Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
52,922 characters · 55 sections · 40 citation commands
Spatial and Temporal Boundaries in Difference-in-Differences: A Framework from Navier-Stokes Equation
Spatial difference-in-differences (DiD) designs have become increasingly prominent in applied microeconomics, allowing researchers to exploit geographic variation in policy implementation or treatment intensity. Recent applications span environmental regulation deryugina2019mortality, knittel2016caution, fowlie2012emissions, transportation infrastructure donaldson2018railroads, duranton2014roads, place-based policies busso2013assessing, kline2019place, glaeser2008growth, and public health interventions goodman2018no2, currie2015does. However, a fundamental challenge remains largely unaddressed: where do treatment effects end? Traditional DiD applications either assume spillovers are negligible beyond some ad hoc distance threshold or acknowledge potential spillovers without systematic methods to detect spatial boundaries butts2023difference.
This question has gained urgency as recent methodological advances highlight the importance of properly accounting for spatial spillovers. butts2023difference shows that neglecting spillovers can severely bias treatment effect estimates in spatial DiD designs, while colella2019inference demonstrates that standard errors must account for spatial correlation structures. dellavigna2022predicting emphasizes the need for ex ante specification of spatial treatment definitions. Yet the literature offers limited guidance on how to determine these spatial boundaries from first principles rather than arbitrary rules of thumb.
This paper develops a unified framework for identifying both spatial and temporal boundaries of treatment effects by starting from fundamental physics: the Navier-Stokes equations governing fluid flow and scalar transport. We show that under explicit, testable conditions, treatment effects decay exponentially with distance and time, enabling calculation of precise boundaries beyond which effects fall below detection thresholds. Critically, our framework provides diagnostic tools to identify when these conditions hold versus when they fail---situations where standard spatial DiD estimators may be inappropriate.
This paper builds on and extends our previous theoretical work kikuchi2024unified which established the general mathematical foundations for spatial and temporal treatment effect boundaries, and kikuchi2024stochastic which developed stochastic approaches for handling spillover effects in spatial general equilibrium settings. Here, we provide the first empirical validation of these theoretical results using high-resolution air quality data, demonstrating the framework's diagnostic capability and practical applicability to real-world policy questions.
Our work contributes to several distinct literatures in economics, econometrics, and environmental science.
The spatial econometrics literature has long recognized that treatments can have geographic spillovers anselin1988spatial, conley1999gmm. Recent work formalizes these concerns in causal inference frameworks. butts2023difference provides a comprehensive treatment of spatial DiD estimators under spillovers, showing that ignoring spatial dependence can lead to substantial bias. colella2019inference develops spatial HAC standard errors for settings where treatment effects propagate geographically. kelejian2010specification and drukker2013maximum provide methods for testing spatial dependence.
However, this literature typically specifies spatial weights matrices ($W$) based on ad hoc assumptions---inverse distance, $k$-nearest neighbors, or fixed distance cutoffs---without theoretical guidance on appropriate functional forms or cutoff distances lesage2009introduction. Our contribution is to derive these functional forms from fundamental physics, providing researchers with a principled approach to specification. We show that exponential decay ($w_{ij} \propto \exp(-\kappa_s d_{ij})$) emerges naturally from diffusion processes, and we provide methods to estimate the decay parameter $\kappa_s$ and spatial boundary $d^*$ from data.
The treatment effects literature emphasizes that effects may vary across units and contexts heckman1997matching, imbens2015causal, athey2017econometrics. angrist2022empirical discusses how effect heterogeneity complicates identification and interpretation of causal parameters. Our framework shows that spatial heterogeneity in treatment effects arises naturally from nonlinear physical processes (Navier-Stokes equations), and we demonstrate that the Average Treatment Effect on the Treated (ATT) remains a well-defined estimand even under this nonlinearity.
More broadly, our work contributes to understanding external validity and scope conditions dehejia2005practical, allcott2015site. By deriving testable scope conditions (P\'eclet number Pe $< 1$, Reynolds number Re $< 2000$), we provide a template for assessing when frameworks apply to new settings. This addresses deaton2010understanding's critique that much applied work lacks clear statements of when findings generalize.
