Multi-Score Regression Discontinuity

• Multi-Score Regression Discontinuity Design (MRD) is an extension of classical RD that uses multiple running variables with complex cutoff rules to determine treatment.
• MRD enhances empirical analysis by addressing heterogeneity and compliance patterns, thus improving external validity in diverse policy and education studies.
• Estimation strategies in MRD include distance-based projections, multivariate local polynomial regression, and machine learning methods to robustly infer causal effects.

A Multi-Score Regression Discontinuity Design (MRD) generalizes the classical univariate RD framework to settings where treatment assignment is determined by multiple running variables (“scores”) and cutoff rules, often of arbitrary logic or geometry. MRD captures a broad array of empirical designs—including multidimensional education and policy interventions, manufacturing control rules, and geographic discontinuity settings—by encoding treatment status as a function of multiple observed criteria. The extension to multiple dimensions poses methodological challenges in identification, estimation, and inference, but also offers an enriched set of estimands, more robust handling of heterogeneity and compliance patterns, and greater external validity.

1. Foundations and Assignment Rules in MRD

The canonical RD design assigns treatment according to whether a scalar running variable $X$ crosses a fixed cutoff $c$: $T_i = \mathbb{1}\{X_i \geq c\}$. In MRD, the running variable becomes a vector $R_i = (R_{i1}, R_{i2}, ..., R_{im})^\top$, each potentially with its own threshold $r_{0k}$. Assignment rules can be constructed in several ways:

• Composite rules: $T_i = \mathbb{1}\{f(R_i) \geq r_0\}$, where $f$ combines information across dimensions (e.g., sum, min, weighted index) (Karabatsos et al., 2014).
• Logical/Boolean rules: $T_i = g(I_1, ..., I_K)$, with $I_k = \mathbb{1}\{R_{ik} > c_k\}$ and $g(\cdot)$ an arbitrary Boolean function (Schwarz et al., 21 Aug 2025).
• Multivariate boundaries: Treatment assignment by region (e.g., $X \in A_1$ vs $A_0$), yielding a discontinuity along a multidimensional boundary $B$ (Cattaneo et al., 8 May 2025).

Assignment may be sharp (deterministic) or fuzzy (with compliance and threshold-induced heterogeneity), and the relevant causal effect can be defined at a point on the boundary, along an entire boundary, or averaged over a distribution of cutoffs (Cattaneo et al., 2023, Bertanha, 2021).

2. Identification and Theoretical Properties

2.1 Definitions and Support

MRD requires clearly specifying how variations in score components affect treatment. The support of the treatment rule, $\operatorname{supp}(T)$, is the span of directions in which changing $R_i$ affects $T_i$ (Schwarz et al., 21 Aug 2025). Each realization $R_i$ can be decomposed orthogonally into $R_i = R_i^T + R_i^\perp$, where $R_i^T \in \operatorname{supp}(T)$ and $R_i^\perp$ lies in the nullspace (nuisance directions).

2.2 Causal Estimands

• Location-Specific Effect: For boundary point $x \in B$, the local treatment effect is $\tau(x) = \mathbb{E}[Y(1) - Y(0) | X = x]$ (Cattaneo et al., 8 May 2025).
• Complier Average Causal Effect (CACE): When assignment is fuzzy and compliance structures are complex, identification results hinge on restrictively defining “compliers” relative to the multi-dimensional assignment rule (Schwarz et al., 21 Aug 2025). For support directions $x^+$, $x^-$, the complier effect at the cutoff can be written

$$\mathbb{E}[Y(1) - Y(0) | X^T = 0,\ \text{complier}] = \frac{1}{\Pr(\text{comp} | X^T=0)} [\lim_{\lambda \to 0}\mathbb{E}[Y | X^T = \lambda x^+] - \lim_{\lambda \to 0}\mathbb{E}[Y | X^T = \lambda x^-]] - C.$$

• Pooled/Weighted Effects: When assignment varies across units (e.g., geographic boundaries, policy variations), effects can be aggregated as weighted averages over the support, cutoffs, or policy-relevant counterfactual distributions (Bertanha, 2021, Cattaneo et al., 2019).

