Spatial Data Analysis
Tobias Rüttenauer∗
UCL Social Research Institute, University College London
March 18, 2025
Abstract
This handbook chapter provides an essential introduction to the field of spatial
econometrics, offering a comprehensive overview of techniques and methodologies for
analysing spatial data in the social sciences. Spatial econometrics addresses the unique
challenges posed by spatially dependent observations, where spatial relationships
among data points can be of substantive interest or can significantly impact statistical
analyses. The chapter begins by exploring the fundamental concepts of spatial depen-
dence and spatial autocorrelation, and highlighting their implications for traditional
econometric models. It then introduces a range of spatial econometric models, particu-
larly spatial lag, spatial error, spatial lag of X, and spatial Durbin models, illustrating
how these models accommodate spatial relationships and yield accurate and insightful
results about the underlying spatial processes. The chapter provides an intuitive guide
on how to interpret those different models. A practical example on London house
prices demonstrates the application of spatial econometrics, emphasising its relevance
in uncovering hidden spatial patterns, addressing endogeneity, and providing robust
estimates in the presence of spatial dependence.
Keywords: Spatial Econometrics, Spatial Data Analysis, Spatial Dependence, Spatial
Spillovers, London House Prices, Handbook
∗I would like to thank Robert Vief for valuable comments on an earlier version of this manuscript. I am
also grateful to workshop participants for their comments on the respective teaching materials.
1
arXiv:2402.09895v2  [econ.EM]  19 Mar 2025

1
Introduction
The availability of spatial data for social sciences has rapidly increased over the past
decade. Various spatial packages have been implemented in standard statistical software
and have steadily been updated (Bivand 2022) in addition to already existing software
in Geo-Information Systems (GIS) such as ArcGIS or QGIS. At the same time, many
empirical social science papers investigate research questions with an explicit spatial focus.
Examples of spatial topics in the social sciences include labour market dynamics (Martén
et al. 2019, Nisic 2017, Zoch 2021), processes of residential segregation (Roberto 2018,
Tóth et al. 2021) and gentrification (Fransham 2020, Zapatka & Beck 2021), the spatial
distribution of environmental goods or bads (Boillat et al. 2022, Jünger 2022, Rüttenauer
2018), the consequences of extreme weather events (Ogunbode et al. 2019, Hoffmann et al.
2022, Rüttenauer 2023) or the access to infrastructural conditions (Moreno-Monroy et al.
2018, Liao et al. 2020, Wiedner et al. 2022).
In general, spatial data is structured like conventional data (e.g. a dataset with variables),
but has one additional dimension: every observation is linked to some geo-spatial information.
Most common types of spatial information are points, lines, or polygons (vector data) or
raster data. Similar to the time dimension in panel data, this adds an additional layer of
information and connectivity between units. As with panel data, we could thus proceed
as if we had conventional data and ignore the spatial dimension. This comes however
with two distinct problems. First, we waste potentially interesting information, that may
help us to understand the underlying social processes. Second, we will end up with biased
inferential statistics and biased point estimates in some cases if we ignore the underlying
spatial dependence.
There are various techniques to model spatial dependence and spatial processes (LeSage &
Pace 2009). Here, we will cover the most common spatial econometric models. Generally,
spatial regression models make some assumptions about the source of spatial dependence
observed in the data and then account for this dependence in the specified model. What
makes spatial regression models more complicated than panel models is the ambiguous
direction and circular nature of the dependence. I may influence my neighbour, but my
neighbour may also influence me (interdependence). Moreover, if someone influences a third
neighbour, they may be neighbours of my neighbour – 2nd order neighbour of me – which
will then influence me as well (diffusion). However, we may also not influence each other
at all but just be affected by the same exogenous shock, thus making our observed values
more similar (common confounding).
The chapter proceeds as follows. First, we will briefly introduce the concept of spatial
2

connectivity W and spatial dependence and clarify why conventional regression techniques
may fail with spatial dependence. We will then provide an overview of the most common
spatial regression models. In a further step, we will demonstrate how to interpret summary
measures of the coefficients of these models, which becomes more complicated in the case of
spatial interdependence. Lastly, we use the relation between neighbourhood characteristics
and house prices in London as an applied example.
2
Spatial weights
Given the geographical information of spatial data (i.e. the location of each unit), we can
form relationships between units: which units are closer or further away from each other.
Similar to network analysis, we have to set up a measure that defines which units are
connected to each other and how they are connected (e.g. the magnitude of connectivity).
There are some obvious measures that can be used to define these relations with spatial
data: adjacency and proximity.
The connectivity between units is usually represented in a matrix denoted W. The spatial
weights matrix W is an N × N dimensional matrix, where each element wij of this matrix
specifies the relation or connectivity between each pair of units i and j.
W =


w11
w12
w13
. . .
w1n
w21
w22
w23
. . .
w2n
w31
w32
w33
. . .
...
...
...
...
...
...
wn1
wn2
wn3
. . .
wnn


(1)
In the example above, w31 describes the relationship between unit 3 and unit 1, while w2n de-
scribes how unit 2 and unit n are connected. The diagonal elements wi,i = w1,1, w2,2, . . . , wn,n
of W are always zero: no unit is a neighbour of itself. This is not true for spatial multiplier
matrices (as we will see later). Contiguity weights are a very common type of spatial weights.
This is a binary specification, taking the value 1 for neighbouring units (queens: sharing a
common edge; rook: sharing a common border), and 0 otherwise. See for instance Pebesma
& Bivand (2023) for more detailed information about spatial relations.
Contiguity weights matrices are usually sparse matrices and keep relations relatively simple
and easy to interpret. However, they often create island, i.e. units without any neighbours,
which can be problematic for spatial regression models. Another common type of connectivity
measures is distance based weights. For instance, inverse distance weights assign higher
3

