Estimation and Inference in Boundary Discontinuity Designs: Distance-Based Methods

1. Problem Statement

The paper addresses the statistical challenges of estimating causal effects in Boundary Discontinuity (BD) Designs. Unlike standard Regression Discontinuity (RD) designs where treatment is assigned based on a univariate running variable crossing a single threshold, BD designs involve units with bivariate scores $X_i = (X_{1i}, X_{2i})^\top$. Treatment is assigned based on whether the unit lies on one side of a one-dimensional boundary curve $\mathcal{B}$ that partitions the support of the score.

The primary goal is to estimate the Boundary Average Treatment Effect Curve (BATEC), denoted as $\tau(x)$, which captures heterogeneous treatment effects at every point $x$ along the boundary $\mathcal{B}$:
$$\tau(x) = E[Y_i(1) - Y_i(0) \mid X_i = x], \quad x \in \mathcal{B}$$

While location-based methods (using the full bivariate score) exist, the paper focuses on distance-based methods, which are widely used in empirical applications (e.g., geographic RD). These methods reduce the bivariate score to a univariate signed distance running variable $D_i(x) = (1(X_i \in A_1) - 1(X_i \in A_0)) \cdot d(X_i, x)$, where $d(\cdot, \cdot)$ is a distance function. The authors investigate the theoretical validity, bias properties, and inference procedures for these distance-based estimators, specifically addressing how the geometric regularity of the boundary $\mathcal{B}$ impacts estimation.

2. Methodology

The authors propose and analyze isotropic local polynomial regression based on the univariate distance score.

• Estimator: For a target point $x \in \mathcal{B}$, the estimator $\hat{\tau}(x)$ is the difference between two local polynomial regressions (one for the treatment region $A_1$ and one for the control region $A_0$) evaluated at the cutoff $c=0$ of the distance variable $D_i(x)$.
  $$\hat{\tau}(x) = \hat{\theta}_{1,x}(0) - \hat{\theta}_{0,x}(0)$$
  where $\hat{\theta}_{t,x}(0)$ minimizes the weighted sum of squared errors using a kernel $K_h(\cdot)$ applied to the distance $D_i(x)$.

• Isotropic Smoothing: The method treats the distance variable as a standard univariate running variable. This implies isotropic smoothing: observations are weighted solely by their Euclidean (or metric) distance to $x$, regardless of the direction from which they approach the boundary.

• Bias Decomposition: The estimation error is decomposed into:

  1. Approximation Bias ($B(x)$): The difference between the true causal parameter $\tau(x)$ and the limit of the induced univariate conditional expectation $\theta_{t,x}(0)$.
  2. Stochastic Error: The sampling variability of the estimator.

3. Key Contributions

The paper makes three major theoretical contributions:

A. Identification Theory

The authors establish conditions under which the distance-based estimand $\theta_{1,x}(0) - \theta_{0,x}(0)$ identifies the true BATEC $\tau(x)$.

• Theorem 1: Shows that identification holds if the boundary $\mathcal{B}$ is a rectifiable curve and the data generating process satisfies standard smoothness conditions. The limit of the distance-based conditional expectation converges to the true conditional expectation at the boundary point.

B. Approximation Bias and Geometric Irregularities

This is the paper's most significant theoretical insight. The authors demonstrate that the geometry of the boundary fundamentally alters the bias properties of distance-based estimators.

• Theorem 2 (Irregular Boundaries): If the boundary $\mathcal{B}$ contains kinks (points where the tangent is undefined or changes abruptly), the approximation bias of the distance-based estimator is irreducibly of order $O(h)$, regardless of the polynomial order $p$ used or the smoothness of the underlying regression functions.
  • Mechanism: Near a kink, the induced univariate conditional expectation $\theta_{t,x}(r)$ becomes non-differentiable (Lipschitz continuous but not smooth) due to the truncation of the level sets of the distance function by the boundary geometry.
  • Implication: Increasing the polynomial order $p$ (e.g., using quadratic instead of linear) does not reduce the bias near kinks.
• Theorem 3 (Smooth Boundaries): If the boundary $\mathcal{B}$ is sufficiently smooth (e.g., $C^{\kappa+2}$), the standard nonparametric bias rate of $O(h^{\kappa \wedge (p+1)})$ is recovered.

C. Minimax Optimality and Convergence Rates

• Theorem 4 & 5 (Convergence & Inference): The authors derive pointwise and uniform convergence rates and construct valid confidence intervals and bands.
  • The variance convergence rate is $(nh^2)^{-1}$, which is slower than standard univariate RD ($nh^{-1}$) because the target is a bivariate function restricted to a curve.
  • Valid uniform inference requires a "small bias" condition. When kinks are present, this requires undersmoothing (choosing $h$ such that bias is negligible relative to variance).
• Theorem 6 (Minimax Lower Bound): The authors prove that for a broad class of isotropic nonparametric estimators on a rectifiable boundary, the uniform minimax convergence rate is $n^{-1/4}$ (up to logarithmic factors).
  • This rate coincides with the optimal rate for estimating Lipschitz continuous bivariate functions.
  • Crucially, no isotropic distance-based estimator can achieve a faster rate than $n^{-1/4}$ if the boundary may contain kinks, regardless of the smoothness of the underlying data. This establishes a fundamental limitation of distance-based methods compared to location-based methods.

4. Results and Numerical Evidence

• Simulation Study: Using data calibrated from the Colombian "Ser Pilo Paga" (SPP) education subsidy program (which has an L-shaped boundary with a kink), the authors show:
  • Standard univariate RD bandwidth selectors (designed for smooth boundaries) lead to significant bias near kinks.
  • Adaptive bandwidth selection (using $h \propto n^{-1/4}$ near kinks and $h \propto n^{-1/(4+2p)}$ elsewhere) effectively mitigates this bias.
  • Ignoring kinks results in biased point estimates, though coverage rates may remain robust due to conservative variance estimation in some settings.
• Empirical Application: The authors re-analyze the SPP data. They find heterogeneous treatment effects along the boundary: the program is more effective for students with lower academic scores and lower wealth (near the kink) compared to those with higher scores and wealth.

5. Significance and Practical Implications

• Theoretical Foundation: This is the first rigorous theoretical justification for distance-based methods in BD designs. It clarifies why these methods are popular (simplicity, data availability) but also highlights their statistical limitations.
• Geometric Awareness: The paper fundamentally changes how researchers should view BD designs. It argues that boundary geometry is not just a nuisance parameter but a central determinant of statistical performance.
• Implementation Guidelines:
  • Bandwidth Selection: Researchers should not blindly apply univariate RD bandwidth rules. If the boundary has kinks, the optimal bandwidth scales as $n^{-1/4}$ globally or adaptively.
  • Bias Correction: Standard robust bias correction (using higher-order polynomials) fails near kinks because the bias is $O(h)$, not $O(h^{p+1})$. Undersmoothing is the recommended remedy.
  • Software: The authors provide the rd2d R package, which implements these distance-based estimators, adaptive bandwidth selectors, and uniform inference procedures.

In summary, the paper provides a comprehensive framework for causal inference in boundary discontinuity designs using distance-based methods. It establishes that while these methods are practical and valid, their performance is strictly bounded by the geometric regularity of the assignment boundary, necessitating specialized estimation and inference strategies to handle irregularities like kinks.