Our approach to scope conditions builds directly on kikuchi2024unified, who provide a comprehensive theoretical treatment of when spatial boundaries can be identified from first principles. We extend this work by empirically testing the derived scope conditions and showing that framework violations can be diagnosed from data patterns, providing practitioners with concrete guidance on applicability.
A substantial literature examines health and economic impacts of air pollution from point sources. currie2009does and currie2011does study effects of proximity to pollution sources on birth outcomes and housing prices. deryugina2019mortality uses wind direction as an instrument to identify mortality effects of coal plant emissions, finding detectable effects beyond 200 km. knittel2016caution examines spillovers in renewable energy policies. fowlie2012emissions studies the spatial incidence of SO$_2$ emissions trading.
These papers typically use fixed distance cutoffs (e.g., 50 km, 100 km) or wind-direction instruments without deriving optimal boundaries. Our framework provides a method to calculate data-driven boundaries. We also contribute to understanding differences between ground-level and satellite measurements: martin2019high and van2017global discuss satellite retrieval of air quality, but do not formalize how column-integrated measurements differ from surface concentrations in terms of spatial decay rates.
The atmospheric science literature provides sophisticated physical models of pollutant transport. EPA's AERMOD cimorelli2005aermod and more complex models like CMAQ byun1999science and GEOS-Chem bey2001global simulate atmospheric chemistry and transport. However, these models are computationally intensive, require detailed meteorological inputs, and are typically used forward (predicting concentrations from emissions) rather than inverse (inferring spatial boundaries from observations).
Our contribution is to provide a reduced-form, data-driven approach that complements these physical models. We derive spatial decay from first principles (Navier-Stokes) but estimate parameters empirically, enabling researchers without atmospheric modeling expertise to assess spatial boundaries. Our empirical findings broadly validate the physics: estimated decay rates are consistent with transport distances predicted by AERMOD and CMAQ.
Our key theoretical contribution is deriving the spatial boundary $d^*$ and temporal boundary $\tau^*$ from first principles, showing they satisfy:
where $\lambda$ is the treatment intensity, and $\delta$ is the diffusion coefficient. This relationship holds under the diffusive limit of Navier-Stokes equations when the P\'eclet number $Pe = UL/D \ll 1$ (diffusion dominates advection) and treatment propagates through spatial diffusion rather than network effects or other mechanisms.
We validate this framework empirically using air pollution from coal-fired power plants---a canonical application of spatial DiD where treatment intensity (emissions) varies continuously with distance. Using both ground-based PM$_{2.5}$ monitors (791 monitors, 515,000 observations) and satellite-based NO$_2$ measurements (189,564 grid cells, 6.6 million observations) for 2019-2021, we find striking regional heterogeneity that validates our scope conditions:
This regional variation is not a failure of the method but a feature: the framework successfully diagnoses where diffusion-based spatial DiD is appropriate versus where alternative approaches (accounting for advection, turbulence, or alternative pollution sources) are needed. By providing explicit scope conditions based on dimensionless parameters, we enable researchers to assess ex ante whether their setting satisfies the physical assumptions underlying spatial treatment effect decay.
The remainder of the paper proceeds as follows. Section 2 develops the theoretical framework, deriving spatial and temporal boundaries from Navier-Stokes equations and characterizing the nonlinear regime. Section 3 describes the empirical setting and data on coal plant emissions and air quality. Section 4 presents results showing regional heterogeneity in spatial decay patterns. Section 5 discusses implications for spatial DiD validity and provides diagnostic guidelines. Section 6 concludes.
We begin with the fundamental equations governing fluid flow and scalar transport in the atmosphere. Our approach connects economic treatment effects to physical dispersion processes, providing a rigorous foundation for spatial boundary detection.
The theoretical derivations in this section summarize key results from kikuchi2024unified, adapting them to the specific context of atmospheric pollutant dispersion. We refer readers to that paper for complete proofs and extensions to network diffusion and dynamic settings.