weights to more proximate units wi,j =
1
dα
ij , where distance is usually discounted by a spatial
decay factor α. It is often recommended to specify a distance threshold (e.g. 100km) to
get rid of very small non-zero weights for very distant units. There is an ongoing debate
about the importance of spatial weights for spatial econometrics and about the right way of
specifying weights matrices (LeSage & Pace 2014, Neumayer & Plümper 2016).
2.1
Normalization
Normalizing ensures that the parameter space of the spatial multiplier in regression models
is restricted to −1 < ρ > 1, and the multiplier matrix is non-singular (more on this later).
The important message is: normalizing the weights matrix is always a good idea. Otherwise,
the spatial parameters may blow up – if they can be estimated at all. Normalising also
ensures an easy interpretation of spillover effects (as we see later). Again, how to normalize
a weights matrix is subject of debate (LeSage & Pace 2014, Neumayer & Plümper 2016).
Row-normalization divides each non-zero weight by the sum of all weights of unit i, which
is the sum of the row i:
wij
Pn
j wij . With contiguity weights and row-normalisation, spatially
lagged variables contain the mean of the respective variable among the neighbours of i.
However, proportions between units such as distances get lost due to row-normalisation,
which can be problematic if one is theoretically interested in using inverse-distance based
weights. It also induces asymmetries, as different units have different numbers of neighbours:
wij ̸= wji.
Another common way of standardization is maximum eigenvalues normalization. Maximum
eigenvalues normalization divides each non-zero weight by the overall maximum eigenvalue of
the entire matrix λmax:
W
λmax. Each element of W is divided by the same scalar value, which
preserves the relations. It keeps proportions of connectivity strengths across rows, which is
relevant for distance based W. I thus recommend maximum eigenvalues normalization for
distance based neighbours weights. However, interpretation may become more complicated.
2.2
Spatial dependence
‘Everything is related to everything else, but near things are more related than distant
things’ (Tobler 1970). Tobler’s first law of geography has been used extensively (13,690
citation in 2025-03) to describe spatial dependence. In practical term, this means that close
observations are more likely to exhibit similar values on some of their characteristics, and
we cannot handle observations as if they were independent.
There is a very easy and intuitive way of detecting spatial autocorrelation: look at the
map. Below we can see three distinct patterns. Figure Figure 1 a) has perfect negative
4

auto-correlation. Every black unit is surrounded by white units, and every white unit is
surrounded by black units. Figure Figure 1 b) has very strong positive autocorrelation.
Most white units are surrounded only by white units, and most black units are surrounded
by only black units. Figure Figure 1 c), by contrast, is generated by a random process,
although even here one is inclined to observe some degree of clustering.
Figure 1: Forms of spatial dependence: a) perfect negative autocorrelation, b) nearly perfect
positive autocorrelation, c) random.
Would our interpretation be the same if we aggregate the data to four larger areas / districts
using the average within each of the four districts? We would actually draw very different
conclusions. It is thus important to keep in mind that spatial dependence is a also a result
of spatial boundaries and potential higher-level processes generating an outcome (Wong
2009). If a variable was measured on the district level and we assign those district-level
measures to the lower neighbourhood level, we will artificially introduce spatial dependence
/ clustering in our data.
Given our spatial data, we can use various statistical measures to test whether there is
spatial dependence. The most common statistic for spatial dependence or autocorrelation is
Moran’s I, which goes back to Moran (1950) and Cliff & Ord (1972). For more extensive
materials on Moran’s I, see for instance Kelejian & Piras (2017), Chapter 11. We first define
a neighbours weights matrix W, and the Global Moran’s I test statistic is calculated as
I = N
S0
P
i
P
j wij(yi −¯y)(yj −¯y)
P
i(yi −¯y)2
, where S0 =
N
X
i=1
N
X
j=1
wij
(2)
In the case of row-standardized weights, S0 = N. Moran’s I measures the correlation
between neighbouring values: how does my yi correlate with the average yj of my neighbours?
Negative values indicate negative autocorrelation, values around zero (not zero exactly)
indicate no autocorrelation, and positive values indicate positive autocorrelation. Moran’s
I can also be calculated for the residuals from an estimated model (e.g. non-spatial OLS),
5

which allows to test for remaining autocorrelation after accounting for potential confounders.
Note that local indicators of spatial autoccorelation (LISA) and clustering – such as local
Moran’s I or Geary’s C – can be a relevant method of analysis by itself (Anselin 1995,
Pebesma & Bivand 2023).
3
Bias in non-spatial OLS
So, why should we care about spatial dependence? First, spatial dependence violates
standard assumptions of common non-spatial estimators. Second, spatial dependence itself
can provide important information about the social processes that generated the data we
observe.
Let us start with a linear model in the non-spatial setting. Here, y is the outcome or
dependent variable (N × 1), X are various exogenous covariates (N × k), and ε (N × 1) is
the error term. We are usually interested in an estimate for the k × 1 coefficient vector β.
y = Xβ + ε
The work-horse for estimating β in the social science is the OLS estimator (Wooldridge
2010), which is given by the form:
ˆβ = (X⊺X)−1X⊺y.
This OLS estimator hinges on a few assumptions, among them that the underlying sample
observations are independent and identically distributed (i.i.d). This assumption is often
violated with spatial data. Another (more important) assumptions is the absence of any
omitted (residual) variables that are related to Y and X: E(ϵi|Xi) = 0. This assumption is
violated when our neighbours’ characteristics influence our covariates and our outcome.
So, does spatial dependence allways induce bias in non-spatial estimators? No, the best
answer is: it depends (Betz et al. 2020, Cook et al. 2020, Pace & LeSage 2010, Rüttenauer
2022). The easiest way to think of it is analogous to the well-kown omitted variable bias
(Betz et al. 2020, Cook et al. 2020):
plim ˆβOLS = β + γ Cov(x, z)
Var(x) ,
where z is some omitted variable, and γ is the conditional effect of z on y. Now imagine
6

