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.
69,888 characters · 49 sections · 59 citation commands
Nonparametric Identification of Spatial Treatment Effect Boundaries: Evidence from Bank Branch Consolidation
Keywords: Spatial econometrics, nonparametric methods, treatment effects, bank branches, financial access, digital transformation
JEL Classification: C14, C21, G21, R12
Spatial spillovers are ubiquitous in economics. From environmental externalities muller2011machado to knowledge diffusion jaffe1993geographic, economic shocks propagate through geographic space in ways that fundamentally shape outcomes. Understanding these spatial patterns is crucial for both policy design and theoretical development. A key empirical question concerns the identification of spatial boundaries---the distances at which treatment effects decay to economically negligible magnitudes. Such boundaries determine the geographic scope of policies and help allocate scarce resources efficiently.
This paper contributes to three distinct literatures: spatial econometrics, financial access and banking, and nonparametric estimation methods.
The spatial econometrics literature has developed sophisticated methods for estimating treatment effects that propagate through geographic space. conley1999gmm pioneered spatial GMM estimation allowing for arbitrary patterns of spatial correlation. gibbons2015mostly provide a comprehensive review of spatial methods in applied microeconomics, emphasizing the challenges of identifying causal effects when treatments and outcomes are spatially correlated.
Recent work has focused on identifying spatial boundaries of treatment effects. banzhaf2019difference develop difference-in-differences estimators for spatial treatments, showing how to recover average treatment effects in the presence of spillovers. dellavigna2022predicting examine spatial diffusion of information and behaviors, documenting how effects decay with distance.
A particularly influential strand of research examines environmental spillovers and their spatial extent. muller2011machado provide comprehensive estimates of external costs from U.S. power plants, documenting air pollution damages that extend 50--100 kilometers from emission sources. Their parametric approach assumes exponential decay with distance, yielding tractable estimates but imposing functional form restrictions. muller2011mendelsohn develop a framework for efficient pollution regulation using damage estimates, while muller2016measuring extend this to measure environmental inequality, showing how pollution damages vary spatially across demographic groups. While their parametric methods facilitate welfare calculations, they may miss non-linearities in damage functions or incorrectly identify boundaries when decay patterns deviate from assumed functional forms.
butts2023machine develops machine learning methods for spatial treatment effect estimation, allowing for heterogeneous treatment effects across space and demonstrating that flexible algorithms can outperform parametric models when spatial relationships are complex. This finding motivates the nonparametric approach developed here, though butts2023machine focuses on treatment effect heterogeneity rather than boundary identification per se.
The standard approach to spatial boundary estimation relies on parametric functional forms---typically exponential or power-law decay functions. While tractable, these parametric restrictions may not hold in practice, potentially leading to biased boundary estimates. This motivates the flexible nonparametric framework developed here.
Building on recent theoretical advances kikuchi2024unified, kikuchi2024stochastic, kikuchi2024navier, I develop a nonparametric framework that avoids imposing functional form assumptions. kikuchi2024unified establishes a unified framework for spatial and temporal treatment effect boundary identification, providing conditions under which boundaries can be consistently estimated without parametric restrictions. kikuchi2024stochastic develops a diffusion-based approach to handle spillover effects in spatial general equilibrium settings, showing how stochastic boundaries arise naturally from economic interactions. kikuchi2024navier extends difference-in-differences methodology by drawing on insights from fluid dynamics (Navier-Stokes equations), demonstrating how treatment effects propagate through both space and time like physical diffusion processes. kikuchi2024nonparametric provides the first large-scale empirical application using 42 million pollution observations, validating the theoretical framework and demonstrating practical implementation with environmental data.
This paper applies and validates this theoretical framework in a new empirical setting---bank branch consolidation---demonstrating its advantages over conventional parametric approaches through both Monte Carlo simulations and real data analysis. Unlike muller2011machado, muller2016measuring, who assume exponential decay, I let data determine the spatial decay function nonparametrically. Unlike butts2023machine, who estimates heterogeneous treatment effects, I focus specifically on boundary detection---identifying distances where effects become negligible. The nonparametric approach proves crucial: I find non-monotonic relationships that parametric models would miss, and correctly identify flat relationships where parametric methods would impose spurious decay patterns.
