Dynamic Spatial Treatment Effect Boundaries: A Continuous Functional Framework from Navier-Stokes

Kikuchi, Tatsuru (2025): Dynamic Spatial Treatment Effect Boundaries: A Continuous Functional Framework from Navier-Stokes.

Abstract

I develop a comprehensive theoretical framework for dynamic spatial treatment effect boundaries using continuous functional definitions grounded in Navier-Stokes partial differential equations. Rather than discrete treatment effect estimators, the framework characterizes treatment intensity as a continuous function $\tau(\mathbf{x}, t)$ over space-time, enabling rigorous analysis of propagation dynamics, boundary evolution, and cumulative exposure patterns. Building on exact self-similar solutions expressible through Kummer confluent hypergeometric and modified Bessel functions, I establish that treatment effects follow scaling laws $\tau(d, t) = t^{-\alpha} f(d/t^\beta)$ where exponents characterize diffusion mechanisms. The continuous functional approach yields natural definitions of spatial boundaries $d^*(t)$, boundary velocities $v(t) = \partial d^*/\partial t$, treatment effect gradients $\nabla_d \tau$, and integrated exposure functionals $\int_0^T \tau \, dt$. Empirical validation using 42 million TROPOMI satellite observations of NO$_2$ pollution from U.S. coal-fired power plants demonstrates strong exponential spatial decay ($\kappa_s = 0.004028$ per km, $R^2 = 0.35$) with detectable boundaries at $d^* = 572$ km from major facilities. Monte Carlo simulations confirm superior performance over discrete parametric methods in boundary detection and false positive avoidance (94\% correct rejection rate versus 27\% for parametric methods). The framework successfully diagnoses regional heterogeneity: positive decay parameters within 100 km of coal plants validate the theory, while negative decay parameters beyond 100 km correctly signal when alternative pollution sources dominate. This sign reversal demonstrates the framework's diagnostic capability---it identifies when underlying physical assumptions hold versus when alternative mechanisms dominate. Applications span environmental economics (pollution dispersion fields), banking (spatial credit access functions), and healthcare (hospital accessibility). The continuous functional perspective unifies spatial econometrics with mathematical physics, connecting to recent advances in spatial correlation robust inference \citet{muller2022spatial} and addressing spurious spatial regression concerns \citet{muller2024spatial}.

• Item Type: MPRA Paper
• Language: English
• Keywords: Dynamic treatment effects, continuous functionals, Navier-Stokes equations, self-similar solutions, spatial boundaries, functional calculus, special functions, satellite remote sensing, spatial econometrics
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C21 - Cross-Sectional Models ; Spatial Models ; Treatment Effect Models ; Quantile Regressions; C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C31 - Cross-Sectional Models ; Spatial Models ; Treatment Effect Models ; Quantile Regressions ; Social Interaction Models; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C65 - Miscellaneous Mathematical Tools; D - Microeconomics > D0 - General > D04 - Microeconomic Policy: Formulation, Implementation, and Evaluation

References

\begin{thebibliography}{99}

\bibitem[Abramowitz and Stegun(1964)]{abramowitz1964handbook} Abramowitz, M., \& Stegun, I.~A. (1964). \textit{Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}. Dover Publications.

\bibitem[Angrist and Kolesár(2022)]{angrist2022empirical} Angrist, J.~D., \& Kolesár, M. (2022). One instrument to rule them all: The bias and coverage of just-ID IV. \textit{Journal of Econometrics}, forthcoming.

\bibitem[Anselin(1988)]{anselin1988spatial} Anselin, L. (1988). \textit{Spatial Econometrics: Methods and Models}. Springer-Verlag.

\bibitem[Athey and Imbens(2017)]{athey2017econometrics} Athey, S., \& Imbens, G.~W. (2017). The econometrics of randomized experiments. In G.~Duranton, J.~V.~Henderson, \& W.~C.~Strange (Eds.), \textit{Handbook of Regional and Urban Economics}, volume 1, pages 73--140.

\bibitem[Butts and Gardner(2023)]{butts2023difference} Butts, K., \& Gardner, J. (2023). Difference-in-differences with spatial spillovers. Working paper.

\bibitem[Conley(1999)]{conley1999gmm} Conley, T.~G. (1999). GMM estimation with cross sectional dependence. \textit{Journal of Econometrics}, 92(1), 1--45.

\bibitem[Deaton(2010)]{deaton2010understanding} Deaton, A. (2010). Understanding the mechanisms of economic development. \textit{Journal of Economic Perspectives}, 24(3), 3--16.

\bibitem[Evans(2010)]{evans2010partial} Evans, L.~C. (2010). \textit{Partial Differential Equations} (2nd ed.). American Mathematical Society.

\bibitem[Gibbons et al.(2015)]{gibbons2015mostly} Gibbons, S., Overman, H.~G., \& Patacchini, E. (2015). Spatial methods. In G.~Duranton, J.~V.~Henderson, \& W.~C.~Strange (Eds.), \textit{Handbook of Regional and Urban Economics}, volume 5, pages 115--168. Elsevier.

\bibitem[Heckman et al.(1997)]{heckman1997matching} Heckman, J.~J., Ichimura, H., \& Todd, P.~E. (1997). Matching as an econometric evaluation estimator: Evidence from evaluating a job training programme. \textit{Review of Economic Studies}, 64(4), 605--654.

\bibitem[Imbens and Rubin(2015)]{imbens2015causal} Imbens, G.~W., \& Rubin, D.~B. (2015). \textit{Causal Inference for Statistics, Social, and Biomedical Sciences}. Cambridge University Press.

\bibitem[Kikuchi(2024a)]{kikuchi2024unified} Kikuchi, T. (2024a). A unified framework for spatial and temporal treatment effect boundaries: Theory and identification. \textit{arXiv preprint arXiv:2510.00754}.

\bibitem[Kikuchi(2024b)]{kikuchi2024stochastic} Kikuchi, T. (2024b). Stochastic boundaries in spatial general equilibrium: A diffusion-based approach to causal inference with spillover effects. \textit{arXiv preprint arXiv:2508.06594}.

\bibitem[Kikuchi(2024c)]{kikuchi2024navier} Kikuchi, T. (2024c). Spatial and temporal boundaries in difference-in-differences: A framework from Navier-Stokes equation. \textit{arXiv preprint arXiv:2510.11013}.

\bibitem[Kikuchi(2024d)]{kikuchi2024nonparametric1} Kikuchi, T. (2024d). Nonparametric identification and estimation of spatial treatment effect boundaries: Evidence from 42 million pollution observations. \textit{arXiv preprint arXiv:2510.12289}.

\bibitem[Kikuchi(2024e)]{kikuchi2024nonparametric2} Kikuchi, T. (2024e). Nonparametric identification of spatial treatment effect boundaries: Evidence from bank branch consolidation. \textit{arXiv preprint arXiv:2510.13148}.

\bibitem[Müller and Watson(2022)]{muller2022spatial} Müller, U.~K., \& Watson, M.~W. (2022). Spatial correlation robust inference. \textit{Econometrica}, 90(6), 2901--2935.

\bibitem[Müller and Watson(2024)]{muller2024spatial} Müller, U.~K., \& Watson, M.~W. (2024). Spatial unit roots and spurious regression. \textit{Econometrica}, 92(5), 1661--1695.

\bibitem[Zhdanov and Zhdanov(2010)]{zhdanov2010selfsimilar} Zhdanov, V.~M., \& Zhdanov, R.~Z. (2010). On the solutions to the Navier-Stokes equations with self-similar structure. \textit{Journal of Mathematical Physics}, 51(9), 093102.

\end{thebibliography}