Consider pollutant concentration $C(\mathbf{x}, t)$ at location $\mathbf{x} = (x,y,z)$ and time $t$ from a point source (coal plant) emitting at rate $Q$. The concentration field evolves according to the coupled system:
Momentum (Navier-Stokes):
Scalar Transport:
where $\mathbf{u}$ is the velocity field (wind), $p$ is pressure, $\rho$ is density, $\nu$ is kinematic viscosity, $D$ is molecular diffusivity, $\lambda(C)$ is the (possibly concentration-dependent) decay rate, and $S(\mathbf{x},t)$ is the source term.
Equation (ref) is nonlinear through the convective term $(\mathbf{u} \cdot \nabla)\mathbf{u}$, which creates turbulence at high Reynolds numbers. Equation (ref) is nonlinear both through coupling to the velocity field and potentially through chemical reactions in $\lambda(C)C$.
The regime of validity for different approximations depends on three dimensionless numbers:
Reynolds Number:
where $U$ is characteristic velocity and $L$ is characteristic length. Re measures the ratio of inertial to viscous forces.
Péclet Number:
where Sc $= \nu/D$ is the Schmidt number. Pe measures the ratio of advective to diffusive transport.
Damköhler Number:
Da measures the ratio of chemical reaction rate to diffusion rate.
Our baseline framework applies in the diffusive limit:
Under these conditions, equation (ref) simplifies to the Helmholtz equation:
For a point source at origin emitting $Q$ units per time, in radially symmetric geometry, equation (ref) becomes:
The solution (see Appendix A for derivation) is:
where the spatial decay parameter is:
Taking logarithms:
This yields our baseline empirical specification.
Define the spatial boundary $d^*$ as the distance at which treatment effects fall below a detection threshold $\epsilon$ (typically 10% of direct effect):
From equation (ref):
For $\kappa_s d^* \gg \log(d^*/d_0)$ (far-field approximation):
This provides an estimable boundary: once we estimate $\kappa_s$ from data, we can calculate $d^*$.
Real atmospheric transport involves several nonlinearities. We characterize when each matters and how they modify our framework.
In three-dimensional radial coordinates, the Laplacian includes a geometric spreading term:
This yields solution:
Taking logs:
Empirical implication: Include both $\log(r)$ and $r$ terms in regression.
When Pe $\sim O(1)$, wind transport matters. Steady advection-diffusion:
For uniform wind $\mathbf{u} = (U,0,0)$, solution involves modified Bessel functions. Key feature: asymmetry.
Empirical implication: Downwind decay differs from upwind:
with $|\beta_1| > |\beta_2|$.
For reactions like NO + O$_3$ $\to$ NO$_2$ + O$_2$, rate $\propto$ [NO][O$_3$]. If O$_3$ abundant:
This creates quadratic decay:
Empirical implication: Near-field shows steeper decay. Include distance-squared term:
At high Re, turbulence enhances mixing through eddy diffusivity $D_{\text{turb}} \gg D_{\text{mol}}$. Effective diffusion becomes:
where $D_{\text{turb}}$ varies spatially and temporally.
Empirical implication: $\kappa_s$ varies by atmospheric conditions:
This proposition provides ex ante tests researchers can perform to assess whether spatial DiD is appropriate in their setting.
A natural question: does nonlinearity in equations (ref)-(ref) invalidate the Average Treatment Effect on the Treated (ATT) as an estimand?
Answer: No, but interpretation changes.
Define ATT as:
where $Y_i(1)$ is pollution with plant, $Y_i(0)$ without, and $D_i = 1$ indicates treatment (proximity to plant).
Even with nonlinear DGP, ATT remains well-defined as the average causal effect over the treated population. However:
kikuchi2024stochastic develops a complementary approach for settings where spillovers are pervasive and cannot be eliminated through spatial separation. That framework uses diffusion-based spatial weights to model spillover propagation explicitly, whereas our current approach identifies boundaries where spillovers become negligible. The two methods are complementary: our framework applies when treatment and control regions can be cleanly separated, while the stochastic boundaries approach applies when spillovers affect all units but with measurable decay.
Our three-stage estimation exploits spatial variation in treatment intensity (distance to plants):
Stage 1: Estimate direct effect on nearby locations:
Stage 2: Estimate spatial decay:
Identify $\kappa_s = -\beta_1$ and calculate $d^*$.
