Spatial Boundaries of Treatment Effects

• Spatial boundaries of treatment effects are defined as the geographic limits where localized interventions become negligible, using decay functions and threshold criteria.
• Researchers utilize parametric, nonparametric, and machine learning approaches to estimate these critical distances, ensuring robust causal inference.
• Accurate estimation of spatial boundaries improves treatment/control classification and informs targeted policy design by addressing geographic spillovers.

The spatial boundaries of treatment effects refer to the precise geographic (or more generally, metric) thresholds beyond which the causal influence of a localized intervention, event, or policy becomes negligible, indistinguishable from zero, or no longer meets a specified threshold of policy or scientific significance. Defining and estimating these boundaries has become central to empirical research and methodological development across economics, epidemiology, environmental science, and related disciplines, especially as high-resolution spatial data and recognition of geographic spillovers have made oversimplified assumptions problematic.

1. Conceptual Foundations and Formal Definitions

The spatial boundary is, in general, a distance $d^*$ such that for a source of intervention at location $x_0$, the treatment effect at any location $x$ with $d(x, x_0) \geq d^*$ satisfies $|\tau(x)| \leq \tau_{\min}$, where $\tau_{\min}$ is a pre-specified negligible effect threshold (for example, regulatory or detection limits). In continuous frameworks, treatment effects are parameterized as a function $\tau(x, t)$ over space and time, capturing dynamic propagation and decay.

Key formalizations include:

• Decay function: $m(d) = \mathbb{E}[Y(d)]$, where $d = d(x, x_0)$ is the distance from the treated unit or location; $m(d)$ typically decreases with $d$.
• Boundary definition: $$d^* = \inf\{d \in [0, \bar{d}]: m(d) \leq \varepsilon m(0)\}$$, where $\varepsilon$ denotes a policy or scientific fraction (e.g., 0.1 for the 10% contour) (Kikuchi, 14 Oct 2025).
• In continuous-diffusive models: the treatment field $\tau(x, t)$ may satisfy partial differential equations such as $$\partial_t\tau + v \cdot \nabla \tau = \nu \nabla^2\tau + S(x, t)$$ (advection-diffusion with sources), with boundaries defined as level sets or first passage contours (Kikuchi, 16 Oct 2025, Kikuchi, 17 Oct 2025).

The spatial boundary thus operationalizes the region in which spillovers or direct effects are nontrivial, facilitating credible treatment/control contrasts and policy targeting.

2. Theoretical and Statistical Frameworks

The literature offers multiple modeling approaches:

• Physical-Process-Based Models: Many spatial boundaries emerge from solutions to advection-diffusion equations (Navier–Stokes, Helmholtz) (Kikuchi, 13 Oct 2025, Kikuchi, 16 Oct 2025, Kikuchi, 17 Oct 2025). For stationary (steady-state) diffusion with linear decay, the concentration profile from a point source is:

$C(r) = \frac{Q}{4\pi D r} \exp(-\kappa_s r)$

where $\kappa_s = \sqrt{\lambda_0 / D}$ is the spatial decay parameter, $Q$ the source rate, and $D$ diffusion. The spatial boundary is:

$d^* = \frac{\log(1/\varepsilon)}{\kappa_s}$

for a detection threshold $\varepsilon$.

• Nonparametric Estimation: Recognizing that structural assumptions may be violated (e.g., due to turbulent atmospheric conditions, heterogeneous terrain, urban confounders), recent work uses local polynomial kernel regression to identify boundaries purely data-driven (Kikuchi, 14 Oct 2025, Kikuchi, 15 Oct 2025). For outcome $Y_i$ at distance $d_i$, estimate

$$\hat{m}(d_0) = \arg\min_{\beta_0,\beta_1} \sum_{i} K_h(d_i - d_0)[Y_i - \beta_0 - \beta_1(d_i - d_0)]^2$$

Then, set $$\hat{d}^* = \inf\{d: \hat{m}(d) \leq \varepsilon \hat{m}(0)\}$$.