that the neighbouring values of the dependent variable Wy are autocorrelated to the focal
unit’s outcome. We denote this correlation with ρ > 0. Imagine further that the covariance
between the focal unit’s exogenous covariate x and my neighbour’s outcome Wy is not zero
(my covariate affects my neighbours’ outcome). Then we will have an omitted variable bias
due to spatial dependence:
plim ˆβOLS = β + ρCov(x, Wy)
Var(x)
̸= β,
4
Spatial Regression Models
Spatial regression models do not only overcome the potential bias, they also help us
to understand the spatial processes happening in the underlying data. Broadly, spatial
dependence in some characteristics can be the result of three different processes: a) Spatial
interdependence, b) Clustering in unobservables, and c) Spillovers from covariates. As
shown in Figure Figure 2, there are three basic ways of incorporating spatial dependence:
the Spatial Autoregressive Model (SAR) accounts for spatial interdependence, the Spatial
Error Model (SEM) for clustering on unobservables, and the Spatially lagged X Model
(SLX) for spillovers from covariates. Moreover, they can be further combined. As before,
the N × N spatial weights matrix W defines the spatial relationship between units.
Figure 2: Spatial regression models and their assumptions about spatial dependence.
4.1
Spatial Autoregressive Model (SAR)
The Spatial Autoregressive Model (SAR) model is by far the most prominent spatial
specification. It assumes spatial interdependence in the outcome Y and incorporates this
interdependence in the model:
7

y = αι + ρWy + Xβ + ε
(3)
Here, ρ denotes the strength of the spatial correlation in the dependent variable (spatial
autocorrelation): your outcome influences my outcome (> 0: positive spatial dependence,
< 0: negative spatial dependence, = 0: traditional OLS model). Given that we have
normalised the weights matrix, ρ is defined in the range of [−1, +1].
4.2
Spatial Error Model (SEM)
A second, also very common spatial model is the Spatial Error Model (SEM). It assumes
Clustering on unobservables, and thus models spatial interdependence in the error term:
y = αι + Xβ + u,
u = λWu + ε
(4)
In this case, λ denotes the strength of the spatial correlation in the errors of the model: your
error influences my errors (> 0: positive error dependence, < 0: negative error dependence,
= 0: traditional OLS model). Again, λ is defined in the range of [−1, +1].
4.3
Spatially lagged X Model (SLX)
A third spatial model is called Spatially lagged X Model (SLX). It assumes spillovers in the
covariates. It specifies a relationship between the covariate values of neighbours and the
outcome of the focal unit:
y = αι + Xβ + WXθ + ε
(5)
In the SLX, θ denotes the strength of the spatial spillover effects from covariate(s) on the
dependent variable: your covariates influence my outcome. In contrast to the previous two
specifications, θ is defined like any other coefficient from a conventional covariate. It is thus
not bound to any range, and its scale depends on the scale of the covariates in X.
The dependence structure assumed in SAR and SEM has a circular element (see Figure
Figure 2). In A SAR model, my outcome influences my neighbours’ outcome, which then
again influences my outcome. In A SEM model, my error term influences my neighbours’
error term, which then again influences my error term. This also means that SAR and SEM
8

models cannot be estimated by conventional OLS estimators, as they would suffer from
simultaneity bias in the spatial autoregressive term:
ˆρOLS = ρ +
h
(Wy)⊤(Wy)
i−1 (Wy)⊤ε
= ρ +
 n
X
i=1
y2
Li
!−1  n
X
i=1
yLiϵi
!
,
(6)
with yLi defined as the ith element of the spatial lag operator Wy = yL. It can further
be shown that the second part of the equation ̸= 0, which demonstrates that OLS would
provide a biased estimate of ρ (Franzese & Hays 2007, Sarrias 2023).
A potential way of estimating SAR-like models is an instrumental variable ap-
proach with 2SLS, where the autoregressive term is instrumented by spatial lags of
H = X, WX, W2X, ..., WlX, (Kelejian & Prucha 1998). SEM-like models can be estimated
using Generalized Method of Moments (Kelejian & Prucha 1999). However, given the
improvements in computational power, it is now common to rely on Maximum Likelihood
estimation of spatial models (Ord 1975, Anselin 1988). They start with some auxiliary
regression to obtain initial estimates, and then update them in further steps. For more
details see Bivand & Piras (2015), LeSage & Pace (2009), and Sarrias (2023). The R
package spatialreg (Bivand & Piras 2015, Bivand et al. 2021, Pebesma & Bivand 2023)
provides a series of functions to calculate the ML estimators for all spatial models considered
here.
Moreover, there are models combining two sets of the above specifications.
4.4
Spatial Durbin Model (SDM)
The spatial Spatial Durbin Model (SDM) integrates spatial interdependence in the outcome
and spatial spillovers in covariates by combining SAR and SLX:
y = αι + ρWy + Xβ + WXθ + ε
(7)
4.5
Spatial Durbin Error Model (SDEM)
The Spatial Durbin Error Model (SDEM) model integrates clustering on unobservables and
spillovers in covariates by combining SEM and SLX:
9