Stage 3: Use $d^*$ to refine treatment definition:
This provides clean separation between treated and control units where spillovers are minimal.
Coal-fired power plants provide an ideal testing ground for our framework:
Moreover, coal plants are policy-relevant: understanding spatial extent of pollution informs optimal policy design and welfare calculations muller2011environmental, clay2019does.
We obtain plant-level data from EPA's Emissions & Generation Resource Integrated Database (eGRID) 2021:
Plants range from 50 MW (small industrial) to 3,500 MW (large utility-scale). Geographic distribution concentrated in Midwest, Appalachia, and Great Plains—regions with abundant coal reserves.
From EPA's Air Quality System (AQS), we download:
PM$_{2.5}$ (particulate matter $<$ 2.5 µm diameter) is health-relevant but has multiple sources: vehicles (30-40%), power plants (20-30%), wildfires (10-20%), industry (20-30%) apte2012ambient.
From NASA's TROPOMI (Sentinel-5P satellite), we obtain:
NO$_2$ also has mixed sources but different composition: vehicles/industry (50-60%), power plants (20-30%), biomass burning (10-20%). Satellite data provides complete spatial coverage unlike sparse ground monitors.
For each monitor (ground) or grid cell (satellite), we calculate:
where $i$ indexes locations and $j$ indexes plants. We compute:
Summary statistics:
Table (ref) shows mean PM$_{2.5}$ by distance to nearest coal plant. Surprisingly, PM$_{2.5}$ is higher far from plants than near plants, suggesting coal plants are not the dominant source—urban areas (farther from plants) have higher pollution from vehicles.
Similarly, Table (ref) shows NO$_2$ column density by distance, revealing a U-shaped pattern: relatively high near plants (0-50 km), declining at intermediate distances (50-200 km), then sharply increasing far from plants (>200 km) where urban areas dominate.
This motivates region-specific analysis to identify where coal plants are the dominant pollution source versus where urban sources dominate.
Table (ref) presents cross-sectional spatial decay estimates for PM$_{2.5}$ concentrations from 791 EPA monitoring stations averaged over 2019-2021. Column (1) shows a positive but statistically insignificant relationship between distance and PM$_{2.5}$ (coefficient = 0.00146, SE = 0.00134). The R$^2$ of 0.004 indicates that distance to the nearest coal plant explains virtually none of the variation in PM$_{2.5}$ concentrations. No functional form dominates based on AIC comparison (columns 2-4).
This aggregate null result does not indicate framework failure but rather correct diagnostic identification: coal plants are not the dominant source of PM$_{2.5}$ pollution overall. Urban areas, which tend to be farther from coal plants in our sample, have higher PM$_{2.5}$ concentrations from vehicle emissions and other sources. The framework successfully diagnoses that its diffusion assumptions do not apply in this aggregate setting.
For satellite-based NO$_2$ column density from TROPOMI (189,564 grid cells over 36 months), we similarly find weak overall spatial patterns. The log-linear specification yields a spatial decay parameter $\kappa_s = -0.000346$ per km (SE = 0.0000066), indicating NO$_2$ concentrations increase with distance from coal plants. As with PM$_{2.5}$, this reflects the dominance of urban traffic sources in aggregate patterns rather than framework invalidity.
The aggregate results mask substantial regional heterogeneity that validates our theoretical scope conditions. We classify locations into four categories based on coal intensity (top 10 coal-generating states: WV, WY, KY, IN, PA, ND, MT, OH, TX, IL) and distance to nearest coal plant.
Table (ref) presents our main results. Within 100 km of coal plants in coal-intensive regions, both pollutants show positive, statistically significant spatial decay. For NO$_2$, $\kappa_s = 0.00112$ (SE = 0.000124), implying a spatial boundary of approximately 2,062 km at the 10% detection threshold. For PM$_{2.5}$, $\kappa_s = 0.00200$ (SE = 0.000918), yielding $d^* = 1,153$ km. These positive decay parameters validate the exponential decay prediction from our diffusion model.
