Catchphrase #1: Tobler’s Law: “I invoke the first law of geography: everything is related to everything else, but near things are more related than distant things.” (Tobler 1970)1
Catchphrase #2: Tobler’s Addendum: “near can take on many meanings in different situations.” (Tobler 2004)2
\(\rightarrow\) “Space is more than geography” (Beck et al. 2006)3
“All things are connected”
A lot of (classic) theories inherently make use of space (e.g., Allport 1954)1
It’s where people interact
It’s what people collectively shape
Space becomes place
Thus, there’s a deep intersection or even embeddedness of space in social science research
It’s what geographers call “human-environment-system”
But often, these links are even only implicit in our data
Geographic information in social science
Exploiting geographic information is not new.
For example, Siegfried (1913)1 used soil composition information to explain election results in France.
The book is often seen as foundational for electoral geography because it demonstrates that political behavior is embedded in place.
Luckily, we moved on from comparing print out maps…
Three (Analytical) Perspectives on Space
The same spatial dependence can be a problem, a finding, or a design feature:
Space as Threat: detect and correct for interdependence Burnett & Lacombe (2012): five predictors of the 2004 US presidential vote change or disappear once spatial dependence is modelled.
Space as Process: model diffusion and spillover as substance Shipan & Volden (2008): cities adopt smoking bans because neighbouring cities did.
Space as Leverage: exploit spatial structure for causal identification
Neighbour list object:
Number of regions: 543
Number of nonzero links: 3120
Percentage nonzero weights: 1.058169
Average number of links: 5.745856
Link number distribution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 9 48 89 137 97 70 43 19 17 4 5 2 1 1
1 least connected region:
69 with 1 link
1 most connected region:
387 with 15 links
Neighbour list object:
Number of regions: 543
Number of nonzero links: 2922
Percentage nonzero weights: 0.9910157
Average number of links: 5.381215
Link number distribution:
1 2 3 4 5 6 7 8 9 10 11 12 13 15
3 14 51 111 145 97 55 32 20 7 4 2 1 1
3 least connected regions:
69 121 196 with 1 link
1 most connected region:
387 with 15 links
Unfortunately, we are not yet done with creating the links between neighborhoods. What we receive is, in principle, a huge matrix with connected observations.
That’s nothing we could plug into a statistical model, such as a regression or the like (see next session).
Step 2: Normalization
Normalization is the process of creating actual spatial weights. There is a debate on how to do it (Neumayer & Plümper, 2016)1. But nobody questions whether it should be done in the first place since, among others, it restricts the parameter space of the weights. Without normalization:
One of the standard procedures is row-normalization. It divides all individual weights (=connections between spatial units) \(w_{ij}\) by the row-wise sum of of all other weights:
Characteristics of weights list object:
Neighbour list object:
Number of regions: 543
Number of nonzero links: 3120
Percentage nonzero weights: 1.058169
Average number of links: 5.745856
Link number distribution:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 9 48 89 137 97 70 43 19 17 4 5 2 1 1
1 least connected region:
69 with 1 link
1 most connected region:
387 with 15 links
Weights style: W
Weights constants summary:
n nn S0 S1 S2
W 543 294849 543 201.1676 2261.458
Tidyverse alternative: sfdep package
The sfdep package provides a more tidyverse-compliant syntax to spatial weights. See:
Global measures consider the average level of spatial autocorrelation across all observations
Bias can be induced through edge effects where important parts of the spatial process fall outside the study area (compare to left- and right-censoring longitudinal data)
Moran I test under randomisation
data: election_results$rent_avg
weights: queens_W
Moran I statistic standard deviate = 33.387, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Moran I statistic Expectation Variance
0.8661742414 -0.0018450185 0.0006759199
Moran scatterplot