The banking literature has extensively studied how physical branch presence affects credit access and local economic outcomes. Early work by petersen2002does shows that distance to lenders matters less in the modern era due to technological improvements in information transmission and credit scoring. However, subsequent research finds that proximity remains important, particularly for relationship-based lending and informationally opaque borrowers.
nguyen2019bank examine bank branch closures following mergers, finding that closures in low-income areas reduce local lending by approximately 10%. The effects are most pronounced for small business loans, which rely heavily on soft information and relationship banking. ergungor2010bank document that branch closures reduce mortgage lending in affected zip codes, with larger effects in minority neighborhoods.
Recent work has focused on the digital transformation of banking and its spatial implications. buchak2018fintech show that fintech lenders have grown rapidly in markets poorly served by traditional banks, potentially mitigating the impact of branch closures. fuster2019predictably document how mortgage processing has become increasingly automated and algorithm-driven, reducing the role of local loan officers in approval decisions.
The Community Reinvestment Act (CRA) provides an important regulatory backdrop. agarwal2012distance find that the CRA increases lending to low-income borrowers near bank branches, suggesting that physical presence matters for regulatory compliance. bhutta2015residential examine how CRA requirements affect branch location decisions, finding that banks maintain presence in low-income areas partly due to regulatory incentives.
My analysis contributes to this literature in three ways. First, I provide comprehensive evidence on spatial decay patterns for both loan volume (demand) and approval rates (supply), revealing that branches primarily affect demand through visibility and convenience rather than supply through underwriting discretion. Second, I document the spatial patterns of branch consolidation during 2010--2023, showing that closures concentrate in wealthy urban areas with redundant coverage rather than poor neighborhoods. Third, I demonstrate that this counterintuitive pattern reflects organizational complexity and strategic decision-making rather than simple economic optimization.
This paper employs local linear regression, a cornerstone of modern nonparametric statistics fan1996local. Local polynomial methods have several advantages over earlier kernel estimators like Nadaraya-Watson: they automatically correct for boundary bias, adapt to local curvature, and achieve optimal convergence rates ruppert1994multivariate.
fan1996local provide comprehensive treatment of local polynomial regression, including asymptotic theory, bandwidth selection, and confidence interval construction. li2007nonparametric extend the framework to handle dependent data, which is particularly relevant for spatial applications where observations are correlated by construction.
Cross-validation bandwidth selection has been extensively studied. hart1997kernel analyze leave-one-out cross-validation, showing that it provides asymptotically optimal bandwidth choice under mild conditions. hall1991cross derive higher-order properties and propose modifications for improved finite-sample performance.
Recent developments have focused on nonparametric estimation with spatial data. robinson2011asymptotic study kernel regression with spatially dependent observations, deriving central limit theorems and establishing consistency. hallin2004local develop local polynomial methods specifically for spatial processes, accounting for the two-dimensional nature of geographic data.
My contribution is to apply these nonparametric methods specifically to spatial treatment effect boundary identification in banking. While kikuchi2024unified, kikuchi2024stochastic, kikuchi2024navier develop the theoretical framework for boundary identification across various settings, and kikuchi2024nonparametric provides validation using environmental data, this paper demonstrates the framework's applicability to financial services and organizational decision-making. The banking context offers distinct advantages: comprehensive administrative data on both treatment (branch locations) and outcomes (mortgage applications), clear policy relevance for financial inclusion, and observable organizational complexity in closure decisions.
Relative to the environmental economics literature muller2011machado, muller2016measuring, my approach offers greater flexibility by avoiding parametric functional form assumptions. This proves essential when spatial relationships are non-monotonic or when no relationship exists---cases where parametric methods would generate biased estimates or false positives. Relative to machine learning approaches butts2023machine, I provide a principled statistical framework with interpretable bandwidth parameters and asymptotic theory, facilitating formal inference about boundary locations.
This paper makes three main contributions. Methodologically, I develop and validate a nonparametric framework for spatial boundary identification that does not impose functional form restrictions. Monte Carlo simulations demonstrate that the approach achieves lower bias than parametric methods and crucially, can detect the absence of boundaries when relationships are flat---avoiding false positives that plague parametric approaches like those in muller2011machado.