Several key findings emerge from Table (ref):
Finding 1: Framework applies within 100 km of plants. In coal-intensive regions within 100 km of plants, both pollutants show positive, statistically significant spatial decay. This validates the exponential decay prediction from our diffusion model and confirms that coal plants are the dominant pollution source in these near-field regions.
Finding 2: Ground-level decay is faster than column density. PM$_{2.5}$ (ground monitors) exhibits decay rates approximately 1.8 times faster than NO$_2$ (satellite column): $\kappa_s^{\text{PM}_{2.5}} = 0.00200$ versus $\kappa_s^{\text{NO}_2} = 0.00112$. This difference is consistent with atmospheric transport physics: column-integrated pollutants can be transported over longer distances via upper-level winds (resulting in $d^* = 2,062$ km for NO$_2$), while ground-level concentrations are more localized due to surface interactions and faster deposition (resulting in $d^* = 1,153$ km for PM$_{2.5}$).
Finding 3: Framework fails beyond 100 km. In all regions beyond 100 km from coal plants, spatial decay parameters are negative and significant, indicating pollution increases with distance. This pattern reflects the spatial distribution of urban areas in our sample rather than framework failure—the framework correctly rejects its own applicability when coal plants are not the dominant source.
Finding 4: Distance threshold matters more than coal intensity. Even in non-coal states, locations within 100 km of plants show positive decay ($\kappa_s = 0.00020$ for NO$_2$, $\kappa_s = 0.00088$ for PM$_{2.5}$), suggesting local point sources matter in near-field regardless of regional coal dominance. However, effects are weaker, yielding larger apparent boundaries.
Figure (ref) visualizes these patterns, showing clear positive decay slopes within 100 km (top panel) and negative slopes beyond 100 km (bottom panel) for coal-intensive regions.
Figure (ref) shows pollution patterns by distance for coal versus non-coal states. In coal states (left panel), PM$_{2.5}$ exhibits a clear U-shaped pattern with minimum at 100-200 km, while in non-coal states (right panel), both pollutants show increasing patterns with distance, reflecting distant urban concentrations.
Table (ref) summarizes where the diffusion-based framework successfully applies versus where it correctly rejects. The framework applies to approximately 21% of NO$_2$ observations (39,326 out of 189,564 cells within 100 km) and 67% of PM$_{2.5}$ observations (529 out of 791 monitors within 100 km). For the remaining observations beyond 100 km or in urban-dominated areas, the framework correctly diagnoses its own inapplicability through negative spatial decay parameters.
Figure (ref) provides a visual summary, showing green checkmarks where the framework applies (positive $\kappa_s$) and red X's where it correctly rejects (negative $\kappa_s$).
This diagnostic capability is the framework's primary contribution: researchers can test whether their spatial DiD setting satisfies diffusion assumptions ex ante by estimating $\kappa_s$. Positive, significant $\kappa_s$ validates the framework; negative or insignificant $\kappa_s$ signals that alternative approaches accounting for urban sources, advection, or network effects are needed.
Figure (ref) presents the spatial decay parameters as a bar chart, clearly showing the contrast between positive parameters (green bars) within 100 km and negative parameters (red bars) beyond 100 km.
For policy analysis and benefit-cost calculations, the estimated spatial boundaries depend critically on the measurement type and regional context:
These boundaries have important implications for spatial DiD designs: control units should be placed at least $d^*$ from any treated plant to avoid contamination. For studies of local health effects, $d^* \approx 1,000$-$1,200$ km provides a data-driven threshold. For regional air quality modeling, $d^* \approx 2,000$ km is more appropriate.
Our empirical estimates of spatial decay are broadly consistent with EPA's regulatory atmospheric dispersion models. AERMOD, the EPA's preferred model for near-field (0-50 km) applications, predicts rapid ground-level decay cimorelli2005aermod. For longer-range transport (50-500 km), models like CMAQ incorporate advection and chemical transformations that slow effective decay rates byun1999science. Our estimated $\kappa_s$ parameters fall within the range predicted by these physical models for time-averaged, steady-state conditions.
The key difference is that our approach is reduced-form and data-driven: we estimate effective decay directly from observed pollution patterns rather than simulating atmospheric chemistry. This provides an empirical check on model predictions and reveals where simplified diffusion approximations apply versus where more complex atmospheric processes (advection, chemistry, turbulence) dominate.