mp <- spdep::moran.plot(x = election_results$rent_avg,listw = queens_W,labels = election_results$district_id,xlab ="Rent average",ylab ="Spatial lag of rent average")x_mean <-mean(election_results$rent_avg, na.rm =TRUE)lag_rent <- spdep::lag.listw(queens_W, election_results$rent_avg)y_mean <-mean(lag_rent, na.rm =TRUE)x_range <-range(mp$x, na.rm =TRUE)y_range <-range(mp$wx, na.rm =TRUE)x_left <-mean(c(x_range[1], x_mean))x_right <-mean(c(x_mean, x_range[2]))text(x_right, y_range[2], "High–High", pos =1, col ="red")text(x_left, y_range[2], "Low–High", pos =1, col ="red")text(x_left, y_range[1], "Low–Low", pos =3, col ="red")text(x_right, y_range[1], "High–Low", pos =3, col ="red")
Moran I test under randomisation
data: x
weights: listw
Moran I statistic standard deviate = 33.387, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Moran I statistic Expectation Variance
0.8661742414 -0.0018450185 0.0006759199
Test of spatial autocorrelation: Geary’s C
Moran’s I is based on covariance between values and their spatial lag and captures overall similarity patterns. It can produce issues when there are only local clusters of spatial interdependence in the data. An alternative is the use of Geary's C:
Geary C test under randomisation
data: election_results$rent_avg
weights: queens_W
Geary C statistic standard deviate = 31.5, p-value < 2.2e-16
alternative hypothesis: Expectation greater than statistic
sample estimates:
Geary C statistic Expectation Variance
0.1291293820 1.0000000000 0.0007643565
Geary C test under randomisation
data: x
weights: listw
Geary C statistic standard deviate = 31.5, p-value < 2.2e-16
alternative hypothesis: Expectation greater than statistic
sample estimates:
Geary C statistic Expectation Variance
0.1291293820 1.0000000000 0.0007643565
Measures of local spatial autocorrelation: LISA clusters
Also referred to the sfdep package because it provides nice functions to calculate local measures of spatial autocorrelation. One popular choice is the estimation of Local Indicators of Spatial Autocorrelation (i.e., LISA clusters). Most straightforwardly, they can be interpreted as case-specific indicators of spatial autocorrelation:
lisa <- lisa |> dplyr::mutate(lisa_cluster_rent = dplyr::if_else( p_folded_sim <0.05,as.character(mean),"Not significant" ),lisa_cluster_rent =factor( lisa_cluster_rent,levels =c("High-High","Low-Low","High-Low","Low-High","Not significant" ) ) )ggplot(lisa) +geom_sf(aes(fill = lisa_cluster_rent), color =NA) +scale_fill_manual(values =c("High-High"="red","Low-Low"="blue","High-Low"="orange","Low-High"="lightgreen","Not significant"="grey90" ),drop =FALSE ) +theme_void() +labs(fill ="LISA cluster",title ="Local Moran clusters for average rent",subtitle ="Significance based on p_folded_sim < 0.05" )
When is space a “threat”
So far: we’ve described spatial structure, i.e. clusters in vote shares or rents and find that nearby units are more equal. What does that mean when we want to run a normal regresison, f.e. to answer the question
Do immigrant shares affect CDU voting shares within voting districts?
One core assumption of OLS: Observations are independent of each other. This is not the case any more and if we do not model the spatial interdependence the residuals are correlated and bias OLS results.
Solution: Spatial Error Model
We can include the spatial interdependence in the error term of our model to correct for deflated standard errors.
Linear Regression: \[\small Y = X\beta + \epsilon\]
Spatial Error Model (SEM): \[\small Y = X\beta + u\]\[\small u = \lambda Wu + \epsilon\]
Linear regression
linear_regression <-lm(cdu_share ~ immigrant_share + inhabitants, data = election_results)summary(linear_regression)
Call:
lm(formula = cdu_share ~ immigrant_share + inhabitants, data = election_results)
Residuals:
Min 1Q Median 3Q Max
-14.7358 -3.1172 -0.1621 2.8060 22.3594
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 28.290554 0.797624 35.469 <2e-16 ***
immigrant_share -0.051758 0.024987 -2.071 0.0388 *
inhabitants -0.085988 0.003849 -22.343 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 4.754 on 540 degrees of freedom
Multiple R-squared: 0.5011, Adjusted R-squared: 0.4992
F-statistic: 271.1 on 2 and 540 DF, p-value: < 2.2e-16
Now we need a spatial weight
Once again, we have to construct a spatial weight as in the analysis of spatial autocorrelation to estimate a spatial regression. In fact, we’ll use the same approach as before.