Empirically, I provide comprehensive evidence on bank branch spatial effects. Branch proximity significantly affects loan applications (8.5% decline per 10 miles) but not approval rates (essentially flat), revealing that branches influence demand through visibility while credit supply remains centralized. Branch survival analysis uncovers a non-monotonic relationship with income, with wealthy areas experiencing more closures due to redundancy and digital substitution.
Substantively, the findings challenge conventional narratives about banking deserts. Rather than abandoning poor areas, banks are consolidating redundant branches in over-banked wealthy markets. This pattern reflects organizational complexity---tensions between cost reduction and customer service, bounded rationality, and local discretion---that cannot be captured by simple parametric models.
The remainder of the paper proceeds as follows. Section 2 develops the nonparametric framework, building on the theoretical foundations in kikuchi2024unified, kikuchi2024stochastic, kikuchi2024navier and the empirical methodology in kikuchi2024nonparametric. Section 3 presents Monte Carlo simulations validating the methodology. Section 4 describes the banking application, covering data construction and main results. Section 5 analyzes branch survival patterns and underlying mechanisms. Section 6 discusses policy implications and concludes.
Consider a geographic space where treatment intensity decays with distance from a source location. Let $d$ denote the distance from the source and $Y(d)$ denote the outcome of interest. I assume that outcomes can be decomposed as:
where $m(d) = \mathbb{E}[Y(d)|d]$ is the conditional expectation function representing the spatial treatment effect, and $\varepsilon(d)$ is a mean-zero error term.
Following kikuchi2024unified, the object of interest is the spatial boundary $d^*$ defined as:
where $\varepsilon \in (0,1)$ is a threshold parameter. The boundary $d^*$ represents the distance at which the treatment effect has decayed to $(1-\varepsilon)$ of its source level. Common choices are $\varepsilon = 0.10$ (10% decay) or $\varepsilon = 0.20$ (20% decay).
The key challenge is estimating $m(d)$ and identifying $d^*$ without imposing parametric functional form assumptions. Parametric approaches typically assume:
for exponential decay, or:
for power-law decay. While tractable, these restrictions may not hold in practice.
Following the methodological framework in kikuchi2024unified, kikuchi2024nonparametric, I employ local linear regression fan1996local to estimate $m(d)$ nonparametrically. For an evaluation point $d_0$, the estimator solves:
where $K_h(u) = K(u/h)/h$ is a scaled kernel function with bandwidth $h$. I use the Gaussian kernel:
The local linear estimator is:
Local linear regression has several advantages over simpler kernel methods like Nadaraya-Watson. It automatically corrects for boundary bias, adapts to local curvature, and achieves the optimal minimax rate of convergence fan1996local.
The bandwidth $h$ controls the bias-variance tradeoff. Small $h$ reduces bias but increases variance; large $h$ increases bias but reduces variance. Following hart1997kernel, I employ leave-one-out cross-validation to select $h$ optimally.
The cross-validation criterion is:
where $\hat{m}_{-i}(d_i)$ denotes the estimator computed excluding observation $i$. The optimal bandwidth minimizes $\text{CV}(h)$:
I search over a grid of candidate bandwidths $h \in \{h_1, \ldots, h_J\}$ and select the value achieving minimum cross-validation score.
Given the nonparametric estimate $\hat{m}(d)$, I identify the spatial boundary following kikuchi2024unified as:
If no such $d$ exists (i.e., $\hat{m}(d) > (1-\varepsilon)\hat{m}(0)$ for all observed $d$), I set $\hat{d}^* = \infty$, indicating no boundary within the observed range.
This approach has two key advantages. First, it does not impose functional form restrictions on $m(d)$. Second, it can correctly identify the absence of boundaries when the relationship is flat, avoiding false positives.
Under standard regularity conditions fan1996local, kikuchi2024unified, the local linear estimator satisfies:
where $f(d)$ is the density of distance observations. The bias term $O_p(h^2)$ arises from approximating $m(d)$ locally by a linear function. The variance term $O_p(1/\sqrt{nhf(d_0)})$ decreases with sample size $n$ and bandwidth $h$.
The optimal bandwidth balances these components:
achieving the minimax optimal rate:
where $\mathcal{M}$ is a smoothness class (typically H\"older or Sobolev).
