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) still 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.). Although spatial interaction models originated at the end of the 19th 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 20th century provided such foundations, and 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 2015; 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: ( ), ij i j ij T kO D f d a b = (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 , 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 origindestination 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 regards to the 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-maximum likelihood estimator proposed by the authors is now one of the most commonly used estimation approaches. Further studies focus on further issues in complementing the above groundbreaking studies. Burger et al. (2009) reviewed alternative estimation approaches focusing on the cases of excess zeros; Baltagi and Egger (2015) proposed a quantile regression approach; Egger (2005), Baier and Bergstrand (2009) and Egger and Staub (2015) proposed estimators for the cross-sectional model, while Egger and Pfaffermayr (2003) discussed panel estimation issues. Many more studies of recent publication could

Arbia, G., Patuelli, R., Spatial econometric interaction modelling, Springer Verlag, Heidelberg 2016: 468. 10.1007/978-3-319-30196-9_1 [http://hdl.handle.net/10807/97414]

### Spatial econometric interaction modelling

#####
*Arbia, Giuseppe*^{Secondo};

^{Secondo};

##### 2016

#### Abstract

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) still 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.). Although spatial interaction models originated at the end of the 19th 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 20th century provided such foundations, and 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 2015; 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: ( ), ij i j ij T kO D f d a b = (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 , 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 origindestination 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 regards to the 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-maximum likelihood estimator proposed by the authors is now one of the most commonly used estimation approaches. Further studies focus on further issues in complementing the above groundbreaking studies. Burger et al. (2009) reviewed alternative estimation approaches focusing on the cases of excess zeros; Baltagi and Egger (2015) proposed a quantile regression approach; Egger (2005), Baier and Bergstrand (2009) and Egger and Staub (2015) proposed estimators for the cross-sectional model, while Egger and Pfaffermayr (2003) discussed panel estimation issues. Many more studies of recent publication couldI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.