Spatial econometric interaction modelling

Advances in Spatial Science

Roberto Patuelli
Giuseppe Arbia Editors

Spatial
Econometric
Interaction
Modelling


Advances in Spatial Science
The Regional Science Series

Series editors
Manfred M. Fischer
Jean-Claude Thill
Jouke van Dijk
Hans Westlund
Advisory editors
Geoffrey J.D. Hewings
Peter Nijkamp
Folke Snickars


More information about this series at />

Roberto Patuelli • Giuseppe Arbia
Editors

Spatial Econometric

Interaction Modelling

123


Editors
Roberto Patuelli
Department of Economics
Rimini Campus
University of Bologna
Rimini, Italy

ISSN 1430-9602
Advances in Spatial Science
ISBN 978-3-319-30194-5
DOI 10.1007/978-3-319-30196-9

Giuseppe Arbia
UniversitJa Cattolica del Sacro Cuore
Rome, Italy

ISSN 2197-9375 (electronic)
ISBN 978-3-319-30196-9 (eBook)

Library of Congress Control Number: 2016947082
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Preface

Is a new book on spatial interaction modelling needed in 2016? Do we need to
update our theoretical and methodological frameworks, about 20 and 30 years away
from landmark books like Gravity Models of Spatial Interaction Behavior (Sen and
Smith 1995) and Gravity and Spatial Interaction Models (Haynes and Fotheringham
1994)? Our answer to this question is ‘yes’!
This book aims to provide a number of convincing reasons—and tools—
for extending the way scientists and practitioners in regional and international
economics, geography, planning and regional science have been implementing, estimating and interpreting spatial interaction models. It does so by collecting a number
of invited contributions by renowned scholars in the field, who propose innovative
interpretative and estimation approaches mostly relying on recent developments in
spatial statistics and econometrics.
The book originates from an International Exploratory Workshop on Advances
in the Statistical Modelling of Spatial Interaction Data held at the University of
Lugano (Switzerland) in September 2011. The papers presented at the workshop
have been published in a special issue of the Journal of Geographical Systems (15:3,

2013). This book collects such articles, as well as additional invited contributions,
in order to provide a broader view on spatial econometric approaches to spatial
interaction modelling.
Thanks are due to many people who made this book happen. We would first like
to express our gratitude to Rico Maggi for supporting our initial idea, to the Swiss
National Science Foundation (SNSF) for funding the International Exploratory
Workshop and to the University of Lugano for kindly hosting it. We would also
like to thank the Editors of the Journal of Geographical Systems for helping us
organize the preceding special issue, as well as Manfred Fischer and the Editorial
Board of the Advances in Spatial Science series and Springer for supporting this
book project. Finally, we are grateful to all contributing authors and to the referees
of both the special issue and the book.

v


vi

Preface

Last but not least, we would like to thank you, the readers. The success of this
project is in your hands. We sincerely hope you will enjoy this collection.
Rimini, Italy
Rome, Italy
January 2016

Roberto Patuelli
Giuseppe Arbia

Contents

1

Spatial Econometric Interaction Modelling: Where
Spatial Econometrics and Spatial Interaction Modelling Meet . . . . . . .
Roberto Patuelli and Giuseppe Arbia

Part I
2

3

4

5

1

General Methodological Issues

Spatial Regression-Based Model Specifications
for Exogenous and Endogenous Spatial Interaction .. . . . . . . . . . . . . . . . . . .
James P. LeSage and Manfred M. Fischer

15

Constrained Variants of the Gravity Model and Spatial
Dependence: Model Specification and Estimation Issues . . . . . . . . . . . . . .
Daniel A. Griffith and Manfred M. Fischer

37

Testing Spatial Autocorrelation in Weighted Networks:
The Modes Permutation Test . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
François Bavaud

67

Effects of Scale in Spatial Interaction Models . . . . . . .. . . . . . . . . . . . . . . . . . . .
Giuseppe Arbia and Francesca Petrarca

Part II

85

Specific Methodological Issues

6

Dealing with Intraregional Flows in Spatial Econometric
Gravity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105
Kazuki Tamesue and Morito Tsutsumi