2.3 Necessary Conditions

Identification of marginal or weighted treatment effects in the presence of multiple assignment variables and possibly multivalued treatments depends on local continuity of outcome functions and independence (stationarity) of compliance type probabilities under small support shifts (Schwarz et al., 21 Aug 2025, Caetano et al., 2020). For multivalued treatment, existence of linearly independent jumps in assignment probabilities (Relevance Assumption) is necessary for identification of all LATEs (Caetano et al., 2020).

3. Estimation Strategies and Inference

3.1 Local Polynomial (Kernel) Methods

Two principal approaches for boundary effect estimation are developed:

• Distance-Based Methods: The multivariate score is mapped to a signed scalar distance from a target boundary point, reducing the problem to a univariate RD (Cattaneo et al., 8 May 2025). Estimation follows via local polynomial regression using this scalar projection. However, distance-based methods are prone to irreducible bias near kinks or irregularities in the assignment boundary, with bias order limited to $O(h)$ (where $h$ is the bandwidth), impeding higher-order bias correction and valid inference (Cattaneo et al., 8 May 2025).
• Bivariate (or Multivariate) Location-Based Methods: The full vector of scores is used directly, with local polynomial regression performed in the original multivariate space around points on the assignment boundary. Bias properties are standard ($O(h^{p+1})$ for a degree $p$ polynomial), and the approach remains robust to irregular boundary geometry (Cattaneo et al., 8 May 2025).

Table: Key distinctions in estimation methods for bivariate MRD

Approach	Uses full score?	Bias near kinks
Distance-based	No	Order $h$; not correctable
Location-based	Yes	Order $h^{p+1}$; standard

3.2 Statistical Inference

Pointwise and uniform inference is built on self-normalized $t$-statistics, robust bias correction, and (for the location-based method) MSE-optimal data-driven bandwidth selectors. Uniform inference over the boundary utilizes the supremum of a Gaussian process approximating the standardized treatment effect estimator (Cattaneo et al., 8 May 2025, Cattaneo et al., 12 May 2025).

3.3 Bayesian and Nonparametric Methods

Flexible approaches based on Bayesian nonparametrics (e.g., mixtures of linear regressions using a restricted Dirichlet Process prior) can cluster observations local to the vector cutoff and propagate uncertainty in cluster structure using posterior sampling (Karabatsos et al., 2014). The clustering is performed in the multivariate assignment variable space, and inference for treatment effects is averaged over posterior draws of the cluster allocations.

3.4 Optimization-Based Minimax Estimation

Convex optimization methods directly produce minimax linear estimators for the discontinuity parameter in MRD, minimizing worst-case MSE over a function class with bounded smoothness (e.g., bounded Hessian norm). Such estimators are robust to the shape of the assignment region—operating in multivariate settings without collapse to a univariate score and providing valid, uniformly covered confidence intervals (Imbens et al., 2017).

3.5 Random Forest and Machine Learning Approaches

• Random forests (honest and local linear forests): These methods handle multivariate assignment variables flexibly, by leveraging data-adaptive partitioning to determine relevant local clusters and boundary structure (Liu et al., 2023). While honest forests do not adjust boundary bias, local linear forests add ridge-penalized adjustment to mitigate this effect. These methods eschew the zero-density pitfalls of scalar projection approaches but may exhibit finite-sample bias at boundaries.
• Post-Lasso, boosting, and flexible covariate adjustment: ML-based adjustment methods can further reduce estimation variance or bias, as illustrated in empirical studies of LED manufacturing using MRD (Schwarz et al., 21 Aug 2025).

4. Compliance Patterns and Unit Typology

MRD settings warrant more elaborate definitions of compliance and noncompliance. Each unit can be classified as:

• Complier: Assignment and actual treatment match for all relevant cutoff shift directions in $\operatorname{supp}(T)$.
• Nevertaker: Always untreated, regardless of cutoff (in any support direction).
• Alwaystaker: Always treated, regardless of cutoff.
• Defier and Indecisive: Systematically disagreeing with or vacillating in response to cutoff direction changes (Schwarz et al., 21 Aug 2025).

These categorizations are local and multi-dimensional, accommodating the richer compliance structures induced by logic-based or geometric assignment rules. Decomposing assignment rules via logical composition (e.g., $T = G \wedge H$) allows recursive identification and removal of non-compliant units, resulting in lower-variance, more policy-relevant effect estimates (Schwarz et al., 21 Aug 2025).