y = αι + Xβ + WXθ + u,
u = λWu + ε
(8)
4.6
Combined Spatial Autocorrelation Model (SAC)
The Combined Spatial Autocorrelation Model (SAC) assumes spatial interdependence in
the outcome and clustering on unobservables to be present at the same time. It combines
SAR and SEM:
y = αι + ρWy + Xβ + u,
u = λWu + ε
(9)
The SAC specification has demonstrated a rather poor performance in Monte Carlo simula-
tions (Rüttenauer 2022). Moreover, it has been argued that the SAC specification has severe
theoretical drawbacks in applied research, and that its popularity (among econometricians)
mainly stems from the fact that it constitutes an interesting estimation problem (LeSage
2014).
4.7
General Nesting Spatial Model (GNS)
Finally, the General Nesting Spatial Model (GNS) nests all three processes: spatial interde-
pendence, clustering on unobservables, and spillovers in covariates. It can be written as a
full combination of SAR, SEM, and SLX:
y = αι + ρWy + Xβ + WXθ + u,
u = λWu + ε
(10)
One could be inclined to think that the General Nesting Spatial Model is superior compared
to the more restricted models with two or one source of spatial dependence. However, in
practice the GNS is rather useless as an estimation model, as it is only weakly identifiable
at best (Gibbons & Overman 2012). This is analogous to Manski’s reflection problem on
neighbourhood effects (Manski 1993): if people in the same group behave similarly, this
can be because a) imitating behaviour of the group (WY), b) members of the same group
are exposed to the same external circumstances (Wε), and c) exogenous characteristics
10

of the group members (WX) influence the behaviour. We just cannot separate those in
observational data.
All of the models above assume different data generating processes (DGP) leading to the
observed spatial pattern. Although there are specifications tests, it is generally not possible
to let the data decide which one is the true underlying DGP (Cook et al. 2020, Rüttenauer
2022). There may however be theoretical reasons to guide the model specification (Cook
et al. 2020). SAR is the most commonly used model, but it is definitely not the best choice
in many applications. Various studies (Halleck Vega & Elhorst 2015, Rüttenauer 2022,
Wimpy et al. 2021) highlight the advantages of the relative simple SLX model. Moreover,
this specification can be incorporated in any other statistical method, such as non-linear
estimators or machine learning algorithms.
Note that missing values create a problem in spatial data analysis. For instance, in a local
spillover model with an average of 10 neighbours, two initial observations with missing
values will lead to 20 missing values in the spatially lagged variable. For global spillover
models, one initial observation with missing values is connect to many other observations
as a higher order neighbour, and thus creates an excess amount of missings. Depending on
the data, units with missings can either be dropped and omitted from the initial weights
creation, or we need to impute the data first, e.g. using interpolation or Kriging. Similarly,
islands (i.e units without neighbours) create problems in the estimation procedure. If this is
a very small number of observations, they can be dropped. Otherwise, distance or k-nearest
neighbours may be alternative options for W that circumvent this problem.
5
Spatial Impacts
As shown in Figure Figure 2, models withe a SAR-like process have a feedback loop in the
outcome: if my x influences my y, this change in my y will influence my neighbour’s y, which
will influence their neighbours’ y and also my own y again (I am second order neighbour
of my neighbour). We thus cannot interpret coefficients as marginal or partial effects in
SAR, SAC, and SDM (Anselin 2003, LeSage & Pace 2009, Kelejian & Piras 2017). This is
similar to auto-regressive time-series models where we have long-term effects due to a one
unit change in xit. We thus differentiate between the effects in SAR-like models and those
in models without an auto-regressive (endogenous) outcome term: while SAR, SAC, and
SDM assume global spatial dependence, SLX and SDEM assume local spatial dependence
(Anselin 2003, Halleck Vega & Elhorst 2015, LeSage & Pace 2009).1 Consequently, also
interpretation of the coefficients differs between models with endogenous feedback loops
1Note that SEM assumes no spatial effects, as all the spatial dependences comes from nuisance
11

and those with only local spillovers.
5.1
Global spillovers
To see the meaning of marginal effects in SAR-like models, we have to consider its reduced
form:
y = (IN −ρW)−1(Xβ + ε),
(11)
where IN is an N × N diagonal matrix (diagonal elements equal 1, 0 otherwise).
If
interpreting regression results, we are usually interested in marginal or partial effects (the
association between a unit change in X and Y ). We obtain these effects by looking at the
first derivative. When taking the first derivative of the explanatory variable xk from the
reduced form in (11) to interpret the partial effect of a unit change in variable xk on y, we
receive
∂y
∂xk
= (IN −ρW)−1
|
{z
}
N×N
βk,
for each covariate k = {1, 2, ..., K}. The partial derivative with respect to xk produces an
N × N matrix, thereby representing the partial effect of each unit i onto the focal unit
i itself and all other units j = {1, 2, ..., i −1, i + 1, ..., N}. The N × N dimensional term
(IN −ρW)−1 is also called spatial multiplier matrix. Intuitively, this multiplier matrix
equals a power series:2
(IN −ρW)−1βk = (IN + ρW + ρ2W2 + ρ3W3 + ...)βk = (IN +
∞
X
h=1
ρhWh)βk,
(12)
where the identity matrix contains the direct effects and the sum over ρhWh represents
the first and higher order indirect effects, including the feedback loops. It implies that a
change in one unit i does not only affect the direct neighbours but passes through the whole
system towards higher-order neighbours, where the impact declines with distance within
the neighbouring system. Global indirect impacts thus are ‘multiplied’ by influencing a)
2A power series of P∞
k=0 Wk converges to (I −W)−1 if the maximum absolute eigenvalue of W < 1,
which is ensured by standardizing W.
12

direct neighbours as specified in W and b) indirect neighbours not connected according to
W, with c) additional feedback loops between those neighbours.
Consider a minimal example with 5 observations, and assume the weights matrix ˜
W and
its row-normalised version W look as follows:
˜
W =











0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0











, W =











0
0.5
0
0.5
0
0.33
0
0.33
0
0.33
0
0.5
0
0.5
0
0.33
0
0.33
0
0.33
0
0.5
0
0.5
0











.
(13)
Assume that we have relatively strong spatial interdependence with ρ = 0.6. If we want to
get the total effect of X on Y , we need to combine the direct effects on the diagonal and
the indirect effects on the off-diagonal.
IN
|{z}
N×N
−ρW
|{z}
N×N
=











