标题： 差分法中的时空边界：来自纳维-斯托克斯方程的框架 显示英文标题

标题： Spatial and Temporal Boundaries in Difference-in-Differences: A Framework from Navier-Stokes Equation

摘要： 本文提出了一种统一的框架，用于在双重差分设计中识别处理效应的空间和时间边界。 从基本的流体动力学方程（纳维-斯托克斯方程）出发，我们推导了处理效应在空间和时间上呈指数衰减的条件，使研究人员能够计算出效应变得不可检测的明确边界。 该框架涵盖了线性（纯扩散）和非线性（带有化学反应的对流扩散）情形，其可检验的范围条件基于物理学中的无量纲数（佩克莱特数和雷诺数）。 我们通过燃煤电厂的空气污染案例展示了该框架的诊断能力。 分析2019-2021年美国西部791个地面PM$_{2.5}$监测站和189,564个基于卫星的NO$_2$网格单元，我们发现显著的区域异质性：在煤矿电厂100公里范围内，两种污染物均表现出正空间衰减（PM$_{2.5}$：$\kappa_s = 0.00200$，$d^* = 1,153$公里；NO$_2$：$\kappa_s = 0.00112$，$d^* = 2,062$公里），验证了该框架。 超过100公里后，负衰减参数正确表明城市源主导，扩散假设失效。 地表PM$_{2.5}$的衰减速率大约是卫星柱状NO$_2$的两倍，这与大气传输物理一致。 该框架成功诊断了其在八个分析区域中的四个区域的有效性，为研究人员提供了基于物理的工具，以评估其空间差分中的差异设置是否在应用估计器之前满足扩散假设。 我们的结果表明，严格的边界检测需要从基本原理进行理论推导，并对基础物理假设进行实证验证。 显示英文摘要

摘要： This paper develops a unified framework for identifying spatial and temporal boundaries of treatment effects in difference-in-differences designs. Starting from fundamental fluid dynamics equations (Navier-Stokes), we derive conditions under which treatment effects decay exponentially in space and time, enabling researchers to calculate explicit boundaries beyond which effects become undetectable. The framework encompasses both linear (pure diffusion) and nonlinear (advection-diffusion with chemical reactions) regimes, with testable scope conditions based on dimensionless numbers from physics (P\'eclet and Reynolds numbers). We demonstrate the framework's diagnostic capability using air pollution from coal-fired power plants. Analyzing 791 ground-based PM$_{2.5}$ monitors and 189,564 satellite-based NO$_2$ grid cells in the Western United States over 2019-2021, we find striking regional heterogeneity: within 100 km of coal plants, both pollutants show positive spatial decay (PM$_{2.5}$: $\kappa_s = 0.00200$, $d^* = 1,153$ km; NO$_2$: $\kappa_s = 0.00112$, $d^* = 2,062$ km), validating the framework. Beyond 100 km, negative decay parameters correctly signal that urban sources dominate and diffusion assumptions fail. Ground-level PM$_{2.5}$ decays approximately twice as fast as satellite column NO$_2$, consistent with atmospheric transport physics. The framework successfully diagnoses its own validity in four of eight analyzed regions, providing researchers with physics-based tools to assess whether their spatial difference-in-differences setting satisfies diffusion assumptions before applying the estimator. Our results demonstrate that rigorous boundary detection requires both theoretical derivation from first principles and empirical validation of underlying physical assumptions.