Dynamic Spatial Treatment Effects as Continuous Functionals: Theory and Evidence from Healthcare Access
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917044456259584 |
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| author | Kikuchi, Tatsuru |
| author_facet | Kikuchi, Tatsuru |
| contents | I develop a continuous functional framework for spatial treatment effects grounded in Navier-Stokes partial differential equations. Rather than discrete treatment parameters, the framework characterizes treatment intensity as continuous functions $τ(\mathbf{x}, t)$ over space-time, enabling rigorous analysis of boundary evolution, spatial gradients, and cumulative exposure. Empirical validation using 32,520 U.S. ZIP codes demonstrates exponential spatial decay for healthcare access ($κ= 0.002837$ per km, $R^2 = 0.0129$) with detectable boundaries at 37.1 km. The framework successfully diagnoses when scope conditions hold: positive decay parameters validate diffusion assumptions near hospitals, while negative parameters correctly signal urban confounding effects. Heterogeneity analysis reveals 2-13 $\times$ stronger distance effects for elderly populations and substantial education gradients. Model selection strongly favors logarithmic decay over exponential ($Δ\text{AIC} > 10,000$), representing a middle ground between exponential and power-law decay. Applications span environmental economics, banking, and healthcare policy. The continuous functional framework provides predictive capability ($d^*(t) = ξ^* \sqrt{t}$), parameter sensitivity ($\partial d^*/\partial ν$), and diagnostic tests unavailable in traditional difference-in-differences approaches. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15324 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamic Spatial Treatment Effects as Continuous Functionals: Theory and Evidence from Healthcare Access Kikuchi, Tatsuru Econometrics Theoretical Economics Applications Methodology I develop a continuous functional framework for spatial treatment effects grounded in Navier-Stokes partial differential equations. Rather than discrete treatment parameters, the framework characterizes treatment intensity as continuous functions $τ(\mathbf{x}, t)$ over space-time, enabling rigorous analysis of boundary evolution, spatial gradients, and cumulative exposure. Empirical validation using 32,520 U.S. ZIP codes demonstrates exponential spatial decay for healthcare access ($κ= 0.002837$ per km, $R^2 = 0.0129$) with detectable boundaries at 37.1 km. The framework successfully diagnoses when scope conditions hold: positive decay parameters validate diffusion assumptions near hospitals, while negative parameters correctly signal urban confounding effects. Heterogeneity analysis reveals 2-13 $\times$ stronger distance effects for elderly populations and substantial education gradients. Model selection strongly favors logarithmic decay over exponential ($Δ\text{AIC} > 10,000$), representing a middle ground between exponential and power-law decay. Applications span environmental economics, banking, and healthcare policy. The continuous functional framework provides predictive capability ($d^*(t) = ξ^* \sqrt{t}$), parameter sensitivity ($\partial d^*/\partial ν$), and diagnostic tests unavailable in traditional difference-in-differences approaches. |
| title | Dynamic Spatial Treatment Effects as Continuous Functionals: Theory and Evidence from Healthcare Access |
| topic | Econometrics Theoretical Economics Applications Methodology |
| url | https://arxiv.org/abs/2510.15324 |