhance statistical power compared to a single-cutoff RD.

The multi-cutoff RD framework provides several advantages:

1. Estimation of Local Heterogeneous Effects
   • Unlike standard RD, which estimates a single treatment effect, multi-cutoff RD allows heterogeneity in effects across groups.
2. Improved Precision
   • More observations across different thresholds can increase statistical power.
3. Policy Implications
   • Useful in settings where policy thresholds vary (e.g., different states setting different income eligibility limits for welfare programs).

27.9.1 Identification

Under the potential outcomes framework, each unit $i$ has:

• A running variable $X_i$.

• A cutoff specific to their group $C_i$.

• A binary treatment indicator:

$$D_i = I(X_i \geq C_i).$$

The observed outcome is:

$$Y_i = D_i Y_{1i} + (1 - D_i) Y_{0i}.$$

The treatment effect is the expected difference in potential outcomes:

$$\tau(x, c) = E[Y_{1i} - Y_{0i} | X_i = x, C_i = c].$$

27.9.2 Key Assumptions

To ensure causal identification, we extend the standard RD assumptions:

1. Continuity of Potential Outcomes
   • The expected potential outcomes $E[Y(0)|X]$ and $E[Y(1)|X]$ are smooth functions of $X$ at each cutoff $C_i$.
   • Formally: $$\lim_{x \uparrow C_i} E[Y(0)|X=x, C_i=c] = \lim_{x \downarrow C_i} E[Y(0)|X=x, C_i=c].$$
2. No Manipulation of the Running Variable
   • The density of $X_i$ must be continuous at each $C_i$, ensuring that individuals cannot selectively sort above or below their assigned cutoff.
3. Local Randomization
   • Near each cutoff, units are as-good-as-randomly assigned to treatment or control.
4. Independence Across Cutoffs
   • The cutoff assignment rule should be exogenous and not correlated with unobserved determinants of $Y$.

If these assumptions hold, each cutoff provides a valid local treatment effect estimate.

27.9.3 Estimation Approaches

27.9.3.1 Pooling Cutoffs with Fixed Effects

A straightforward way to estimate multi-cutoff RD is to include cutoff fixed effects:

$$Y_i = \alpha + \beta (X_i - C_i) + \tau D_i + \gamma C_i + \epsilon_i.$$

where:

• $\tau$ captures the average treatment effect across all cutoffs.

• $C_i$ is included as a fixed effect to account for different intercepts across groups.

27.9.3.2 Separate RD Estimation for Each Cutoff

Instead of pooling, we can estimate separate RD effects for each $C_i$:

$$\tau_c = \lim_{x \downarrow C_i}E[Y|X = x, C_i = c] - \lim_{x \uparrow C_i} E[Y|X = x, C_i = c].$$

This approach allows for heterogeneous treatment effects.

27.9.3.3 Interaction Model for Heterogeneous Effects

To estimate how treatment effects vary with $C_i$, we interact $D_i$ with $C_i$:

$$Y_i = \alpha + \beta (X_i - C_i) + \tau D_i + \lambda D_i C_i + \epsilon_i.$$

where:

• $\lambda$ captures how the treatment effect varies with the cutoff.

• A significant $\lambda$ implies that $\tau(x, c)$ is not constant across cutoffs.

27.9.3.4 Nonparametric Local Estimation

A fully flexible approach estimates $\tau(x, c)$ separately at each cutoff using kernel-based methods:

$$\hat{\tau}(c) = \frac{\sum_{i=1}^{n} K_h (X_i - C_i) D_i Y_i}{\sum_{i=1}^{n} K_h (X_i - C_i) D_i} - \frac{\sum_{i=1}^{n} K_h (X_i - C_i) (1 - D_i) Y_i}{\sum_{i=1}^{n} K_h (X_i - C_i) (1 - D_i)}.$$

where:

• $K_h(\cdot)$ is a kernel function (e.g., Epanechnikov).

• $h$ is the bandwidth, chosen via cross-validation.

27.9.4 Robustness Checks

1. Covariate Balance at Each Cutoff

• Test whether pre-treatment covariates show jumps at each $C_i$.

• Run placebo RD regressions on covariates:

  $$W_i = \alpha + \beta (X_i - C_i) + \gamma D_i + \epsilon_i.$$

• A significant $\gamma$ suggests that RD assumptions are violated.

3. McCrary Density Test

• Perform a McCrary test separately at each cutoff to check for manipulation:

  $$f(X) \text{ should be continuous at } X = C_i.$$

• If discontinuities exist, individuals may be sorting around cutoffs.

3. Placebo Cutoffs

• Implement fake cutoffs and re-estimate $\tau(x, c)$.
• If significant effects appear, the RD estimates may be biased.

4. Varying Bandwidths

• Re-estimate treatment effects using different bandwidths.
• If $\hat{\tau}(x,c)$ changes drastically, it suggests sensitivity to bandwidth choice.