7

A Bayesian Spatial Interaction Model Variant
of the Poisson Pseudo-Maximum Likelihood Estimator . . . . . . . . . . . . . . . 121
James P. LeSage and Esra Satici

vii


viii

Contents

8

The Space of Gravity: Spatially Filtered Estimation
of a Gravity Model for Bilateral Trade .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 145
Roberto Patuelli, Gert-Jan M. Linders,
Rodolfo Metulini, and Daniel A. Griffith

9

A Spatial Interaction Model with Spatially Structured
Origin and Destination Effects .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171
James P. LeSage and Carlos Llano

10 Bayesian Variable Selection in a Large Vector
Autoregression for Origin-Destination Traffic Flow Modelling . . . . . . . 199
Minfeng Deng
11 Double Spatial Dependence in Gravity Models: Migration
from the European Neighborhood to the European Union . . . . . . . . . . . . 225
Michael Beenstock and Daniel Felsenstein
12 Multilateral Resistance and the Euro Effects on Trade Flows.. . . . . . . . 253
Camilla Mastromarco, Laura Serlenga, and Yongcheol Shin
Part III

Applications

13 The Effects of World Heritage Sites on Domestic Tourism:
A Spatial Interaction Model for Italy . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 281
Roberto Patuelli, Maurizio Mussoni, and Guido Candela
14 Testing Transport Mode Cooperation and Competition
Within a Country: A Spatial Econometrics Approach . . . . . . . . . . . . . . . . . 317
Jorge Díaz-Lanchas, Nuria Gallego, Carlos Llano, and Tamara
de la Mata
15 Modeling the Effect of Social-Network on Interregional
Trade of Services: How Sensitive Are the Results
to Alternative Measures of Social Linkages . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 365
Carlos Llano and Tamara de la Mata
16 On the Mutual Dynamics of Interregional Gross
Migration Flows in Space and Time . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 415
Timo Mitze
17 Residential Relocation in a Metropolitan Area: A Case
Study of the Seoul Metropolitan Area, South Korea . . . . . . . . . . . . . . . . . . . 441
Monghyeon Lee and Yongwan Chun
18 Conclusions: The Future of Spatial Interaction Modelling . . . . . . . . . . . . 465
Giuseppe Arbia and Roberto Patuelli


Contributors

Giuseppe Arbia Department of Statistical Sciences, Università Cattolica del Sacro
Cuore, Roma, Italy
François Bavaud Department of Language and Information Sciences, Institute of
Geography and Sustainability, University of Lausanne, Lausanne, Switzerland
Michael Beenstock Faculty of Social Sciences, The Hebrew University of

Jerusalem, Jerusalem, Israel
Guido Candela Department of Economics, University of Bologna, Bologna, Italy
Yongwan Chun School of Economic, Political and Policy Sciences, University of
Texas at Dallas, Richardson, TX, USA
Tamara de la Mata Departamento de Análisis Económico: Teoría e Historia
Económic, Universidad Autónoma de Madrid, Madrid, Spain
CEPREDE, Madrid, Spain
Minfeng Deng Department of Psychological Sciences and Statistics, Faculty of
Health, Arts and Design, Swinburne University of Technology, Melbourne, VIC,
Australia
Jorge Díaz-Lanchas Departamento de Análisis Económico: Teoría Económica e
Historia Económica, Facultad de Ciencias Económicas y Empresariales, Universidad Autónoma de Madrid, Madrid, Spain
CEPREDE, Madrid, Spain
Daniel Felsenstein Department of Geography, Faculty of Social Sciences, The
Hebrew University of Jerusalem, Jerusalem, Israel
Manfred M. Fischer Institute for Economic Geography and GIScience, Vienna
University of Economics and Business, Vienna, Austria

ix


x

Contributors

Nuria Gallego Departamento de Análisis Económico: Teoría Económica e Historia Económica, Facultad de Ciencias Económicas y Empresariales, Universidad
Autónoma de Madrid, Madrid, Spain
CEPREDE, Madrid, Spain
Daniel A. Griffith School of Economic, Political and Policy Sciences, University
of Texas at Dallas, Richardson, TX, USA

