J Geogr Syst DOI 10.1007/s10109-017-0256-z ORIGINAL ARTICLE

A spatial error model with continuous random effects and an application to growth convergence Ma´rcio Poletti Laurini1

Received: 14 February 2016 / Accepted: 10 July 2017 Ó Springer-Verlag GmbH Germany 2017

Abstract We propose a spatial error model with continuous random effects based on Mate´rn covariance functions and apply this model for the analysis of income convergence processes (b-convergence). The use of a model with continuous random effects permits a clearer visualization and interpretation of the spatial dependency patterns, avoids the problems of defining neighborhoods in spatial econometrics models, and allows projecting the spatial effects for every possible location in the continuous space, circumventing the existing aggregations in discrete lattice representations. We apply this model approach to analyze the economic growth of Brazilian municipalities between 1991 and 2010 using unconditional and conditional formulations and a spatiotemporal model of convergence. The results indicate that the estimated spatial random effects are consistent with the existence of income convergence clubs for Brazilian municipalities in this period. Keywords Spatial effects  Mate´rn covariance  Growth convergence  Spatiotemporal models JEL Classification O47  C21  C23  C11

I thank the valuable comments and suggestions from the editor Manfred M. Fischer and three anonymous referees. The author acknowledges the financial support of CNPq and FAPESP. Electronic supplementary material The online version of this article (doi:10.1007/s10109-017-0256-z) contains supplementary material, which is available to authorized users. & Ma´rcio Poletti Laurini [email protected] 1

Department of Economics, FEARP - University of Sa˜o Paulo, Av. dos Bandeirantes 3900, Ribeira˜o Preˆto 14040-950, Brazil

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1 Introduction The literature measuring spatial effects has focused on the use of spatial econometrics methods, in particular simultaneous autoregression models (SAR), spatial lag models (SLM), and spatial error models (SEM). A description of these models can be found in Anselin (1988), Arbia (2006), LeSage and Pace (2014), and Anselin and Rey (2014). These three models rely on some spatial weight matrix, commonly used in economic analysis, capturing spillover and neighborhood effects. However, these models have a number of limitations, as discussed in Arbia (2006), Anselin (2002), LeSage and Fischer (2008) and Elhorst (2014). A major criticism is that the estimation results can be affected by the choice of the weighting matrix. This is a fundamental problem in spatial econometrics, as discussed, for example, in Arbia and Fingleton (2008), Corrado and Fingleton (2012), and Elhorst and Vega (2013). LeSage and Pace (2014) indicate that several problems associated with spatial weighting matrices are in fact caused by incorrect specification problems. An issue is the use of discrete observation units, such as territorial divisions, generating spatial discontinuities in geographic boundaries, while true economic process is continuous in nature. The discrete spatial structure is essential areal and lattice models (Cressie 1993; Bivand et al. 2013), like CAR (conditional autoregressive) and SAR (spatial autoregressive) models. The use of discrete approximations in spatial models, such as lattices and adjacency models, is only a simplification to approximate the true continuous generating process of location and spatial dependence effects. The spatial weight matrix can represent an irregular lattice structure, with most regions being fully disconnected. As pointed by Wall (2004), the use of lattice designs with very irregular structures can lead to counterintuitive results and the imposition of non-stationary covariance structures. See also Cressie and Wikle (2011) for a detailed discussion of the problems associated with irregular lattice structures. To avoid this problem, Wall (2004) recommends using geostatistical models based on geographic centroids as a measure of location. We propose to use a continuous representation of geographic space, using the formulation of a continuous Gaussian Markov random field to represent a spatial error model structure. This spatial error model allows us to represent the spatial component of the model as a continuous process, using a representation based on the equivalence between a Mate´rn class covariance function and the solution of a stochastic partial differential equation, as suggested by Lindgren et al. (2011), that refines the concept of continuous spatial models (Whittle 1954) and Gaussian random fields (Rozanov, 1977; Rue and Held 2005). The main innovation is the use of a continuous covariance matrix to represent the spatial error structure of the model. The use of a continuous spatial covariance matrix is important because it allows parameterizing the dependency and decay structure and estimating the pattern of spatial persistence and the decay patterns of spillover effects from the data. This avoids ad hoc specifications for spatial dependence or the use of fixed weighting matrices, as discussed in Arbia and Fingleton (2008), Corrado and Fingleton (2012), Elhorst and Vega (2013), and LeSage and Pace (2014). This structure also allows