Call:spatialreg::errorsarlm(formula = cdu_share ~ immigrant_share +
inhabitants, data = election_results, listw = queen_W)
Residuals:
Min 1Q Median 3Q Max
-8.54327 -2.24926 -0.31318 1.87871 23.47900
Type: error
Coefficients: (asymptotic standard errors)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 25.2771710 1.1671855 21.6565 < 2.2e-16
immigrant_share -0.1301300 0.0310444 -4.1917 2.768e-05
inhabitants -0.0320731 0.0049987 -6.4163 1.397e-10
Lambda: 0.78004, LR test value: 216.59, p-value: < 2.22e-16
Asymptotic standard error: 0.03193
z-value: 24.429, p-value: < 2.22e-16
Wald statistic: 596.8, p-value: < 2.22e-16
Log likelihood: -1507.176 for error model
ML residual variance (sigma squared): 12.967, (sigma: 3.6009)
Number of observations: 543
Number of parameters estimated: 5
AIC: 3024.4, (AIC for lm: 3238.9)
Spatial Autocorrelation Is Not the Only Threat
Beyond autocorrelation in residuals: spatial data is often hierarchically structured.
Respondents nest within municipalities. Municipalities nest within regions. Regions differ systematically in history, culture, and politics. Ignoring this hierarchy can produce findings that are artefacts of unmodelled spatial heterogeneity — not substantive effects.
A vivid example: Does living near a former Nazi concentration camp make Germans more intolerant today?
Spatial Autocorrelation Is Not the Only Threat
Beyond autocorrelation in residuals: spatial data is often hierarchically structured.
Respondents nest within municipalities. Municipalities nest within regions. Regions differ systematically in history, culture, and politics. Ignoring this hierarchy can produce findings that are artefacts of unmodelled spatial heterogeneity — not substantive effects.
A vivid example: Does living near a former Nazi concentration camp make Germans more intolerant today?
The Original Claim: Homola, Pereira & Tavits (APSR 2020)
“Legacies of the Third Reich: Concentration Camps and Out-group Intolerance”
Design: distance from a survey respondent’s municipality to the nearest former Nazi concentration camp as a continuous treatment; outcome: out-group intolerance, anti-immigrant attitudes, AfD vote share.
Finding: Germans living closer to former camps display significantly higher levels of out-group intolerance and are more likely to vote for the AfD → mechanisms of historical persistence: local exposure to Nazi terror left a lasting imprint on regional political culture.
The Problem
The Problem: Camps Are Not Randomly Distributed Across Space
Pepinsky, Goodman & Ziller (APSR 2024) show that the result is an artefact of unobserved regional heterogeneity:
Concentration camps cluster heavily in what became East Germany after 1945
East German Bundesländer differ from West German states on a wide range of political and cultural dimensions (for reasons mostly unrelated to camp proximity)
Respondents are nested within Bundesländer. Failing to model that hierarchy means state-level confounders load onto the camp-proximity coefficient.
The Fix: State Fixed Effects and Mulilevel Models
PGZ replicate HPT’s results using European Values Survey + 2017 election returns, then extend with:
State fixed effects to absorb all between-Bundesland variation, isolating within-state variation in camp proximity
Multilevel models to model the nested structure explicitly, allowing intercepts (and optionally slopes) to vary across Bundesländer
Result: once state-level heterogeneity is accounted for, no consistent evidence that distance to camps predicts contemporary intolerance or AfD support.
::::
The Rebuttal: Post-Treatment Effect
HPT’s counter-argument (Homola, Pereira & Tavits 2021, 2024): post-war Bundesland borders were partly shaped by Allied occupation decisions that were themselves correlated with Nazi-era geography → state fixed effects introduce post-treatment bias.
Better: pre-treatment spatial fixed effects (e.g., historical Holy Roman Empire districts) are available and do not carry the same risk (?)