Monghyeon Lee School of Economic, Political and Policy Sciences, University of
Texas at Dallas, Richardson, TX, USA
James P. LeSage Department of Finance and Economics, McCoy College of
Business Administration, Texas State University – San Marcos, San Marcos, TX,
USA
Gert-Jan M. Linders Ministry of Social Affairs and Employment, The Hague,
The Netherlands
Carlos Llano Departamento de Análisis Económico: Teoría Económica e Historia Económica, Facultad de Ciencias Económicas y Empresariales, Universidad
Autónoma de Madrid, Madrid, Spain
CEPREDE, Madrid, Spain
Camilla Mastromarco Department of Economics and Quantitative Methods,
Università del Salento, Lecce, Italy
Rodolfo Metulini IMT Institute for Advanced Studies Lucca, Lucca, Italy
Timo Mitze Department of Border Region Studies, University of Southern
Denmark, Odense, Denmark
Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Essen, Germany
The Rimini Centre for Economic Analysis, Rimini, Italy
Maurizio Mussoni Department of Economics, University of Bologna, Bologna,
Italy
The Rimini Centre for Economic Analysis (RCEA), Rimini, Italy
Roberto Patuelli Department of Economics, University of Bologna, Bologna, Italy
The Rimini Centre for Economic Analysis (RCEA), Rimini, Italy
Francesca Petrarca Department of Methods and Models for Economics, Territory
and Finance, Sapienza University of Rome, Rome, Italy
Esra Satici Productivity Analysis, Transportation Costs and Productivity, Turkish
Highways, Ankara, Turkey
Laura Serlenga Department of Economics and Mathematical Methods, University
of Bari Aldo Moro, Bari, Italy

Contributors

xi

Yongcheol Shin Department of Economics and Related Studies, University of
York, York, UK
Kazuki Tamesue College of Policy and Planning Sciences, University of Tsukuba,
Tsukuba, Japan
Morito Tsutsumi Faculty of Engineering, Information and Systems, University of
Tsukuba, Tsukuba, Japan


Chapter 1

Spatial Econometric Interaction Modelling:
Where Spatial Econometrics and Spatial
Interaction Modelling Meet
Roberto Patuelli and Giuseppe Arbia

Keywords Gravity • Spatial econometrics • Spatial interaction

JEL Classifications: C18, C51, R11

1.1 The Spatial Interaction Model: An Established Regional
Economics Workhorse
The present book is concerned with spatial interaction modelling. In particular, it
aims to illustrate, through a collection of methodological and empirical studies, how
estimation approaches in this field recently developed, by including the tools typical
of spatial statistics and spatial econometrics (Anselin 1988; Cressie 1993; Arbia
2006, 2014), into what LeSage and Pace (2009) deemed as ‘spatial econometric

interaction models’.
It is no surprise to scientists and practitioners in regional science, planning,
demography or economics that spatial interaction models (or gravity models, following the traditional Newtonian denomination, still popular in fields like international
trade) are, after a long time, some of the most widely used analytical tools in
studying interactions between social and economic agents observed in space. Spatial
interaction indeed underlies most processes involving individual choices in regional
economics, and can apply to all economic agents (firms, workers or households,
public entities, etc.).

R. Patuelli ( )
Department of Economics, University of Bologna, Bologna, Italy
The Rimini Centre for Economic Analysis (RCEA), Rimini, Italy
e-mail:
G. Arbia
Department of Statistics, Università Cattolica del Sacro Cuore, Rome, Italy
e-mail:
© Springer International Publishing Switzerland 2016
R. Patuelli, G. Arbia (eds.), Spatial Econometric Interaction Modelling, Advances
in Spatial Science, DOI 10.1007/978-3-319-30196-9_1

1


2

R. Patuelli and G. Arbia

Although spatial interaction models originated at the end of the nineteenth
century following the Newtonian paradigm relating two masses and the distance
between them (for a more detailed review, see Sen and Smith 1995), they now have

solid theoretical economic foundations grounded on probabilistic theory, discrete
choice modelling and entropy maximization. The works of, among others, Stewart
(1941), Isard (1960) and Wilson (1970) during the twentieth century provided such
foundations, allowed to see spatial interaction models not just as mechanical tools
for empirical analysis, but also as a framework for theoretical and structural analyses
(see, e.g., Baltagi and Egger 2016; Egger and Tarlea 2015).
A spatial interaction model describes the movement of people, items or information (the list of possible applications is long) between generic spatial units. We can
loosely write it as a multiplicative model of the type:
ˇ

Tij D kO˛i Dj f dij ;