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the simultaneous estimation of the full posterior distribution of the parameters of the conditional mean and the spatial random effects, allowing to calculate the uncertainty associated with these measures. In this paper, we use a specification similar to that employed in geostatistical models (Cressie and Wikle 2011; Bivand et al. 2013), and we assume that the measurement takes place in specific locations that represent a sample of the mean effects for each location. Due to the continuous representation used in the procedure of Lindgren et al. (2011), these effects are projected continuously through a covariance matrix associated with the process, performing a process similar to kriging and avoiding the discontinuity problems associated with areal/lattice discrete models. As discussed in Simpson et al. (2012), the use of continuous models allows a better representation of spatial effects related to random fields, since the method approximates the true continuous spatial process rather than a possibly ad hoc discrete approximation. For illustration purposes, we apply this approach to the analysis of income bconvergence, using Brazilian data. Income convergence is one of the most studied topics in economics, and a broad class of economic and econometric methods have been developed in the past, e.g., Durlauf and Johnson (1995), Galor (1996), and Durlauf et al. (2005). The importance of spatial location in the dynamics of economic growth has been emphasized in several works, including Sala-i Martin (1996b), Abreu et al. (2005), Rey and Janikas (2005), Corrado et al. (2005), Arbia (2006), Quah (2006), Fischer and Stirbo¨ck (2006), Fischer and Stumpner (2008), Elhorst et al. (2010), and Fischer and LeSage (2015). We performed a b-convergence analysis for municipalities using data on per capita income for the years 1991 and 2010, analyzing unconditional, conditional and spatiotemporal models of income convergence, compared with SEM and proper CAR models. This paper is organized as follows. The continuous spatial random effects model is presented in Sect. 2. The used data and the results are described in Sect. 3, and the final section closes.

2 SPDE-Mate´rn spatial model The objective of this study is to suggest a spatial error model using a continuous representation of the spatial random effects. The main problem by using continuous spatial models is how to obtain a representation of the spatial continuum that is computationally efficient and also allows incorporating a flexible dependence structure that can be represented by a limited set of parameters. The approach proposed in Lindgren et al. (2011) solves these problems by using a structure of spatial random effects based on the covariance matrix of the Mate´rn class, which can be represented in the continuous space through a simple discretization, that is obtained as the solution of a stochastic partial differential equation associated with a spatial error process. The fundamental structure in the definition of continuous spatial models used in this study is continuously indexed Gaussian Markov random fields (GMRF), e.g.,

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Rue and Held (2005), defined on a graph structure. An undirected graph is given by a tuple G ¼ fV; Eg, with V denoting a set of nodes (edges) and E a set of fi; jg vertices, with i; j 2 V. A random vector y ¼ ðy1; y2 ; . . .; yn Þ0 2 Rn is called a Gaussian Markov random field with respect to G ¼ fV; Eg with mean EðyÞ ¼ l and precision matrix Q if and only if its density is characterized by pðyÞ ¼ ð2pÞn=2 jQj1=2 expð 12 ðy  lÞ0 Qðy  uÞ, with Qij 6¼ 0 () fi; jg 2 G_i 6¼ j. The class of continuous spatial models used in this work was proposed in Lindgren et al. (2011) and originates from Whittle (1954) and Rozanov (1977), defining an equivalence between certain stochastic partial differential equations (SPDE) and spatial covariance functions. The main characteristic is the relationship between a Mate´rn covariance function in a random field x(u) and a particular SPDE. This representation is the fundamental element in continuous modeling, since it permits using a basis expansion based on the solution of the SPDE to represent the continuous spatial random effects. The spatial covariance function is associated with the spatial random effects in a hierarchical representation of a continuously indexed random field Y(s), given by   ð1Þ Y ðsÞ ¼ yðsÞÞ : s 2 D  R2  R where s is a vector of locations characterized by a spatial covariance function CovððsÞðs0 ÞÞ ¼ r2 CððsÞðs0 ÞÞ defined for each s and s0  R2  R. We assume that this process is stationary and the covariance depends only on the distance between locations via distances h ¼ jjs  s0 jj. The hierarchical representation of the model is given by yðsÞ ¼ zðsÞb þ nðsÞ þ eðsÞ nð s Þ ¼ x ð s Þ

ð2Þ

2

CovðxðsÞÞ ¼ r CðhÞ with z representing a set of observed variables in location s, n the latent component of spatial random effects defined for each possible location in the spatial continuum, xðsÞ a spatial white noise process with covariance given by a Mate´rn function CðhÞ with marginal variance r2 and e denoting the component of non-spatial idiosyncratic innovations, with eðsÞ  Nð0; r2e Þ. This model can be viewed as a hierarchical representation of a spatial error model, but the error is decomposed into a spatial error component (n) and a non-spatial error (e). The covariance matrix represents the observed points in each location s embedded in a fully continuous spatial structure. In this formulation, we use the representation of the Mate´rn function to CðhÞ, which is a continuous random field characterized by CðhÞ ¼

