27.10 Multi-Score Regression Discontinuity Design

The Multi-Score Regression Discontinuity Design extends the standard single-score RD and the multi-cutoff RD by introducing multiple running variables that simultaneously determine treatment eligibility. Instead of relying on a single threshold for assignment, treatment now depends on a combination of multiple continuous scores crossing predetermined cutoffs.

Multi-score RD is relevant when policy eligibility is based on multiple criteria, such as:

• Education: Honors program admission based on both math and English scores.

• Healthcare: Medical trial eligibility based on both BMI and blood pressure levels.

• Taxation: Tax incentives based on income level and household size.

27.10.1 General Framework

Each individual $i$ has:

• Two running variables, $X_{1i}$ and $X_{2i}$.

• Two predetermined cutoffs, $C_1$ and $C_2$.

• A binary treatment indicator $D_i$, assigned based on whether the individual’s scores exceed both thresholds.

The treatment effect is defined as:

$$\tau (x_1, x_2) = E[Y_{1i} - Y_{0i} | X_{1i} = x_1, X_{2i} = x_2].$$

This represents the local average treatment effect in a two-dimensional RD setting.

27.10.2 Identification

Under the potential outcomes framework, for each individual $i$, we define:

• $Y_{1i}$: Potential outcome under treatment.

• $Y_{0i}$: Potential outcome under control.

• $D_i$: Treatment assignment rule.

The observed outcome is:

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

The treatment assignment mechanism follows:

$$D_i = \begin{cases} 1 & \text{if } X_{1i} \geq C_1 \text{ and } X_{2i} \geq C_2, \\ 0 & \text{otherwise}. \end{cases}$$

27.10.3 Key Assumptions

To ensure valid causal inference, the multi-score RD framework extends the standard RD assumptions:

1. Continuity of Potential Outcomes in Both Running Variables
   • The expected potential outcomes $E[Y(0) | X_1, X_2]$ and $E[Y(1) | X_1, X_2]$ are smooth in both $X_1$ and $X_2$.
   • Formally: $$\lim_{(x_1, x_2) \to (C_1, C_2)^-} E[Y(0) | X_1 = x_1, X_2 = x_2] = \lim_{(x_1, x_2) \to (C_1, C_2)^+} E[Y(0) | X_1 = x_1, X_2 = x_2].$$
   • Ensures that any observed discontinuity in $E[Y | X_1, X_2]$ is attributable to treatment.
2. No Manipulation of Running Variables
   • The density of $(X_1, X_2)$ must be continuous at $(C_1, C_2)$.
   • No agents should be able to precisely manipulate both scores to cross the threshold.
3. Local Randomization
   • Near $(C_1, C_2)$, units are as good as randomly assigned to treatment or control.
4. No Interaction Effects in Running Variables (optional)
   • In some models, we assume that the effect of crossing $C_1$ does not depend on $C_2$ and vice versa.

If these assumptions hold, the treatment effect is identified as the discontinuity in $E[Y | X_1, X_2]$ at $(C_1, C_2)$.

27.10.4 Estimation Approaches

27.10.4.1 Local Linear Regression in Two Dimensions

The simplest approach is to estimate separate regressions on each side of the cutoff in both dimensions:

For observations below the threshold $(C_1, C_2)$:

$$Y_i = \alpha + \beta_1 (X_{1i} - C_1) + \beta_2 (X_{2i} - C_2) + \epsilon_i.$$

For observations above the threshold $(C_1, C_2)$:

$$Y_i = \gamma + \delta_1 (X_{1i} - C_1) + \delta_2 (X_{2i} - C_2) + \tau D_i + \nu_i.$$

The treatment effect $\tau$ is estimated as:

$$\hat{\tau} = \hat{E}[Y | X_1 = C_1^+, X_2 = C_2^+] - \hat{E}[Y | X_1 = C_1^-, X_2 = C_2^-].$$

This approach assumes local linearity, but higher-order polynomials can be used:

$$Y_i = \alpha + \sum_{k=1}^{K} \beta_k (X_{1i} - C_1)^k + \sum_{k=1}^{K} \gamma_k (X_{2i} - C_2)^k + \tau D_i + \epsilon_i.$$

27.10.4.2 Kernel-Weighted Estimation

A more flexible approach estimates $\tau(x_1, x_2)$ using nonparametric local regression:

$$\hat{\tau}(x_1, x_2) = \frac{\sum_{i=1}^{n} K_h (X_{1i} - x_1) K_h (X_{2i} - x_2) D_i Y_i}{\sum_{i=1}^{n} K_h (X_{1i} - x_1) K_h (X_{2i} - x_2) D_i} - \frac{\sum_{i=1}^{n} K_h (X_{1i} - x_1) K_h (X_{2i} - x_2) (1 - D_i) Y_i}{\sum_{i=1}^{n} K_h (X_{1i} - x_1) K_h (X_{2i} - x_2) (1 - D_i)}.$$

where:

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

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

27.10.4.3 Interaction Model for Heterogeneous Effects

To assess interaction effects between running variables, estimate:

$$Y_i = \alpha + \beta_1 (X_{1i} - C_1) + \beta_2 (X_{2i} - C_2) + \tau D_i + \lambda D_i (X_{1i} - C_1)(X_{2i} - C_2) + \epsilon_i.$$

• $\lambda$ captures whether the treatment effect depends on both $X_1$ and $X_2$.

27.10.5 Robustness Checks

1. Covariate Balance in Both Dimensions
   • Test whether pre-treatment covariates jump at $(C_1, C_2)$.
2. McCrary Density Test in Two Dimensions
   • Verify that density of $(X_1, X_2)$ is smooth at $(C_1, C_2)$.
3. Placebo Cutoffs
   • Implement fake cutoffs and re-estimate $\tau(x_1, x_2)$.
4. Varying Bandwidths
   • Re-estimate using different bandwidths for robustness.