For boundary estimation, consistency requires that the true boundary $d^*$ is an interior point and that $m(d)$ crosses the threshold transversally muller1989kernel. Under these conditions:
To validate the nonparametric approach, I conduct Monte Carlo simulations across four data generating processes (DGPs) that span different spatial relationships.
The first DGP features strong spatial decay:
where $\varepsilon \sim N(0, 0.1)$ and $d \sim \text{Uniform}(0, 100)$. The true boundary for 10% decay is:
This DGP represents settings where treatment effects dissipate rapidly with distance, such as local air pollution from point sources.
The second DGP features weaker spatial decay:
The true boundary is:
This represents settings with longer-range spillovers, such as highway access or hospital availability.
The third DGP features a non-monotonic relationship with a peak at intermediate distance:
Outcomes peak at $d = 20$ miles and decay in both directions. The boundary is defined as the distance beyond which outcomes fall below 90% of the maximum. This pattern might arise from, for example, commuting patterns where moderate distances are optimal.
The fourth DGP features no spatial relationship:
There is no true boundary ($d^* = \infty$). This DGP tests whether methods incorrectly detect boundaries when none exist---a critical test for avoiding false positives.
For each DGP, I compare two estimation approaches:
I simulate $n = 5{,}000$ observations per replication and run 500 replications per DGP.
Table (ref) presents the simulation results. For DGP 1 (strong decay), both methods perform well, with the nonparametric approach showing slightly lower bias (0.3 miles versus 0.5 miles) and comparable RMSE (1.2 miles versus 1.3 miles).
For DGP 2 (weak decay), the nonparametric method continues to perform well, while the parametric approach shows increased bias when the decay rate is misspecified. The nonparametric bias is 1.1 miles compared to 2.3 miles for parametric.
The critical test is DGP 3 (non-monotonic). Here, the exponential parametric model is fundamentally misspecified. Parametric estimation yields large bias (8.7 miles) and RMSE (12.4 miles), while the nonparametric approach adapts to the hump shape with much lower bias (2.1 miles) and RMSE (4.3 miles).
Most importantly, for DGP 4 (flat), the nonparametric method correctly identifies the absence of a boundary in 94% of replications (by setting $\hat{d}^* = \infty$). In contrast, the parametric method incorrectly detects spurious boundaries in 73% of replications, with an average false boundary at 43.2 miles. This demonstrates the nonparametric method's ability to avoid false positives---a crucial advantage over parametric approaches.
Figure (ref) visualizes one realization from each DGP, showing the true function (green), observed data (black dots), parametric fit (red dashed), and nonparametric fit (blue solid). The nonparametric method closely tracks the true function across all DGPs, while the parametric approach fails notably for the non-monotonic and flat cases.
The simulations establish three key results. First, nonparametric methods achieve comparable or superior performance to parametric approaches even when the parametric model is correctly specified (DGPs 1--2). Second, nonparametric methods substantially outperform parametric approaches when functional forms are misspecified (DGP 3). Third, and most importantly, nonparametric methods avoid false boundary detection when no relationship exists (DGP 4), whereas parametric methods frequently impose spurious patterns.
These findings motivate the empirical application to bank branches, where the true spatial relationship is unknown and likely complex due to organizational decision-making, competition, and technological change.
The U.S. banking industry has undergone substantial transformation over the past two decades. Driven by technological innovation, changing consumer preferences, and cost pressures, banks have dramatically reduced physical branch networks. From 2010 to 2023, the industry experienced approximately 17,000 branch closures against 6,000 openings---a net decline of 11,000 branches.
This consolidation raises concerns about spatial inequality in financial access. Physical proximity to banks matters for credit access through several channels petersen2002does, nguyen2019bank. First, relationship banking relies on personal interactions between loan officers and borrowers, particularly for small businesses and complex mortgages. Second, branches serve as visible signals of bank presence, increasing consumer awareness and applications. Third, while online banking has grown, certain populations---particularly elderly, low-income, and rural residents---continue to rely on physical branches.
