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

Abstract

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 (Péclet 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$_{2.5}$: $κ_s = 0.00200$, $d^* = 1,153$ km; NO$_2$: $κ_s = 0.00112$, $d^* = 2,062$ 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.

Figures (6)

• Figure 1: Regional Spatial Decay in Coal-Intensive States. Top panel: Within 100 km of coal plants, NO$_2$ shows positive spatial decay ($\kappa_s = +0.00112$), validating the diffusion framework. Bottom panel: Beyond 100 km, NO$_2$ increases with distance ($\kappa_s = -0.00123$), indicating urban sources dominate and the framework correctly rejects. Each point represents a grid cell's time-averaged NO$_2$ column density. The 100 km threshold emerges as a natural boundary separating coal-dominated from urban-dominated spatial patterns.
• Figure 2: Pollution Patterns by Distance: Coal vs Non-Coal States. Pollutant levels normalized to 0-100 scale within each type for comparability. Left panel (Coal States): PM$_{2.5}$ (green) shows clear U-shaped pattern with minimum at 100-200 km, while NO$_2$ (blue) remains relatively flat. Right panel (Non-Coal States): Both pollutants show increasing pattern with distance, with sharp increases beyond 200 km reflecting distant urban areas. These contrasting patterns demonstrate that spatial decay depends on the dominance of point sources (coal) versus distributed sources (urban traffic).
• Figure 3: Framework Validity Assessment. Green checkmarks ($\checkmark$) indicate regions where the framework applies ($\kappa_s > 0$, positive spatial decay). Red X's ($\times$) indicate regions where the framework correctly rejects ($\kappa_s \leq 0$, negative or zero decay). The framework successfully applies to both pollutants within 100 km of plants in both coal and non-coal states, but fails beyond 100 km where urban sources dominate. This demonstrates the framework's diagnostic capability: it identifies when diffusion assumptions are appropriate versus when alternative approaches are needed.
• Figure 4: Regional Spatial Decay Parameters: PM$_{2.5}$ vs NO$_2$. Green bars represent positive $\kappa_s$ (framework applies), red bars represent negative $\kappa_s$ (framework rejects). Error bars show 95% confidence intervals. Within 100 km of coal plants, both pollutants exhibit positive spatial decay, with PM$_{2.5}$ showing faster decay ($\kappa_s = 0.00200$) than NO$_2$ ($\kappa_s = 0.00112$). This difference is consistent with atmospheric physics: ground-level pollutants (PM$_{2.5}$) decay faster due to surface interactions, while column-integrated pollutants (NO$_2$) can be transported over longer distances via upper-level winds. Beyond 100 km, both show negative decay as urban sources dominate.
• Figure 5: Geographic Distribution of Coal-Fired Power Plants (2021). Point sizes proportional to nameplate capacity (MW). Concentration in Midwest, Appalachia, and Great Plains reflects proximity to coal deposits. Data from EPA eGRID 2021.

Theorems & Definitions (1)

• Proposition 2.1: Validity of Diffusion Approximation