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.

  1. 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.

  1. Placebo Cutoffs
  • Implement fake cutoffs and re-estimate \(\tau(x, c)\).
  • If significant effects appear, the RD estimates may be biased.
  1. Varying Bandwidths
  • Re-estimate treatment effects using different bandwidths.
  • If \(\hat{\tau}(x,c)\) changes drastically, it suggests sensitivity to bandwidth choice.