1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1











−











0
0.3
0
0.3
0
0.2
0
0.2
0
0.2
0
0.3
0
0.3
0
0.2
0
0.2
0
0.2
0
0.3
0
0.3
0











=











1
−0.3
0
−0.3
0
−0.2
1
−0.2
0
−0.2
0
0.3
1
0.3
0
−0.2
0
−0.2
1
−0.2
0
−0.3
0
−0.3
1











.
(14)
Finally, we take the inverse and calculate the spatial multiplier matrix
(IN −ρW)−1
|
{z
}
N×N
=











1.1875
0.46875
0.1875
0.46875
0.1875
0.3125
1.28125
0.3125
0.28125
0.3125
0.1875
0.46875
1.1875
0.46875
0.1875
0.3125
0.28125
0.3125
1.28125
0.3125
0.1875
0.46875
0.1875
0.46875
1.1875











.
(15)
13

The N × N multiplier matrix (IN −ρW)−1 has diagonal elements > 1: these include direct
effects and also feedback loops, which amplify the direct impact: my x influences my y
directly, but my y then influences my neighbour’s y, which then influences my y again (and
other neighbour’s ys). The influence of my x on my y includes a spatial multiplier effect.
To get the partial effect of a change in x1, we need to multiply the coefficient estimate ˆβ1
from the SAR model with the spatial multiplier matrix. Assume we have ˆβ1 = 0.1, then the
partial effect is given by N × N matrix
∂y
∂x1
= (IN −ρW)−1
|
{z
}
N×N
ˆβ1 =











0.11875
0.046875
0.01875
0.046875
0.01875
0.03125
0.128125
0.03125
0.028125
0.03125
0.01875
0.046875
0.11875
0.046875
0.01875
0.03125
0.028125
0.03125
0.128125
0.03125
0.01875
0.046875
0.01875
0.046875
0.11875











= Ω. (16)
The partial effects matrix Ωcontains the effect of each unit i on itself on the diagonal
(including feedback loops) and the effect on each other unit j on the off-diagonal. In theory,
m31 = 0.01875 tells us that a one-unit change of x1 in observation 1 correlates with a 0.01875
unit change in the outcome of observation 3. The ith row of the matrix Ωrepresent the
impacts on individual observation i, whereas the jth column contains the impacts from an
individual observation j (Anselin 2003, LeSage & Pace 2009, LeSage 2014). However, the
variation across these individual effects depends foremost on the weights matrix W. They
are not individual estimates, and it is advisable to not interpret these individual effects, but
rather refer to their summary measures (see below).
Substantively Interpreting these global spillover effects can be a bit tricky. The global
spillover effects can be understood as a diffusion process. For example, an exogenous event
may increase the house prices in one district of a city, thus leading to an adaptation of
house prices in neighbouring districts, which then leads to further adaptations in other units
(the neighbours of the neighbours), thereby globally diffusing the effect of the exogenous
event due to the endogenous lag of y term. Yet, those processes happen over time. In
a cross-sectional framework, Anselin (2003) proposes an interpretation as an equilibrium
outcome, where the partial impact represents an estimate of how this long-run equilibrium
would change due to a change in xk (LeSage 2014).
14

5.2
Local spillovers
In contrast, the spatial spillover effects of SLX and SDEM are local spillover effects. They can
be interpreted as the effect of a one unit change of xk in the spatially weighted neighbouring
observations on the dependent variable of the focal unit. It is the effect of the weighted
average value among neighbours. When using a row-normalised contiguity weights matrix,
Wxk is the simple mean of xk in the neighbouring units.
Assume we have k = 2 covariates, then
W
|{z}
N×N
X
|{z}
N×2
θ
|{z}
2×1
=











0
0.5
0
0.5
0
0.33
0
0.33
0
0.33
0
0.5
0
0.5
0
0.33
0
0.33
0
0.33
0
0.5
0
0.5
0






















3
120
4
140
1
200
8
70
5
250












θ1
θ2


=











6
105
3
190
6
105
3
190
6
105












θ1
θ2


(17)
Only direct neighbours – as defined in W – contribute to those local spillover effects. The
ˆθ coefficients only estimate how my direct neighbour’s X values influence my own outcome
y. There are no higher order neighbours involved as long as we do not explicitly specify
such higher order processes, nor are there any feedback loops due to interdependence.
In consequence, local and global spillover effects represent two distinct kinds of spatial
spillover effects (LeSage 2014). The interpretation of local spillover effects is straightforward:
it is the effect of a change in xj among local neighbours on the outcome of the focal unit yi.
Global spillover effects are a bit more complicated: it is the effect that a change in one unit
xj has on the entire system of neighbours, bringing y on a new equilibrium outcome.
5.3
Summary measures
Marginal or partial effects in SAR-like models are given by an N × N matrix of effects.
However, since reporting the individual partial effects is usually not of interest, LeSage &
Pace (2009) proposed to average over these effect matrices. While the average diagonal
elements of the effects matrix (IN −ρW)−1 represent the so called direct impacts of variable
15