(1.1)

where Tij is the flow (physical or not) moving from unit i to unit j, k is a
proportionality constant, Oi and Dj are sets of potentially different variables (e.g.,
population, income, jobs) measured at the origin and the destination, respectively,
and dij is the distance (possibly measured according to different metrics) between
units i and j. The latter is solely an example of different types of deterrence variables
accounting for factors which impede or favour pairwise interaction. Different
functional forms—most frequently power or exponential—have been tested over
the years to model the effect of distance on spatial interaction. The parameters ˛, ˇ
and those involved by the deterrence functions need to be properly estimated.
Such a simple specification is described as an unconstrained model, because it
does not fix the total number of outgoing or incoming flows (the marginal sums of
the origin–destination matrix). Singly- or doubly-constrained model specifications
impose such limitations by including sets of balancing factors, which are nonlinear
constraints requiring iterative calibration (Wilson 1970). Constrained approaches,
which are often seen as the correct way of estimating the model, are, however,
only seldom used in applied work, mostly because of the computational complexity

involved.
Although spatial interaction models have been used for decades by researchers
and practitioners in many fields, several authors have shown a renewed interest
in them over the last 10–15 years, both with regard to their theoretical foundations and to the estimation approaches, the latter being greatly facilitated by
the wider computing power availability. The contributions by Anderson and van
Wincoop (2003, 2004) pushed the envelope in trade-related research by proposing
a theory-consistent interpretation of the balancing factors, relabelled as multilateral
resistance terms. Santos Silva and Tenreyro (2006) provided a stepping stone in the
discussion on the estimation of spatial interaction (and in general multiplicative)
models. They suggested that, because of Jensen’s inequality and of overdispersion,
these models should not be estimated in their loglinear transformation, but rather
using the count data (such as the Poisson) regressions framework. The pseudo-


1 Spatial Econometric Interaction Modelling: Where Spatial Econometrics. . .

3

maximum likelihood estimator proposed by the authors is now one of the most
commonly used estimation approaches. Some recent studies focus on issues in
complementing the above groundbreaking studies. Burger et al. (2009) reviewed
alternative estimation approaches focusing on the cases of excess zeros. Krisztin
and Fischer (2015) suggested spatial filtering variants of the Poisson gravity model
(e.g., with zero-inflation) along with pseudo-ML estimation. Baltagi and Egger
(2016) proposed a quantile regression approach; Egger (2005), Baier and Bergstrand
(2009), and Egger and Staub (2016) proposed estimators for the cross-sectional
model, while Egger and Pfaffermayr (2003) discussed panel estimation issues. Many
more studies of recent publication could be cited in what appears to be—once
more—a very active and developing field.

1.2 Spatial Interaction and Flow Dependence
One of the dimensions along which the literature on spatial interaction modelling
recently developed is the explicit consideration of flow data correlation due to the
spatial configuration of the units involved. Explicitly incorporating spatial structure
(and its consequences) in spatial interaction is not a novel research question. It had
actually emerged among regional scientists much earlier. One of the first examples
in the literature is the lively debate appeared in the 1970s, generated by a paper
by Leslie Curry (1972), and then involving a series of quick-reply papers, by Cliff
et al. (1974, 1976), Curry et al. (1975), Sheppard et al. (1976), and Griffith and Jones
(1980). This discussion layed the ground for what was actively picked up only 30
years later.
In particular, this initial debate was centred on the misspecification of spatial
interaction models when it comes to the interpretation of the estimated distance
deterrence parameter. As Curry et al. (1975, p. 294) pointed out, problems of
interpretation of the distance parameter occur when in the model specified as a ratio
the numerator (the origin- and destination-specific variables) and the denominator
(the distance, when a power function is considered) are not independent. The authors
went as far as to state that ‘it will not be possible to determine how much of
the often observed variations in the distance exponent is due to aggregate spatial
pattern as opposed to that explainable by differences in spatial interaction’. The
authors refer in particular to the case in which origin or destination characteristics
(examples being population or income) can be seen as functions of distance (i.e.,
they are spatially autocorrelated). In other words, the spatial structure of the units
matters, so much that Sheppard et al. (1976, p. 337) state that ‘in the spatial
context misspecification will almost always occur’. This type of issue will later
be formalized more generally, giving birth to spatial econometrics. In a more
recent contribution, Tiefelsdorf (2003) picked up the topic of spatial autocorrelation
and spatial interaction, providing evidence in favour of the above statements.
Similarly, Fotheringham and Webber (1980) used a simultaneous equations model
to show how spatial structure and spatial interaction are ‘inextricably and mutually