However, the rise of digital banking complicates this picture. Modern mortgage underwriting is increasingly centralized and algorithm-driven fuster2019predictably, potentially reducing the importance of local branch presence for credit approval decisions. This creates a testable distinction: if branches matter primarily for loan demand (through awareness and convenience) rather than supply (through underwriting), we should observe spatial patterns in application volume but not approval rates.
Understanding branch location decisions requires recognizing organizational complexity march1976ambiguity. Chief Financial Officers (CFOs) prioritize cost reduction, pushing for aggressive branch closures. Sales and relationship managers emphasize customer engagement and revenue preservation, advocating to retain branches. The resulting decisions reflect compromise, context-dependence, and bounded rationality rather than simple optimization.
Branch data come from the Federal Deposit Insurance Corporation (FDIC) Summary of Deposits, which provides annual snapshots of all FDIC-insured branches. For each branch, the data include:
I construct a panel covering 2010--2023, identifying branch openings and closures through year-to-year comparisons of branch identifiers. This yields 5,743 new branches opened during 2015--2020 in five major states: California (2,187 branches), Texas (1,421), Florida (982), New York (746), and Pennsylvania (407). These states account for approximately 40% of U.S. population and banking activity.
Mortgage data come from the Home Mortgage Disclosure Act (HMDA) database, which covers over 90% of U.S. mortgage applications. For 2019 (the midpoint of the treatment period), I obtain 5.9 million applications across the five study states. Each record includes:
I aggregate applications to the census tract level, calculating:
This yields 25,571 tract-year observations. I focus on 2019 to maximize data quality while maintaining temporal proximity to branch openings.
I merge tract-level income and demographic data from the American Community Survey (ACS) 2019 5-year estimates. Key variables include:
Geographic information comes from the U.S. Census Bureau's Gazetteer files, providing tract centroids (latitude, longitude) and land area.
For each census tract, I calculate the Euclidean distance to the nearest branch opened during 2015--2020:
where $\mathcal{B}$ denotes the set of new branches and the factor 69 converts decimal degrees to miles at mid-latitudes. I restrict analysis to tracts within 100 miles of a new branch, yielding 14,209 tracts.
Table (ref) presents summary statistics. The median tract is 2.5 miles from the nearest new branch, with substantial variation (standard deviation 15.3 miles). Mortgage approval rates average 52.3% with modest variation (standard deviation 9.2%), while application volume is highly skewed (mean 171, median 137).
Tracts vary considerably in socioeconomic characteristics. Median household income averages \$64,700 with a range from \$24,000 to \$180,000. Population density ranges from rural (fewer than 100 people per square mile) to highly urban (over 50,000 per square mile).
I begin by examining whether branch proximity affects loan application volume. Figure (ref) presents the core results. Panel A shows a clear spatial decay pattern: tracts farther from new branches experience fewer applications. The nonparametric estimate (blue solid line) reveals non-linearities, with steeper decline in the first 25 miles followed by gradual flattening.
Simple linear regression yields:
The coefficient implies an 8.5% decline in applications per 10 miles ($[\exp(-0.0089 \times 10) - 1] \times 100 = -8.5\%$), statistically significant at $p < 0.001$. However, the linear model explains only 0.75% of variation ($R^2 = 0.0075$), suggesting substantial heterogeneity.
The nonparametric estimate provides richer detail. Applications decline sharply within the first 10 miles (approximately 15% decline), then more gradually from 10--50 miles (additional 20% decline), and flatten beyond 50 miles. This suggests a spatial boundary around 50 miles for loan demand effects.
In contrast to application volume, approval rates show no meaningful spatial pattern. Figure (ref) presents the analysis. Linear regression yields:
The coefficient is statistically significant ($p = 0.002$) but economically negligible: a 10-mile increase corresponds to only a 0.25 percentage point change in approval rates. The $R^2$ is 0.0007, indicating distance explains essentially none of the variation.
The nonparametric estimate (Figure (ref)) remains essentially flat across the entire distance range. The Spearman correlation between distance and approval rates is 0.028 ($p = 0.0008$), confirming an extremely weak relationship.
This flat pattern has important interpretation. Modern mortgage underwriting relies on centralized algorithms using applicant credit scores, income, and property characteristics fuster2019predictably. Local branch presence does not affect these standardized criteria. Thus, branches influence who applies (the demand side) but not who gets approved (the supply side).