xk, the average column-sums of the off-diagonal elements represent the so called indirect
impacts (or spatial spillover effects).
Model
Direct Impacts
Indirect Impacts
type
OLS/SEM
βk
–
–
SAR/SAC
Diagonal elements of
(I −ρW)−1βk
Off-diagonal elements of
(I −ρW)−1βk
global
SLX/SDEM
βk
θk
local
SDM
Diagonal elements of
(I −ρW)−1 [βk + Wθk]
Off-diagonal elements of
(I −ρW)−1 [βk + Wθk]
global
Note that impacts in SAR and SAC are bound to a common ratio between direct and
indirect impacts. SAR and SAC models only estimate one single spatial multiplier coefficient.
Thus direct and indirect impacts have a common ratio (say ϕ) across all covariates: if
βdirect
1
= ϕβindirect
1
, then βdirect
2
= ϕβindirect
2
, βdirect
k
= ϕβindirect
k
. For specifications including
a lagged version of X, in contrast, we estimate a local spatial effect for each unique covariate,
plus an additional spatial multiplier in case of an SDM. SLX-like specification are thus much
more flexible. Usually, impact measures come with simulation based inferential statistics
(Bivand & Piras 2015).
6
Model selection
Various spatial model specifications can be used to account for the spatial structure of the
data. Selecting the correct model specification remains a crucial task in applied research.
There are two empirical strategies for model selection: a specific-to-general or a general-
to-specific approach (Florax et al. 2003, Mur & Angulo 2009). However, both come with
severe drawbacks.
The specific-to-general approach is more common in spatial econometrics. This approach
starts with the most basic non-spatial model and tests for possible misspecification due to
omitted autocorrelation in the error term or the dependent variable. Anselin et al. (1996)
has proposed to use Lagrange multiplier (LM) tests for the hypotheses H0: λ = 0 and
H0: ρ = 0, which are robust against the alternative source of spatial dependence. The
specific-to-general approach based on the robust LM test offers a good performance in
distinguishing between SAR, SEM, and non-spatial OLS (Florax et al. 2003). Still, the test
disregards the presence of spatial dependence from local spillover effects (θ is assumed to be
zero), as resulting from an SLX-like process. Cook et al. (2020) show theoretically that an
16

SLX-like dependence structure leads to the rejection of both hypotheses H0: λ = 0 and H0:
ρ = 0, although no autocorrelation is present (Elhorst & Halleck Vega 2017, Rüttenauer
2022).
The general-to-specific approach follows the opposite direction. It starts with the most
general model and stepwise imposes restrictions on the parameters of this general model. In
theory, we would 1) start with a GNS specification and 2) subsequently restrict the model to
simplified specifications based on the significance of parameters in the GNS (Halleck Vega
& Elhorst 2015). The problem with this strategy is that the GNS is only weakly identified
and, thus, is of little help in selecting the correct restrictions (Burridge et al. 2016). The
most intuitive alternative would be to start with one of the two-source models SDM, SDEM,
or SAC. This, however, bears the risk of imposing the wrong restriction in the first place
(Cook et al. 2020). Furthermore, Cook et al. (2020) show that more complicated restrictions
are necessary to derive all single-source models from SDEM or SAC specifications.
Some argue that the best way of choosing the appropriate model specification is to exclude
one or more sources of spatial dependence – autocorrelation in the dependent variable,
autocorrelation in the disturbances, or spatial spillover effects of the covariates – by design
(Gibbons & Overman 2012, Gibbons et al. (2015)). Natural experiments would be the
best way of making one or more sources of spatial dependence unlikely, thereby restricting
the model alternatives to a subset of all available models. However, the opportunities
to use natural experiments are restricted in social sciences, making it a favourable but
often impractical way of model selection. Cook et al. (2020) and Rüttenauer (2022) argue
that theoretical considerations should guide the model selection. 1) Rule out some sources
of spatial dependence by theory, and thus restrict the specifications to a subset, and 2)
theoretical mechanisms may guide the choice of either global or local spillover effects.
A recent simulation study (Rüttenauer 2022) has shown that SLX, SDM, and SDEM are
preferable if all sources of dependence may be present. Besides that, the SLX is the most
simple specification, as it can easily be estimated by OLS. Given that WX is just another
variable, SLX can easily be combined with non-linear models or other more complicated
model specifications, such as panel estimators or machine learning algorithms. Similar
conclusions are supported by Wimpy et al. (2021), and also Jeffrey Wooldridge argued for
SLX as the only reasonable spatial specification in a Tweet from 2021 called “I will use
spatial lags of X, not spatial lags of Y” 3.
3Tweet on using SLX by J. Wooldridge on Twitter: https://twitter.com/jmwooldridge/status/13694605
26770753537
17

7
House prices in London
As an example to compare the different spatial model specifications, we estimate the effect
of local characteristics such as green space and public transport connectivity on the median
house price. The relation between environmental characteristics and housing choice and
prices has been investigated in several studies (Anselin & Lozano-Gracia 2008, Kley &
Dovbishchuk 2021, Liebe et al. 2023). The data for the current example was retrieved
from the London Datastore4, the 2011 Census5 and OpenStreetMaps and combined at the
Middle Layer Super Output Areas (MSOA). There are 983 MSOAs in London with an
average population size of around 8,000 residents. The script for compiling and preparing
the data can be found in the Supplementary Materials. All data preparation and analysis
were performed with the statistical software R. For a comprehensive overview of spatial
software see Bivand et al. (2021) or Pebesma & Bivand (2023).
Figure 3: Spatial distribution of log-transformed median house prices and transport accessi-
bility across London.
Figure Figure 3 shows an unclassified choropleth map of house prices and public transport
access across London, both log-scaled for mapping. As we would expect, both indicators
follow a relatively strong spatial of positive autocorrelation: house prices first decrease
with increasing distance to the centre, and then seem to increase again in suburban areas.
Moreover, there seems to be a pattern of higher prices towards the west and particularly high
prices around Hyde Park. Public transport accessibility steadily decreases with distance to
the city centre. Spatial regression models thus seem to be important here for two reasons:
a) observations are not independent of each other but follow clear spatial patterns, and
4For house prices, see: https://data.london.gov.uk/dataset/average-house-prices. For London
accessibility scores see: https://data.london.gov.uk/dataset/public-transport-accessibility-levels
5For UK demographics, see: https://www.nomisweb.co.uk/sources/census_2011
18