4

R. Patuelli and G. Arbia

linked’ (Roy and Thill 2003, p. 353). Furthermore, Getis (1991) demonstrated
mathematically how spatial autocorrelation measures are confounded in spatial
interaction models, while Andersson and Gråsjö (2009) suggested how accessibility
(closely tied to spatial interaction) is a prime determinant of spatial autocorrelation
in areal data as well.
These contributions prompted new interest in the issue of spatial autocorrelation
and spatial autocorrelation (see, e.g., Griffith 2007, 2011). Although still not a
wide field of research, the general question of how to measure and treat spatial
dependence in flow (dyadic) data (the so-called network autocorrelation) is now
discussed in several papers. Black (1992) was an early contributor in this field,
relying on graph theory to define network autocorrelation providing empirical
evidence of it in real cases. Moran’s I statistic was used for this purpose. More
studies followed, for instance in Black and Thomas (1998), which focuses on
highway accidents. More recently, Peeters and Thomas (2009) provided a review on
network autocorrelation. In the context of political science, Neumayer and Plümper
(2010) discuss the problem of quantifying the correlation between flow (dyadic)
data and how public policies set contagion forces to work.
When it comes to indicators of network autocorrelation (or, so to say, spatial autocorrelation on networks), an extension of the Getis-Ord Gi statistic for measuring
local spatial autocorrelation was proposed by Berglund and Karlström (1999), and
Fischer et al. (2010). Further attempts of measuring spatial/network autocorrelation
in flow data could be based on some recently developed spatial autocorrelation
indices implemented for generalized linear models and primarily for count data (see,
e.g., Jacqmin-Gadda et al. 1997; Lin and Zhang 2007; Griffith 2010).

1.3 Towards a New Class of Spatial Interaction Models
The renewed interest in spatial autocorrelation and spatial interaction modelling
documented above is manifest in the recent econometric contributions that collectively form a new class of augmented spatial interaction models that are now
often referred to as spatial econometric interaction models. Such models are based
on the spatial statistics and econometrics techniques typically employed with areal
(and sometimes point) data, and aim to ‘cure’ spatial/network autocorrelation and
reinterpret the model parameters accordingly. A few initial attempts (see Bolduc
et al. 1992; Porojan 2001; Neumayer and Plümper 2010) were made to include
spatial relations in flow modelling. Bolduc et al. (1992), dealing with travel flow
data, proposed spatially structured error terms, which emerged as the sum of an
origin-level and a destination-level components, in addition to a non-spatial error.
Porojan (2001) used both spatial lag and spatial error models (and generalized
spatial models, with both terms), although without fully outlining how the rowstandardized spatial weights matrix used could be adapted for the task. Neumayer
and Plümper (2010) presented a similar approach, discussing contagion phenomena
between countries and their influence on the signing of bilateral investment treaties.


1 Spatial Econometric Interaction Modelling: Where Spatial Econometrics. . .

5

They estimated spatial lag models where the weights matrix was alternatively
constrained to model only origin- or destination-level dependence (contagion), or
both.
Further and critical contributions to the above debate, providing a more in-depth
view on the estimation of spatial econometric interaction models, were published
between the years 2008 and 2012 (most notably by Fischer and Griffith 2008;
LeSage and Pace 2008; Behrens et al. 2012) and gained immediate recognition in the
scientific community. Methodologically, we can distinguish between three strains of
the literature, proposing:

• spatial interaction model estimated in their log-linearized form;
• spatial interaction models estimated using Poisson-type regressions;
• spatial interaction models augmented with spatial filters (estimated as Poisson as
well).
The most common spatial econometric-aware applications of the spatial interaction model belong to the first of the above categories. Estimating multiplicative
models in their loglinear form has long been a widely employed approach (e.g.,
in economics), although the publication of Santos Silva and Tenreyro (2006),
advocating Poisson-type regressions and the pseudo-Poisson maximum likelihood
estimator (PPML), has greatly reduced the diffusion of this approach, in particular
in the trade literature.
Fischer and Griffith (2008) proposed two competing models. The first was
based on a spatial error autoregressive model specification estimated within a
maximum likelihood (ML) framework. Along the same line, LeSage and Pace
(2008) presented a further ML-based spatial econometric model, encompassing
several alternative specifications, and providing additional economic motivations
for its use. Their model included simultaneously potential origin, destination and
network dependence elements, resulting in an equation of the type:
yD

o Wo y

C

d Wd y

C

w Ww y

C Zı C ";