Practical implication for us: Whenever your geographic treatment clusters at a spatial level above the individual (region, state, labour market area), ask whether that cluster-level unit is also a confounder. And if so, whether controlling for it is pre- or post-treatment.
Space as a Process
Is it meaningful or just nuisances?
Space can be important in our analysis in two ways.
It’s meaningful in our theory, and we thus interpret it accordingly after estimation
It can distort our empirical estimates, producing bias, inconsistency, and inefficiency
We can address these different perspectives in our analysis with spatial econometric methods.
What are spatial econometrics?
Classic econometrics:
Using statistics to model (complex) theories, esp. causal thinking
As default, we think about regression analysis
One core assumption: Observations are independent of each other. However, we just learnt that is often not the case.
Where does spatial dependence and spatial processes enter our models and affect our outcome of interest?
Spatial Diffusion
\(y_i\) affects \(y_j\) through \(w_{ij}\)
\(y_j\) affects \(y_i\) through \(w_{ji}\)
endogenous by design!
Examples:
tax competition: if a state cuts corporate tax, neighbours respond by cutting theirs too
civil war onset: conflict in one country raises the probability of onset in neighbours
Spatial Spill-Over
\(x_i\) affects \(y_j\) through \(w_{ij}\)
\(x_j\) affects \(y_i\) through \(w_{ij}\)
Examples:
trade and export: a neighbour region’s GDP (their X) raises your export volumes or wages (your Y)
crime displacement: increased policing in one neighbourhood raises crime in your neighbourhood, as criminals relocate
Formulas…
Linear Regression: \[\small Y = X\beta + \epsilon\]
Spatial Error Model (SEM): \[\small Y = X\beta + u\]\[\small u = \lambda Wu + \epsilon\]
Spatial Lag Y / Spatial Autoregressive Model (SAR, Diffusion): \[\small Y = \rho WY + X\beta + \epsilon\]
Spatial Lag X Model (SLX, Spillover): \[\small Y = X\beta + WX\theta + \epsilon\]
Flavors and extensions
But what if….
… you have interdependence and spillovers in covariates?
There are a lot of models you could estimate to explain spatial autocorrelation. And there’s a vast body of literature on the best choice for which application. Important for us: theory-grounded reasoning for the underlying data generating process.
Getting this wrong has real consequences:
Misspecifying SAR as OLS → biased β (omitted variable: Wy)
Misspecifying SEM as SAR → spurious diffusion, inefficient β
Misspecifying SAR as SLX → underestimates spillovers (captures first-order neighbour effect)
I’d explicitly like to recommend the work of Tobias Rüttenauer for us social scientists. Here are some really nice workshop materials.
Another practical application
We will use the same example and test if one of our spatial regression models helps further investigate the data generation process. We may ask:
Do immigrant shares affect CDU voting shares between neighborhoods? (=spillover)
Do CDU voting shares affect CDU voting shares between neighborhoods? (=diffusion)
Again, controlling inhabitant numbers within the voting districts might also be a good idea.
Linear regression
linear_regression <-lm(cdu_share ~ immigrant_share + inhabitants, data = election_results)summary(linear_regression)
Call:
lm(formula = cdu_share ~ immigrant_share + inhabitants, data = election_results)
Residuals:
Min 1Q Median 3Q Max
-15.0379 -3.3415 -0.3242 3.2834 24.7445
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 28.066680 1.050168 26.726 < 2e-16 ***
immigrant_share -0.077070 0.014311 -5.385 1.08e-07 ***
inhabitants -0.083491 0.004932 -16.928 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 5.174 on 540 degrees of freedom
Multiple R-squared: 0.409, Adjusted R-squared: 0.4068
F-statistic: 186.8 on 2 and 540 DF, p-value: < 2.2e-16
Call:spatialreg::lagsarlm(formula = cdu_share ~ immigrant_share +
inhabitants, data = election_results, listw = queen_W)
Residuals:
Min 1Q Median 3Q Max
-10.48876 -2.36875 -0.21506 1.93747 23.78484
Type: lag
Coefficients: (asymptotic standard errors)
Estimate Std. Error z value Pr(>|z|)
(Intercept) 9.0224178 1.0909437 8.2703 2.22e-16
immigrant_share -0.0295856 0.0101910 -2.9031 0.003695
inhabitants -0.0323508 0.0038451 -8.4135 < 2.2e-16
Rho: 0.71173, LR test value: 311.67, p-value: < 2.22e-16
Asymptotic standard error: 0.033348
z-value: 21.342, p-value: < 2.22e-16
Wald statistic: 455.49, p-value: < 2.22e-16
Log likelihood: -1505.609 for lag model
ML residual variance (sigma squared): 13.319, (sigma: 3.6495)
Number of observations: 543
Number of parameters estimated: 5
AIC: 3021.2, (AIC for lm: 3330.9)
LM test for residual autocorrelation
test value: 29.403, p-value: 5.8781e-08
A 1pp increase in immigrant share decreases CDU vote share in this unit by 0.0341pp (direct), and decreases CDU vote share in the whole neighbourhood system by a further 0.0685pp (indirect).