5. Boundary Geometry, Extrapolation, and Software Implementations

5.1 Assignment Boundary Geometry

Arbitrary assignment boundary geometry is central for MRD analysis—especially in geographic RDs or when boundaries have kinks. When the assignment boundary is irregular, any estimation method that discards the full multivariate structure (e.g., enforcing projection onto a single distance) suffers from unavoidable bias. Direct modeling in multivariate score space using location-based methods is necessary for uniform bias control and valid inference (Cattaneo et al., 8 May 2025).

5.2 Extrapolation Using Multiple Cutoffs

In multi-cutoff MRD, identification of treatment effects away from the local cutoff (extrapolation) is attainable using information from subpopulations with higher or lower cutoffs, under a “constant bias” assumption analogous to parallel trends in DiD (Cattaneo et al., 2018). Extrapolated effects are constructed by imputing counterfactuals for one group using observable outcomes from another, adjusted by cutoff-specific bias estimands.

5.3 Software Implementations

The R package rd2d operationalizes both location-based and distance-based methods, implements robust bias-corrected local polynomial estimation, MSE-optimal data-driven bandwidth selection, and constructs both pointwise and uniform inference (including simulation-calibrated confidence bands). “rdmulti” and “rdrobust” packages similarly provide tools for estimation in multiple cutoff or score settings with robust inference (Cattaneo et al., 12 May 2025, Cattaneo et al., 2019).

6. Empirical Applications and Practical Implications

Empirical validation of MRD methods exists across several domains:

• Education: Eligibility for programs depending on multiple test scores or need indexes. For instance, effects of the Ser Pilo Paga (SPP) program in Colombia where assignment depends on both academic and wealth scores, with results showing sensitivity to boundary geometry and compliance structure (Cattaneo et al., 8 May 2025, Cattaneo et al., 2023).
• Manufacturing: Production and rework decisions based on multi-score cutoff rules; MRD analysis yields lower estimation variance and more interpretable treatment effects after removing non-compliant units (Schwarz et al., 21 Aug 2025).
• Healthcare/Insurance: Multivalued treatment assignment at discrete age cutoffs (e.g., Medicare eligibility at 65) analyzed using MRD tools, with distinct extensive and intensive margin effects identified under suitable assumptions (Caetano et al., 2020).
• Geographic Discontinuity: Policy assignment using geographic boundaries (e.g., media markets, school assignment zones) illustrated with bivariate RD estimation along curved or irregular boundaries (Cattaneo et al., 8 May 2025).

Instrumental for causal inference in these contexts is the choice of estimation method adapted to boundary properties, precise compliance definitions, use of robust bias correction, and bandwidths chosen according to the structure of the assignment region. Under appropriate conditions, these methodologies yield low-bias, efficient estimators, and valid confidence intervals for effect estimates both pointwise and uniformly over the assignment boundary.

7. Extensions: Interference, Heterogeneity, and Advanced Designs

MRD designs have recently been further generalized to permit:

• Unit Interference: Treatment assignment and outcomes influenced by both own and neighbors’ scores, leading to multidimensional boundaries in networked settings. Effects can be estimated using modified local polynomial procedures and distance transformations, with variance estimates accounting for network correlations (Torrione et al., 3 Oct 2024).
• Heterogeneous Effects/Ever-Compliers: Identification in fuzzy/multivalued MRD requires either finite-dimensional parameterizations of effect heterogeneity or weighting strategies that aggregate over the observed compliance patterns (Bertanha, 2021).
• Randomization-Based Approaches: Bayesian nonparametric clustering, as well as local randomization, accommodates discrete as well as continuous running variables and enables fully probabilistic assessment of uncertainty (Karabatsos et al., 2014, Cattaneo et al., 2023).

The field continues to extend the MRD framework for richer assignment rules, multidimensional treatments, overlapping assignment mechanisms (such as in market design or tie-breaking), and designs facing practical challenges such as support overlap, boundary misspecification, and high-dimensional running variables.

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This overview consolidates theoretical and practical advances in MRD, highlighting key assignment and estimation principles, robust analytical strategies, compliance typologies, and frontier methodological extensions. MRD stands as a unifying framework for modern causal inference in the presence of multidimensional policy assignment mechanisms, with ongoing research devoted to expanding its flexibility, efficiency, and empirical scope.