Figure (ref) directly compares the two outcomes, normalizing both to 100 for the 0--10 mile baseline. Loan volume declines monotonically with distance, falling to 64 at 50--100 miles. Approval rates remain essentially constant, fluctuating narrowly around 100.
This divergence confirms distinct mechanisms: branches affect demand (through visibility, convenience, and awareness) but not supply (through underwriting standards). The finding has policy implications: branch closures may reduce loan originations in affected areas by reducing applications, even if qualified borrowers continue to receive approval.
Having established that branch proximity affects loan demand, I turn to branch survival during the digital transformation era (2010--2023). This period saw massive consolidation as banks adapted to technological change and cost pressures. Understanding which branches survive provides insight into strategic decision-making and has implications for financial inclusion.
A natural hypothesis, suggested by the banking deserts literature nguyen2019bank, ergungor2010bank, is that banks disproportionately close branches in low-income areas. I test this by examining the relationship between census tract income and branch survival rates.
I focus on branches operating in 2015 and track their survival through 2023. Of 84,305 branches in 2015 (with valid tract matches), 65,008 remained open in 2023, yielding an overall survival rate of 77.1%.
I match each 2015 branch to its census tract using geographic coordinates and merge with ACS 2019 income data. The analysis sample contains 53,312 branches with complete income information.
The outcome variable is binary:
I examine survival rates across income quartiles and estimate logistic regression models controlling for branch density, urbanization, and competition.
Table (ref) presents the core finding. Contrary to the banking deserts hypothesis, high-income areas experience lower survival rates than low-income areas:
The highest-income quartile exhibits the lowest survival rate (73.9%), while the second quartile shows the highest (78.8%). A chi-squared test strongly rejects independence ($\chi^2 = 96.00$, $p < 0.001$), confirming a statistically significant relationship.
Figure (ref) visualizes this pattern, showing survival rates declining for the wealthiest quartile despite conventional expectations.
To understand this counterintuitive pattern, I test three potential mechanisms: branch density, urbanization, and competition. Figure (ref) presents comprehensive results.
High-income areas may have had more branches initially, creating redundancy that banks are now consolidating. Table (ref) reveals that tracts with more branches show somewhat higher survival (78.1% for 3--5 branches versus 73.5% for isolated branches).
Importantly, the correlation between income and branch density is weak ($\rho = 0.065$), suggesting that density per se, rather than income, drives survival patterns.
Wealthy areas tend to be more urban, and urban areas may experience more closures. Table (ref) decomposes survival by income quartile within urban versus rural/suburban areas.
Urban areas show substantially lower survival (74.5%) than rural/suburban areas (79.0%). Notably, the income pattern persists within both urban and rural areas, suggesting that urbanization alone does not fully explain the paradox.
Table (ref) presents multivariate logistic regression results with all controls. I standardize all continuous variables to enable comparison of effect magnitudes.
Key findings emerge. First, after controlling for other factors, the income effect remains negative but modest ($-7.4\%$ per standard deviation). Second, branch density exerts the largest negative effect ($-13.4\%$), consistent with consolidation of redundant branches. Third, surprisingly, more banks in the tract predicts higher survival ($+18.6\%$), suggesting that competitive areas are strategically important markets.
The model achieves modest explanatory power (AUC-ROC = 0.532), indicating substantial unexplained variation reflecting organizational complexity march1976ambiguity.
The multivariate analysis reveals that the negative income-survival relationship operates primarily through branch density and urbanization rather than direct income effects. High-income areas had more branches initially, creating redundancy that banks are now rationalizing. The direct income effect, while statistically significant, is economically modest compared to density effects.
This finding challenges the banking deserts narrative nguyen2019bank. Branch closures in wealthy areas reflect strategic consolidation of over-banked markets rather than discrimination. Poor areas may benefit from regulatory protection (Community Reinvestment Act) and necessity (being the only branch in remote locations).
This paper establishes three main results. First, methodologically, nonparametric boundary estimation outperforms parametric approaches by avoiding functional form misspecification and correctly identifying the absence of boundaries when relationships are flat. Monte Carlo simulations validate this advantage across multiple data generating processes.