So far, spatial dependence has appeared in two roles:
Threat: autocorrelation in residuals biases OLS inference → we detect (Moran’s I, Geray’s C and LISA) and correct it (SEM)
Process: clustering and spillover are the substantive phenomenon → we model it explicitly (SAR, SDM)
But the same spatial structure ( borders, distances, geographic discontinuities) can also become a design feature. If treatment assignment is plausibly driven by geography, we can exploit that variation to identify causal effects we could not otherwise recover.
Any ideas?
A short note on causal designs and the POF
The Potential Outcomes Framework defines the causal effect of treatment \(D_i\) on unit \(i\) as:
\[\tau_i = Y_i(1) - Y_i(0)\]
where \(Y_i(1)\) is the outcome if treated and \(Y_i(0)\) the outcome if untreated. The fundamental problem is that we only ever observe one of the two.
All causal designs solve this by finding a situation in which treatment assignment is as-if random conditional on observed covariates, i.e., treated and control units would have had the same outcome absent treatment.
Geography in the POF
Why geography helps: administrative rules, historical events, and physical features assign treatment based on where a unit is and this often without regard to the outcome. The border is not drawn because of what will happen to voters. The river did not choose which side would receive EU funds.
Difference-in-differences with place-based interventions
Spatial regression discontinuity designs
Geographic features as instruments
Matching on spatial covariates / synthetic controls
Difference-in-Differences Designs
DiD compares the change in outcomes for treated units to the change for control units across two periods:
\[\hat{\tau}^{DiD} = \underbrace{(\bar{Y}_{treated,post} - \bar{Y}_{treated,pre})}_{\text{change for treated}} - \underbrace{(\bar{Y}_{control,post} - \bar{Y}_{control,pre})}_{\text{change for controls}}\]
The central assumption is parallel trends: absent treatment, treated and control units would have followed the same trajectory.
Who Is Treated? Space as Assignment Mechanism
In place-based applications: \(\text{Treated}_i\) is a geographic indicator. For example, some municipalities, districts, or regions receive a policy intervention, others serve as counterfactuals.
Place-based treatments assign intervention at the level of geographic units.
The key design question is always the same: why did some places get treated and not others? If assignment correlates with pre-existing trends in the outcome, parallel trends fails.
Who Is Treated? Space as Assignment Mechanism
Treatment
Geographic unit
Control
EU structural funds
NUTS 2 regions below GDP threshold
Regions just above threshold
Wolf reintroduction
Municipalities with attacks
Neighboring municipalities without
Minimum wage increase
States/provinces adopting policy
Adjacent states/provinces
Refugee reception centers
Municipalities with facilities
Municipalities without
What Can Go Wrong? SUTVA and Spatial Spillovers
SUTVA (Stable Unit Treatment Value Assumption) requires:
No interference: Does unit \(i\)’s outcome depends only on \(i\)’s own treatment?
No hidden treatment variation: Only one version of treatment for all units?