Table 2: Spatial regression models. Outcome variable: median house price.
OLS
SAR
SEM
SLX
SDM
SDEM
(Intercept)
0.000
−0.012
0.022
0.007
0.002
0.007
(0.027)
(0.017)
(0.139)
(0.024)
(0.015)
(0.103)
Green space
0.204∗∗∗
0.133∗∗∗
0.100∗∗∗
0.136∗∗∗
0.106∗∗∗
0.111∗∗∗
(0.029)
(0.018)
(0.015)
(0.026)
(0.016)
(0.020)
Public transport access
0.366∗∗∗
0.097∗∗∗
−0.054
−0.152∗∗
−0.100∗∗
−0.042
(0.033)
(0.021)
(0.033)
(0.054)
(0.034)
(0.033)
Population density
0.189∗∗∗
0.055∗
−0.094∗∗∗
−0.112∗
−0.111∗∗∗
−0.099∗∗∗
(0.037)
(0.023)
(0.027)
(0.044)
(0.028)
(0.029)
Percent non-UK
−0.033
−0.050∗
−0.250∗∗∗
−0.235∗∗∗
−0.262∗∗∗
−0.232∗∗∗
(0.033)
(0.020)
(0.033)
(0.053)
(0.033)
(0.032)
Percent social housing
−0.402∗∗∗
−0.202∗∗∗
−0.260∗∗∗
−0.306∗∗∗
−0.266∗∗∗
−0.282∗∗∗
(0.032)
(0.020)
(0.022)
(0.035)
(0.022)
(0.024)
W Green space
0.249∗∗∗
−0.029
0.050
(0.040)
(0.026)
(0.043)
W Public transport access
0.696∗∗∗
0.239∗∗∗
0.273∗∗∗
(0.069)
(0.045)
(0.073)
W Population density
0.455∗∗∗
0.136∗∗∗
−0.043
(0.065)
(0.041)
(0.075)
W Percent non-UK
0.304∗∗∗
0.300∗∗∗
0.242∗∗∗
(0.066)
(0.042)
(0.070)
W Percent social housing
−0.352∗∗∗
0.119∗∗∗
−0.135∗
(0.053)
(0.035)
(0.063)
Num. obs.
983
983
983
983
983
983
R2
0.263
0.439
Adj. R2
0.259
0.433
LR test: statistic
789.480
934.291
732.684
695.097
LR test: p-value
0.000
0.000
0.000
0.000
AIC
2502.492
1715.012
1570.201
2244.135
1513.451
1551.038
∗∗∗p < 0.001; ∗∗p < 0.01; ∗p < 0.05
b) surrounding / adjacent urban characteristics likely play a role for housing demand and
prices in the focal unit as well.
In Table 1, we regress the median house price in 2011 on the area (in kmˆ2) covered by
green space according to OpenStreetMaps, an index of public transport access (ranging from
0-low accessibility to 100-high accessibility), and several population characteristics from the
2011 census such as population density, the percent of non-UK residents and the percent of
social housing. Reported are results form (1) non-spatial OLS, (2) Spatial Autoregressive
(SAR), (3) Spatial Error Model (SEM), (4) Spatial Lag of X (SLX), (5) Spatial Durbin
Model (SDM), (6) and Spatial Durbin Error Model (SDEM). All variables were standardized
before estimation, and we thus interpret coefficients in standard deviations. Note that we
do not estimate results for Spatial Autoregressive Combined (SAC) models because of its
severe drawbacks for applied research (LeSage 2014).
Compared to results from conventional non-spatial models, Table 1 comes with several
additions: First, variables starting with a “W” (or “lag”) indicate the spatially lagged
variable or in the case of row-normalized weights matrices the average value of the respective
variable across the local neighbours. Moreover, there are two auto-regressive parameters:
“rho” for the estimated auto-correlation in the dependent variable and “lambda” for the
estimated auto-correlation in the error term. In case of the SAR, a highly significant ˆρ
19

coefficient of 0.786 indicates strong positive spatial auto-correlation in the median house
price: the house price in adjacent areas positively impacts the focal house prices. A ˆλ of
0.89 in the SEM however indicates that there is very strong spatial auto-correlation among
the (remaining) error variance. The likelihood ratio test in the goodness-of-fit statistics are
highly significant in both cases, rejecting the NULL of no spatial auto-correlation.
Given the strong positive auto-correlation in the dependent variable in SAR and SDM,
we cannot directly interpret the coefficients as marginal effects. Similar to auto-regressive
temporal models, we need to account for the spatial multiplier effect. For SEM, SLX and
SDEM, we could directly interpret the coefficients of Table 1. However, we plot the impacts
of all five models in Figure Figure 4 for reasons of comparison. Note that SEM only has
direct and no indirect impacts.
Figure 4: Direct and indirect impacts from spatial regression models. Dependent variable:
mean house prices. All vairables are standardised.
We start with the results of the SAR model in Figure Figure 4. A one standard-deviation
increase of green space in the focal unit is associated with a 0.161 standard deviation increase
in house prices within the same spatial unit. However, there are also highly significant
20