Second, empirically, bank branch proximity significantly affects loan application volume (8.5% decline per 10 miles) but not approval rates (essentially flat). This divergence reveals that branches influence credit access through demand-side channels (awareness, convenience) rather than supply-side channels (underwriting standards), consistent with the centralization of mortgage processing fuster2019predictably.
Third, branch survival during 2010--2023 follows a non-monotonic relationship with area income. High-income areas experience more closures due to strategic consolidation of redundant branches in over-banked wealthy urban areas. After controlling for branch density, urbanization, and competition, the direct income effect diminishes substantially.
The findings have several policy implications. First, physical branch presence continues to matter for loan demand even in the digital age, suggesting that branch closures may reduce credit access by decreasing applications rather than approvals. Policymakers concerned about financial inclusion should monitor application volumes, not just approval rates.
Second, the branch survival analysis complicates conventional banking deserts narratives. Current consolidation patterns do not appear to disproportionately harm poor neighborhoods. Indeed, low-income areas maintain relatively high survival rates, possibly reflecting Community Reinvestment Act protections agarwal2012distance, bhutta2015residential.
Third, optimal policy should be heterogeneous. Wealthy urban areas can likely sustain more consolidation given digital alternatives and redundant coverage. Rural low-income areas require continued physical presence due to limited alternatives. The findings suggest that current regulatory approaches may already be achieving this differentiation.
The paper makes several methodological contributions to spatial econometrics. First, it develops and validates a principled nonparametric approach to boundary identification, building on the theoretical frameworks in kikuchi2024unified, kikuchi2024stochastic, kikuchi2024navier. The cross-validation bandwidth selection provides an objective way to balance bias and variance.
Second, the method can detect the absence of spatial boundaries when relationships are flat---demonstrated by the approval rate analysis. This capability avoids false positives that plague parametric approaches.
Third, the application extends the framework beyond environmental settings kikuchi2024nonparametric to organizational decision-making in financial services. This demonstrates the framework's broad applicability across contexts where spatial spillovers matter: from pollution diffusion to credit access to branch network optimization.
Fourth, the non-monotonic income-survival relationship illustrates how the framework handles complex spatial patterns that would be missed by standard parametric models, potentially leading to incorrect policy conclusions. The ability to detect such patterns without imposing functional form assumptions represents a key practical advantage.
Several limitations warrant discussion. First, the analysis focuses on a single sector (banking) during a specific period (2010--2023). While the methodological framework applies broadly, empirical generalization requires caution.
Second, I observe outcomes but not underlying mechanisms directly. While patterns are consistent with proposed channels, causal identification would benefit from exogenous variation in branch locations. Future research could exploit regulatory discontinuities or merger-induced closures for stronger causal inference.
Third, the cross-sectional design limits temporal inference. Panel methods exploiting within-tract variation would strengthen identification. The substantial unexplained variation in branch survival suggests that organizational factors not captured by observable characteristics play major roles.
Fourth, the analysis predates full manifestation of COVID-19's impact. Accelerated digital adoption during 2020--2023 may have fundamentally altered spatial relationships. Future research should examine pandemic-induced changes.
Future methodological extensions could develop formal inference procedures (confidence intervals, hypothesis tests) for nonparametric boundary estimates, incorporate multivariate treatments, and allow for time-varying boundaries to capture evolving spatial relationships during structural transformations.
This paper demonstrates the value of flexible nonparametric methods for understanding spatial economic phenomena. Economic geography is complex---shaped by technology, regulation, competition, organizational politics, and historical accidents. Imposing rigid parametric structures risks missing essential features of these landscapes.
The empirical application to bank branches reveals patterns that parametric models would miss: non-monotonic relationships, differential effects by outcome type (volume versus approvals), and organizational complexity in survival decisions. These findings challenge simplistic narratives about banking deserts and demonstrate how spatial consolidation reflects strategic optimization rather than discrimination.
As digital technology continues reshaping economic geography, flexible empirical methods will prove increasingly valuable. The nonparametric framework developed here---validated through simulations and applied to substantively important questions---provides a template for future spatial analysis that lets data speak without predetermined functional forms.
This research was supported by a grant-in-aid from Zengin Foundation for Studies on Economics and Finance.