Place-based interventions almost always put SUTVA under pressure:
A policy in municipality A affects commuters, consumers, and voters in adjacent municipality B: near treated areas are partially treated through spillovers
This attenuates DiD estimates: the control group is no longer a clean counterfactual and, in the extreme, spillovers can reverse the estimated sign
Fictional example: Reform in Cologn
SUTVA: Districts Are Contaminated?
Robustness: Donut vs Ring
Donut DiD removes the spillover zone from the control group. Ring DiD keeps it and estimates a separate coefficient per ring, tracing spillover decay with distance.
The Good or the Bad Wolf?
The original claim by Clemm von Hohenberg & Hager (PNAS 2023): Wolf attacks on livestock in German municipalities predict a rise in AfD vote share and a decline in Green votes. DiD design: municipalities experiencing first wolf attack vs. never-attacked neighbors.
The Good or the Bad Wolf?
The Good or the Bad Wolf?
The original claim by Clemm von Hohenberg & Hager (PNAS 2023): Wolf attacks on livestock in German municipalities predict a rise in AfD vote share and a decline in Green votes. DiD design: municipalities experiencing first wolf attack vs. never-attacked neighbors.
The replication by Sonntag (Electoral Studies 2024): Wolf attacks cluster overwhelmingly in East Germany which are the same region with structurally different political trajectories since reunification. Pre-treatment voting trends predict the probability of a wolf attack. Splitting by East/West or including region-specific time trends nullifies or reverses the effect.
The methodological lesson: Spatial clustering of treatment might not be random clustering: Pre-trend tests, placebo treatments, and heterogeneity checks by region are essential, not optional.
Regression Discontinuity Designs
In a standard RD, units are assigned to treatment based on whether a running variable\(R_i\) exceeds a threshold \(c\):
\[D_i = \mathbf{1}[R_i \geq c]\]
The identifying assumption is local continuity: near the threshold, potential outcomes are continuous in \(R_i\), so units just above and below \(c\) are comparable except for treatment status. Treatment is as-if randomly assigned in a neighbourhood around \(c\).
The estimand is the local average treatment effect at the cutoff
Finding a Threshold in Space
In spatial RD (Keele & Titiunik 2015): the running variable is distance to a geographic boundary, and the boundary itself creates the discontinuity in treatment. What makes a geographic boundary a valid discontinuity?
Sharp and clean: treatment changes discretely at the boundary, not gradually.
No compound treatment: the boundary of interest should not perfectly coincide with other boundaries ( media markets, county borders).
No sorting: units should not be able to select which side of the boundary they fall on based on anticipated treatment effects.
Historical stability: the longer a border has existed, the more likely it is that units on either side have diverged along unobserved dimensions which undermines comparability while also strengthening the case that the boundary is exogenous to current outcomes.
RDD: The Rhine as a Natural Boundary
RDD: Distance to the Rhine as Running Variable
“Real life example”: EU Funds and Euroscepticism
Crescenzi, Di Cataldo & Giua (Journal of Public Economics 2021) — “It’s not about the money. EU funds, local opportunities, and Euroscepticism”
Design: EU structural funds are allocated to NUTS 2 regions with GDP per capita below 75% of the EU average. In East vs. West Wales that created a sharp geographic discontinuity in funding. Regions just below the cutoff receive substantial transfers; regions just above receive little.
“Real life example”: EU Funds and Euroscepticism
“Real life example”: EU Funds and Euroscepticism
Crescenzi, Di Cataldo & Giua (Journal of Public Economics 2021) — “It’s not about the money. EU funds, local opportunities, and Euroscepticism”
Design: EU structural funds are allocated to NUTS 2 regions with GDP per capita below 75% of the EU average. In East vs. West Wales that created a sharp geographic discontinuity in funding. Regions just below the cutoff receive substantial transfers; regions just above receive little.
Finding: Funds do not reduce Euroscepticism in recipient regions. What matters is whether funds translate into local economic opportunity — employment, wages, firm creation. Where they do, attitudes toward the EU improve; where they do not (i.e., funds are absorbed without local economic multipliers), Euroscepticism persists or worsens.
On the implementation of DiD and Spatial RDD
The main “spatial task” is defining your treatment (and the robustness checks for your treatment). The calculation of the models follows standard modelling:’