(1.2)

where y is an n2 1 vector containing all Tij flows, " are IID errors, and Wo ,
Wd and Ww are (row-normalized) spatial weights matrices obtained by means of
Kronecker products of the regular n n spatial weights matrix W with an identity
matrix In or itself (in the case of Ww ) (LeSage and Fischer 2010). An alternative
to this specification, encompassing the above-mentioned one in Fischer and Griffith
(2008), is a model employing the same three spatial dependence terms of Eq. (1.2)
in a spatial error framework of the type:
y D Zı C uI
u D o Wo u C

d Wd u

C

w Ww u

C ":

(1.3)

LeSage and Fischer (2010) also suggest a simpler approach, consisting in
employing one spatial weights matrix only in u, as the row-normalized sum of Wo
and Wd . Finally, Behrens et al. (2012) presented a formal theoretical justification for


6

R. Patuelli and G. Arbia

the use of spatial econometrics in trade modelling, within a framework controlling
for multilateral resistance terms and heterogeneity. They eventually estimated an
empirical spatial error model similarly to the one specified by Fischer and Griffith
(2008). Recent papers provide further extensions of these approaches (Egger and
Pfaffermayr 2016; Koch and LeSage 2015), and more heterodox approaches to
estimation, for example based on entropy maximization like in Bernardini Papalia
(2010). Finally, a development that is worth mentioning is the recent application of
the impact measures popularized by LeSage and Pace (2009) to spatial interaction
models, provided by Thomas-Agnan and LeSage (2014), and LeSage and ThomasAgnan (2015). The authors demonstrate that, similarly to the standard case of areal
data, the model’s estimated parameters cannot be directly interpreted as marginal
effects or elasticities, and that the bidimensionality of spatial interaction models,
where both origin- and destination-level effects are often estimated for a variable,
introduces further complications.
Although the path of research described above is the one that is most often followed in the current literature (perhaps due to the simplicity of linear approaches),
two more heterodox strategies deserve consideration: the first based on Poissontype regression and the second on spatial filtering. Poisson-type regression models
incorporating spatial dependence or heterogeneity followed the wake traced by
Flowerdew and Aitkin (1982). However, while in statistics and in some applied
fields (like ecology or epidemiology) researchers have proposed various spatial
extensions of count models (see, e.g., Congdon 2010), mostly based on point data
and spatially correlated random effects, the explicit (parametric) inclusion of spatial
dependence into the models still represented a major obstacle until recent years. In
fact, it wasn’t until very recently that some contributions have emerged, tackling
this problem. For example, outside the spatial interaction modelling paradigm,
Lambert et al. (2010) proposed a two-step spatial lag model for count data providing
adaptations of the direct and indirect effects (LeSage and Pace 2009), while LeSage
et al. (2007) presented a first Poisson-based model grounded on Bayesian estimation
and Markov chain Monte Carlo (MCMC) methods. In this last contribution, originand destination-level random effects with an autoregressive structure were estimated
in a hierarchical Poisson specification. A similar approach was recently followed

by Sellner et al. (2013), who developed a Poisson SAR estimator, based on twostage nonlinear least squares. In their model, the flow variable y is assumed to be
determined by a spatially random component y* , and by a residual spatial component y˜ , where only the latter is assumed to be Poisson distributed. Consistently
with most of the literature cited above, the spatial component y* is expressed as a
linear combination of the origin- and destination-level spatial lags expressed by the
products Wo y and Wd y, respectively. Therefore, the dependent variable is defined as:
y D yQ C y D

o Wo y

C

d Wd y

Cy :

(1.4)

A second alternative approach to modelling spatial and network autocorrelation
in flow data employs eigenvector spatial filtering within Poisson-type regression
models. This technique was first introduced by Griffith (2000, 2003, 2006) for