diffusion processes. This increase in green space in the focal unit will also increase house
prices in neighbouring units and the neighbours of these neighbours. This indirect impact
will add up to a 0.458 standard deviation increase in house prices across neighbouring units
connected through the spatial weights system. Similarly, an increase in public transport
accessibility is associated with a 0.118 standard-deviation higher median house price in the
unit itself and an additional 0.334 deviation increase diffusing though the neighbouring
regions. Note that direct and indirect effects are bound to a common ration, as SAR
only estimates one single spatial parameter ˆρ. In our case, every indirect impact equals
approximately 2.83 times the direct impact. This is a very restrictive conditions and a
severe drawback of the SAR model.
The SLX - similar to SAR - estimates a positive impact of green space in the focal and also in
adjacent neighbourhoods on house prices in the focal unit. A one standard deviation in the
focal unit is associated with 0.136 standard-deviations higher house price in the focal unit.
If green spaces in adjacent neighbourhoods increase on average by one standard deviation,
this would increase house prices in the focal unit by 0.249 standard deviations. Note that
the SLX tells a different story about the effect of public transport access than SAR: there
is a negative direct and a very strong and positive indirect effect. A one standard deviation
increase in public transport access in the focal unit is associated with -0.152 standard
deviations lower house prices. In contrast, more public transport in the local surrounding
(the average neighbours) is associated with 0.696 standard deviations higher prices. This is
in line with the idea that public transport facilities are usually not particularly attractive:
it is good to have them close but not too close. The same is true for population density: it
is good to live in a broader area with high population density as indicated by the indirect
impacts (likely indicating high centrality), but the local neighbourhood should have a low
population density as indicated by the negative direct impact.
We could go further with the other models. However, interpretation in SDM follows the
same logic as SAR, and interpretation in SDEM aligns to SLX. Interpretation in SEM is
analogous to non-spatial OLS, as there are no indirect impacts. Moreover, it is important to
keep in mind that the indirect impacts are summary measures which sum over all impacts
from or onto neighbouring regions. The indirect public transport effect of 0.696 in SLX
would occur if the average public transport access across neighbours would increase by one
standard deviation. This only occurs if all neighbours would simultaneously increase public
transport access by one standard deviation.
21

8
Conclusions
Interest in spatial research topics has witnessed a surge within the social sciences, and it is
paralleled by the increasing availability of geo-referenced data. This growing availability
carries immense potential for delving into the analysis of spatial phenomena, such as
spillovers and diffusions. However, it also presents challenges for statistical estimators.
Notably, utilizing non-spatial techniques with spatial data results in the loss of valuable
information. This chapter offers an extensive overview of common spatial econometrics
models that permit the explicit testing of spatial relationships. For those keen on exploring
spatial panel data, consider the works of Elhorst (2014) and Cook et al. (2023). Meanwhile,
those intrigued by non-linear spatial models should delve into LeSage & Pace (2009) and
Franzese et al. (2016).
In the regression framework of this chapter, spatial dependence can be integrated as three
distinct processes: a) Spatial interdependence in the outcome, b) Clustering of unobservable
factors, and c) Spillovers originating from covariates. In any practical application, it is
crucial to first contemplate potential theoretical underpinnings for spatial dependence. In
the case outlined above, it is plausible to anticipate dependence in the outcome, as house
prices in adjacent neighbourhoods directly influence prices in the focal units, given that
agents or home-owners rely on price information from surrounding areas. Clustering of
unobservable factors is also evident; attributes like distance to the city centre or housing
age are spatially clustered and likely exert an influence on house prices and other covariates,
potentially causing an omitted variable bias. Moreover, there are likely spillover effects from
the covariates, where factors like parks, population density, and public transport access in
surrounding neighbourhoods have a direct impact across neighbourhood borders. Thus, all
the models discussed here are theoretically plausible.
The choice of the correct model specification is often arbitrary, especially in cases like house
price modelling. It is advisable to steer clear of the Spatial Autoregressive (SAR) and
Autoregressive Conditional (SAC) models, as they come with drawbacks in applied research
as highlighted by LeSage (2014) and Rüttenauer (2022). Models with only one estimated
spatial parameter across all covariates, like SAR and SAC, impose heavy restrictions on
indirect impacts, potentially leading to biased estimates when multiple covariates are
involved. Consequently, it is generally sensible to consider more flexible specifications such
as SLX, Spatial Durbin Model (SDM), and Spatial Durbin Error Model (SDEM). In our
example above, the conclusions derived from SLX, SDM, and SDEM are fairly consistent,
with SDEM being the most conservative regarding indirect spatial impacts. This aligns with
the model’s accounting for spatial clustering among errors, which encompasses potential
22

confounders. For instance, the indirect impact of population density diminishes significantly
when controlling for the distance to the city center, explaining why the indirect positive
effect of population density vanishes in SDEM—it is largely confounded by distance to the
city center. Note that controlling for the distance to the city center would already account
for some proportion of the auto-correlation.
The Spatial Lag Model (SLX) stands out for several reasons: 1) It is straightforward in
its simplicity; 2) Estimation can be performed using least squares; 3) It can be seamlessly
integrated into panel data models, non-linear models, and machine learning techniques,
treating WX as just another set of covariates; 4) SLX can be globalised by incorporating
higher-order neighbours such as W2X and so forth, allowing for a broader assessment of
spatial impacts.
A topic deserving more attention is the necessity for spatial econometric models when
working with individual-level survey data merged with geographic context information. Do
we need to account for spatial structure when adding neighbourhood information to survey
data? A common approach involves multi-level models, which address error dependence.
However, this approach assumes that units living very close to each other but separated
by an arbitrary spatial border are independent —- a strong assumption. An alternative
approach is a spatial error model, which accommodates spatially clustered errors. For
instance, Diekmann et al. (2023) presents a compelling example in the field of environmental
inequality, where error models seem more plausible since it is unlikely that randomly
sampled survey respondents directly influence each other (as assumed in SLX and SAR),
but very likely that neighbouring respondents are exposed to similar unobservable factors.
Nevertheless, one may still wish to investigate the influence of context effects and their
spatial patterns. In such cases, SLX-like specifications for the context appear reasonable, as
demonstrated by Haußmann & Rüttenauer (2023), who employed spatial SLX specifications
to explore the impact of regional deprivation on right-wing votes at various spatial scales.
For further exploration in spatial data analysis, I recommend Pebesma & Bivand (2023)
as an open-science book on Spatial Data Science, offering a comprehensive overview of
handling and processing spatial data. LeSage & Pace (2009) and Kelejian & Piras (2017)
provide comprehensive introductions to spatial econometrics, complete with the necessary
mathematical foundations. Ward & Gleditsch (2008) offers an intuitive introduction to
spatial regression models, while Elhorst (2012), Halleck Vega & Elhorst (2015), LeSage
(2014), and Rüttenauer (2022) present article-length introductions to spatial econometrics.
23

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