• Jump-Diffusion and Structural Breaks: In contexts with regime shifts—where effects propagate slowly until a critical threshold and then jump—the spatial spillover process is modeled as a Lévy process. Boundaries are detected using sequential change-point (e.g., CUSUM) methods (Kikuchi, 8 Aug 2025).
• Unified Spatial–Temporal Boundaries: In reaction-diffusion models, boundaries in space and time are tied via the same physical parameters. For example, in certain models the spatial and temporal boundaries are $$d^* = \frac{\lambda^2}{\sqrt\delta} / \log(\kappa/(2\pi\lambda^2 K_{min}))$$ and $$\tau^* = (1/\delta) \log(\kappa / (\delta K_{min}))$$, revealing shared dynamics (Kikuchi, 1 Oct 2025).

3. Identification and Estimation Methodologies

Parametric Approaches

• Exponential decay model: Fit $m(d) = A\exp(-\kappa d)$ under structural assumptions derived from physics, and infer $d^*$ from estimated $\kappa$.
• Boundary diagnostics: Empirical sign and magnitude of $\kappa$ used to test model scope (positive decay parameters validate diffusion model; negative or zero reject it and signal confounding) (Kikuchi, 13 Oct 2025, Kikuchi, 16 Oct 2025).

Nonparametric Approaches

• Local Linear Regression: Avoids strong functional form restrictions by estimating $m(d)$ flexibly. Adaptive to non-monotonic or complex decay profiles; robust to cases with no effect or multiple inflection points (Kikuchi, 14 Oct 2025, Kikuchi, 15 Oct 2025).
• Cross-Validated Bandwidth Selection: Ensures optimally balanced bias–variance trade-off.
• Asymptotic properties: For optimal bandwidth $h_n = n^{-1/5}$, boundary estimator is $d^* \to_p d^*$, with convergence rate $n^{-2/5}$ (Kikuchi, 14 Oct 2025).

Machine Learning and Spatial Modeling

• Counterfactual Construction via CNNs: For spatial treatments, convolutional neural networks generate counterfactual locations matched on spatial covariates, allowing estimation of effect curves as a function of spatial proximity (Pollmann, 2020).
• Low-Rank and Sparse Decomposition: Detects direct and interference effects spatially using lasso and nuclear-norm regularization, providing both estimation and neighbor detection under sparse heterogeneity (Zhang et al., 7 Sep 2024).

Boundary Discontinuity and Geographic RD

• Local polynomial estimation on assignment boundaries (bivariate and distance-based), with results showing that using only distance to the boundary suffers from misspecification bias (especially at kinks); bivariate estimators are comparatively robust (Cattaneo et al., 8 May 2025).

4. Empirical Evidence and Case Studies

• Pollution from Coal-Fired Power Plants: Satellite NO₂ data (42 million observations) shows exponential decay of pollution, with nonparametric methods outperforming exponential models near sources ($\kappa_s=0.004$ per km, $d^*=572$ km at 10% threshold) (Kikuchi, 16 Oct 2025, Kikuchi, 14 Oct 2025). Nonparametric kernel regression improved prediction accuracy by up to 3.7 percentage points at long distances (Kikuchi, 14 Oct 2025).
• Bank Branch Openings and Mortgage Markets: Flexible nonparametric estimation reveals substantial spatial decay in loan application volume (8.5% reduction per 10 miles) but not in approval rates (no boundary detected), highlighting the value of nonparametric boundary detection in the presence of potentially flat or non-monotonic decay (Kikuchi, 15 Oct 2025).
• Healthcare Access: Exponential decay parameter for hospital influence was estimated at $\kappa = 0.002837$/km, with the effective influence boundary at $d^* = 37.1$ km. Regional and demographic heterogeneity sharply increase gradients for elderly and low-education groups; log-linear decay is empirically preferred ($\Delta AIC > 10,000$ over exponential) (Kikuchi, 17 Oct 2025).