1 Spatial Econometric Interaction Modelling: Where Spatial Econometrics. . .

7

analysing areal and grid/raster data and it is based on the mathematical relationship
between Moran’s I (Moran 1948) and spatial weights matrices. In synthesis,
having defined M as the projection matrix In – 110 /n and 1 as an n 1 vector of
ones, the eigenvectors of matrix MWM represent all possible independent and

orthogonal spatial patterns implied by W. Using (parsimoniously) such eigenvectors
as additional covariates in regression models allows to filter spatial autocorrelation
similarly to SAR models (Griffith 2000). Fischer and Griffith (2008) presented an
application of spatial filtering to the spatial interaction model, where separate originand destination-level spatial filters (again, similarly to what has been mentioned
above for fixed effects or spatially correlated random effects) were obtained by
means of n2 1 eigenvectors piled by means of Kronecker products. Chun (2008)
extended this approach to network autocorrelation, by using the eigenvectors of
MWw M, while Griffith and Chun (2015) recently showed that both origin/destination
and network dependence are relevant and stressed the need to incorporate both
of them in spatial interaction models. Finally, Krisztin and Fischer (2015) applied
this approach within the framework of a PPML estimation (as in Santos Silva and
Tenreyro 2006), directly addressing the trade modelling literature.
As it is evident from the work reported in this section, the last 10 years have
shown a resurgence of the gravity/spatial interaction modelling literature, and the
emergence of innovative estimation approaches making use of spatial statistics and
econometrics. Due to the extreme relevance of these new contributions, this book
aims at providing the state-of-the-art of such developments, and collects methodological and empirical contributions authored by some of the main contributors to
the field.

1.4 The Structure of the Book
The book is divided into three parts. Part I (General Methodological Issues) contains
general contributions on spatial econometric interaction modelling, pertaining
to coefficient interpretation, constrained specifications, scale effects and spatial
weights matrix specification. Part II (Specific Methodological Issues) concerns
in particular the phase of estimation and focuses on innovative estimators and
approaches, such as the treatment of intraregional flows, Bayesian PPML or VAR
estimation, and Pesaran-type cross-sectional dependence. Finally, Part III (Applications) contains a number of empirical studies ranging from interregional tourism
competition, domestic trade, to space-time migration modelling and residential
relocation. In what follows, we will describe in detail the content of the various
papers to orientate the reader.

Part I is composed of four chapters. In Chap. 2, James LeSage and Manfred
Fischer provide an extension of the recent article by LeSage and Thomas-Agnan
(2015) discussing impact measures for the case of spatial interaction modelling.
The authors focus on the distinction between endogenous and exogenous spatial
interactions, drawing an analytical framework consistent with the one differentiating


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R. Patuelli and G. Arbia

conventional spatial models and models including only spatial lags of the independent variables (ultimately similar to spatial error models, as suggested in LeSage
and Pace 2009). As a consequence, this chapter provides the reader with further
guidance in correctly interpreting the parameters of spatial interaction models.
In Chap. 3, Daniel Griffith and Manfred Fischer compare the (main) different
approaches to estimating spatial interaction models, that is specifications including
balancing factors, fixed effects (by means of sets of indicator variables), or random
effects. An empirical example of the equivalence between them is provided, also
employing spatially structured random effects obtained through eigenvector spatial
filtering. Chapter 4 is written by François Bavaud, who develops a new approach
to spatial autocorrelation testing for weighted networks by means of Moran’s I.
He provides an example based on Swiss migratory flows, showcasing a modes
autocorrelation test based on the transformation of spatial weights matrices into
exchange matrices. In Chap. 5, Giuseppe Arbia and Francesca Petrarca analyse
the implications of the modifiable areal unit problem (MAUP, see Arbia 1989)
for spatial interaction models. They focus on the ‘scale’ dimension of the MAUP,
which pertains to the different (hierarchical) levels of geographical aggregation of
flows. They illustrate their theoretical analysis with a set of simulations, and show in
particular how negative spatial autocorrelation in the origin and destination variables
affects mean interaction flows.