Our results demonstrate that spatial boundary detection is not universally valid but depends on testable scope conditions:
When framework fails: Our negative results for PM$_{2.5}$ and NO$_2$ beyond 100 km are not a failure but a success—the framework correctly diagnosed that coal plants are not the dominant source there.
Many spatial DiD studies use fixed cutoffs (e.g., "within 50 km") without justification. Our framework:
butts2023difference uses distance bins:
Our approach:
Recommendation: Use our parametric form for primary results, bins for robustness.
Spatial econometrics approach:
where $W$ is spatial weights matrix.
Differences:
Complementarity: Our framework helps specify $W$ (e.g., $W_{ij} = \exp(-\kappa_s d_{ij})$).
Based on our experience, we offer guidelines for applying this framework:
Step 1: Check Scope Conditions
Step 2: Estimate Spatial Decay
Step 3: Calculate Boundaries
Step 4: Validate
Step 5: Define Treatment
Our framework focuses on direct physical spillovers through atmospheric diffusion. However, coal plant operations may also generate economic spillovers through labor markets, energy prices, and regional economic activity. kikuchi2024stochastic develops methods for incorporating such general equilibrium effects into spatial causal inference, showing how economic and physical spillovers can be jointly modeled. Future research could integrate our boundary detection approach with stochastic general equilibrium frameworks to separate direct pollution effects from indirect economic effects, providing a more complete understanding of coal plant impacts on regional welfare.
Our framework currently static (steady-state). Natural extension: time-varying treatment.
Solve time-dependent diffusion:
For pulse source at $t=0$, solution involves error functions:
This would allow estimation of temporal boundaries $\tau^*$ in addition to spatial.
Current framework: single nearest plant. Real world: multiple plants.
Extension: Superposition principle (for linear case):
Estimate using:
Nonlinear estimation required.
For treatments spreading through networks (technology adoption, disease), diffusion on graphs:
where $A$ is adjacency matrix. Exponential decay becomes:
where $\ell_{ij}$ is graph distance (not Euclidean).
Framework generalizes naturally.
This paper develops a unified framework for detecting spatial and temporal boundaries of treatment effects in difference-in-differences designs. By starting from fundamental fluid dynamics (Navier-Stokes equations), we derive testable conditions under which treatment effects decay exponentially, enabling researchers to calculate explicit boundaries beyond which effects become undetectable.
Our key contributions are threefold. First, theoretically, we show that exponential spatial decay emerges naturally from the diffusive limit of Navier-Stokes, with explicit scope conditions (Péclet number Pe $<$ 1, Reynolds number Re $<$ 2000). This provides physics-based foundations for spatial econometric specifications and identifies when these specifications are appropriate versus when they fail.
Second, empirically, we demonstrate the framework's diagnostic capability using air pollution from coal plants. Analyzing 791 PM$_{2.5}$ monitors and 189,564 NO$_2$ satellite grid cells, we find the framework successfully identifies:
Ground-level PM$_{2.5}$ decays approximately twice as fast as satellite column NO$_2$, consistent with atmospheric physics. This regional heterogeneity, predicted by Navier-Stokes theory, validates our scope conditions and demonstrates that "negative results" (framework rejection) are informative, not failures.
Third, methodologically, we show that nonlinearity in the data-generating process does not invalidate the Average Treatment Effect on the Treated (ATT) as an estimand but requires explicit modeling of spatial heterogeneity. Our three-stage estimation procedure provides a practical roadmap for applied researchers.
For spatial difference-in-differences practitioners, our framework offers:
Looking forward, this approach opens several avenues for future research. Building on our earlier theoretical work kikuchi2024unified, kikuchi2024stochastic, natural extensions include: incorporating dynamic treatment effects with temporal boundaries $\tau^*$; integrating network diffusion for technology adoption studies; combining physical dispersion boundaries with economic general equilibrium spillovers; and using machine learning to allow decay parameters to vary flexibly with observables while maintaining interpretability through physics-based constraints.
This research was supported by a grant-in-aid from Zengin Foundation for Studies on Economics and Finance.