5. Theoretical Advances and Robustness

• Dynamic and Continuous Functional Perspective: Formalizing treatment effects as $\tau(x, t)$ unifies spatial econometrics and mathematical physics, with boundaries derived from self-similar solutions (e.g., $\tau(d, t) = t^{-\alpha} f(d/t^\beta)$, where $\alpha$ and $\beta$ depend on diffusion regime) and allows for explicit boundary evolution over time (Kikuchi, 16 Oct 2025).
• Stochastic Regime Shifts: Jump-diffusion models (Lévy process) provide the first rigorous boundaries between partial and general equilibrium, with boundary crossing detected via first passage times; ignoring such stochastic boundaries results in substantial underestimation of treatment effects (by 28–67%) (Kikuchi, 8 Aug 2025).
• Diagnostic Scope Conditions: Assessment of regime validity using physical (Peclet, Reynolds, Damköhler) numbers prevents misapplication and provides self-testing diagnostics (sign and value of spectral decay parameters) (Kikuchi, 13 Oct 2025).
• Asymptotic Theory for Nonparametric Boundary Estimation: Explicit bias and variance formulas facilitate confidence set construction for $d^*$ even under weak smoothness, with negligible efficiency loss in massive datasets (Kikuchi, 14 Oct 2025).

6. Practical and Policy Implications

• Improved Causal Inference: Defining spatial boundaries more accurately enables valid definition of treatment and control units, robustly corrects for spillovers, and enhances comparability in difference-in-differences and related designs.
• Policy Design: Quantifying how far and for how long interventions matter supports cost–benefit analysis, targeted resource allocation, and the design of buffer zones or eligibility regions for subsidies.
• Flexible Methods Adapt to Complexity: Nonparametric and functional approaches avoid bias and specification errors, handle irregular, non-monotonic, or null patterns, and are diagnostic for confounding or inappropriate model assumptions.
• Applicability Across Domains: Although many leading examples are in environmental and public health settings (e.g., pollution, healthcare access), identical frameworks generalize to banking (catchment areas), infrastructure (project spillovers), and any spatially-mediated intervention.

7. Summary Table: Estimation Strategies for Spatial Boundaries

Method/Model	Functional Form	Handles Spillover Complexity?
Exponential decay (parametric)	$A\exp(-\kappa d)$	✗ Only if true exponential decay
Nonparametric local polynomial (kernel)	$\hat{m}(d)$ data-driven	✓ Non-monotonic, flat, nonlinear
Jump-diffusion/Lévy (stochastic process + CUSUM)	Stochastic, jump processes	✓ Regime shift, discontinuities
Reaction–diffusion PDEs (e.g., Navier–Stokes-based)	$\tau(x, t)$, self-similar forms	✓ Dynamic, continuous, multi-scalar
Low-rank/sparse detection (lasso/FISTA)	Sparse matrix/tensor models	✓ Heterogeneous, complex interference

Across methods, clear diagnostics—such as parameter sign, rate of decay, and robustness to bandwidth—are essential for valid boundary detection.

Conclusion

Modern spatial boundary estimation combines rigorous theoretical underpinnings (often drawing from physics-based or stochastic process models), flexible nonparametric and machine learning methods, and empirical diagnostic checks to reliably identify the extent of treatment effects. The move from rigid, parametric assumptions to continuous, boundary-aware and data-adaptive frameworks has substantially improved the accuracy, robustness, and interpretability of spatial treatment effect estimation, making these methods essential for credible causal inference and policy design in settings with geographic spillovers and interference (Kikuchi, 16 Oct 2025, Kikuchi, 14 Oct 2025, Kikuchi, 8 Aug 2025, Kikuchi, 13 Oct 2025, Kikuchi, 17 Oct 2025, Kikuchi, 1 Oct 2025, Pollmann, 2020, Kikuchi, 15 Oct 2025).