Part II of the book deals in particular with estimation issues and consists of seven
chapters. In Chap. 6, Kazuki Tamesue and Morito Tsutsumi focus on how to define
internal distances and to estimate models in the case of missing intraregional flows.
They propose approaches based on an expectation-maximization (EM) algorithm
and Heckman’s two-step estimator, evaluating the sample-selection bias caused
by missing intraregional flows data. Chapter 7, by James LeSage and Esra Satici
describes a spatial econometric extension of the abovementioned PPML estimator
by Santos Silva and Tenreyro (2006), though based on a Bayesian approach.
The authors deal with intraregional flows as well, by including a separate set of
parameters to be estimated for the diagonal of the origin–destination matrix. In
Chap. 8, Roberto Patuelli, Gert-Jan Linders, Rodolfo Metulini and Daniel Griffith
focus on the gravity model of trade, and discuss the links between the multilateral
trade resistance terms popularized by Anderson and van Wincoop (2003), and
the spatial filtering approach already suggested in Fischer and Griffith (2008).
Moreover, they provide an empirical comparison with the approaches of Feenstra
(2004) and Baier and Bergstrand (2009) within a negative binomial estimation
framework. Chapter 9 is by James LeSage and Carlos Llano. They propose a
spatial interaction model augmented with origin and destination spatially structured
latent factors. The model is estimated in a Bayesian hierarchical framework, with
spatially autoregressive random effects which incorporate the latent effects. Also
Chap. 10, by Minfeng Deng, makes use of Bayesian estimation. In particular, the
author develops a spatial vector autoregressive (VAR) model to predict traffic flows
within a simulated system. He uses temporally and spatially lagged traffic flows
as predictors, and sets up a Bayesian variable selection procedure to deal with the
large VAR at hand. In Chap. 11, Michael Beenstock and Daniel Felsenstein build


1 Spatial Econometric Interaction Modelling: Where Spatial Econometrics. . .

9

on the definition of spatial dependence in flow (trade) data. They first consider the
case of spatial interaction models where the spatial units (regions, countries) are not
simultaneously repulsing and attracting, but exert their effect only in one direction,
so that bilateral relationships are one-way. Then, they propose a Lagrange multiplier
spatial autocorrelation test, and a further test for spatial autoregressive conditional
heteroscedasticity (SpARCH), that is, the case of spatially autocorrelated variances.
Finally, in Chap. 12, Camilla Mastromarco, Laura Serlenga and Yongsheol Shin
provide an empirical comparison between a trade model estimated following
Behrens et al. (2012) and an alternative specification using a common factor-based
approach. They stress that strong cross-sectional dependence is better accounted
for by factor models. In addition, they augment their factor model by instrumental
variables, in order to evaluate the effect of possibly endogenous trade determinants,
such as trade barriers. They provide an application looking at the trade effects of the
Euro area.
Part III of the book contains a number of applied contributions, showcasing
the potential of the new class of spatial interaction models discussed in Sect. 1.3.
In Chap. 13, Roberto Patuelli, Maurizio Mussoni and Guido Candela set up a
model for analysing domestic tourism flows in Italy and the effect of the inclusion
of landmark sites in UNESCO’s World Heritage List. They model intervening
opportunities in the tourists’ travelling choices, and demonstrate the emergence
of spatial competition between regions on the basis of their cultural amenities
using spatially lagged evaluations of origin/destination characteristics. Chapter 14
is written by Jorge Díaz-Lanchas, Nuria Gallego, Carlos Llano and Tamara de
la Mata. The authors focus on domestic trade in Spain, and in particular on the
phenomenon of ‘ambushed flows’, that is, recorded flows of goods that emerge
solely as a result of the multimodal operations in logistic/transport hubs. They
develop a spatial interaction model for studying this issue that accounts for the
related artificially generated cross-sectional dependence. In Chap. 15, Carlo Llano
and Tamara de la Mata once again study domestic trade flows in Spain, but from

a different perspective focusing on services. They investigate the role of social
networks in influencing such flows and they proxy social linkages by means of a
set of differentiated measures (such as past migration, second homes, past tourism
patterns), so as to provide a sensitivity analysis. In their models, they account for
social linkages by modelling spatial and network autocorrelation, and find that the
resulting deterrence effect of distance is diminished, while the relevance of borders
is increased. In Chap. 16, Timo Mitze estimates a spatial interaction model for
domestic migration by means of a dynamic panel spatial Durbin specification. His
results are consistent with a traditional neoclassical migration model, while pointing
at the need to separate temporal and spatial dynamics. In addition, he computes
cumulative multipliers up to 10 years ahead, in order to evaluate labour market
response over a longer run. Finally, in Chap. 17, Monghyeon Lee and Yongwan
Chun study residential relocation in the Seoul metropolitan area. They estimate a
spatial filtering-augmented spatial interaction model, and show that accounting for
network autocorrelation, even in a limited spatial domain like a metropolitan area,
is needed and leads to a reduction in the estimated overdispersion.


10

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Part I

General Methodological Issues