1 ðkhÞm Km ðkhÞ CðmÞ2m1

ð3Þ

with Km denoting a modified Bessel function of the second kind, Cð:Þ the Gamma function, k and m parameters and h ¼ jjs  s0 jj the Euclidean distance between locations s and s0 , and marginal variance given by

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A spatial error model with continuous random effects...

r2 ¼

C ð mÞ Cðm þ d=2Þð4pÞd=2 j2m s2

ð4Þ

with j and s parameters which define the structure of spatial dependence. Lindgren et al. (2011) show how to obtain a computationally efficient representation of the continuous spatial random effect n in the model, based on the fact that the covariance function of the spatial random effect is related to solution of the following SPDE for a process x(u) ðj  DÞa=2 xðuÞ ¼ WðuÞ; u 2 Rd ; a ¼ m þ d=2; j [ 0; m [ 0

ð5Þ

where ðj  DÞa=2 is a pseudo-differential operator, j is a parameter of scale and a, m are fixed parameters controlling the smoothness of the realizations, while W(u) being an innovation process corresponding to a spatial Gaussian white noise with unit variance. D is a Laplacian given by D¼

d X o2 : ox2i i¼1

ð6Þ

The key aspect of the approach is to use this relationship to obtain a numerical representation of this solution of the SPDE through a finite elements representation (Brenner and Scott 2007) in a given triangulation in the form xðuÞ ¼

n X

wðuÞx

ð7Þ

i¼1

for a choice of bases wðuÞ with weights x given by a Gaussian distribution, n is the number of vertices in the triangulation, controlling the precision of the approximation to the true random field. This representation allows a continuous approximation to the underlying spatial process and thus avoids discontinuities generated by the areal/lattice structure in spatial econometric models based on spatial weight matrices. The weights determine the value of the random field at each vertex, and within the triangle the values are obtained by interpolation. The finite-dimensional representation of the SPDE solution is obtained through the distribution of weights that solves the equation for a particular set of test functions /k , determining the approximation properties. Two possible choices are /k ¼ ðj  DÞ1=2 wk for a ¼ 1 and /k ¼ wk for a ¼ 2, representing, respectively, least squares and Galerkin solutions (Brenner and Scott, 2007). The choice of a ¼ 2 imposes a smoothness structure on the driving field and is therefore used in this work. Note that this basis representation is a numerical device to obtain a computational approximation to a continuous spatial covariance, allowing the incorporation of this structure in spatial models and the consequent use of inference methods for the model parameters. To facilitate the inference procedure, note that we can write the model in Eq. (2) as

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y ¼ zb þ n þ e n ¼x e  Nð0; r2e Id Þ

ð8Þ

x  Nð0; r2 CðhÞÞ with y ¼ ðyi ðs1 Þ; . . .; yðsd ÞÞ0 , z ¼ ðzðs1 Þ; . . .; zðsd ÞÞ0 , n ¼ ðnðs1 Þ; . . .; nðsd ÞÞ0 , Id a indentity matrix of dimension d and CðhÞ defined by the Mate´rn covariance. The continuous spatial component is estimated using the basis expansion structure given by Eq. (7), which allows to estimate the values of spatial random effect for each point in the continuum, using the triangulated mesh approximation. In the vertices of the mesh, the values are obtained as solution of the SDPE for the process x(u) equivalent to the spatial Mate´rn covariance matrix for the process x, and within the triangles by interpolation.   The parameter vector to be estimated is given by h ¼ b; r2e ; s; j , with b representing the parameters for the conditional mean, r2e representing the variance of the non-spatial error component, and the parameters s and j defining the Mate´rn spatial covariance. Using only two parameters, we can represent a continuous spatial covariance matrix in our spatial error model, obtaining a compact representation for a complex spatial structure. For computational motivations in the Bayesian estimation procedure, we estimated the precision (inverse of the variance) 1=r2e and the log of parameters s and j. In this model, we simultaneously estimate the conditional mean structure, the spatial random effects, and the non-spatial error component, which are treated as not observed parameters and estimated by Bayesian estimation, as detailed below. Bayesian estimation allows to recover the full posterior distribution of the parameters and spatial random spatial effects through an additive Gaussian Markov random field structure. This GMRF representation permits to identify and estimate these components, as discussed in Rue and Held (2005) and Rue et al. (2009). The error structure composed of a spatial error component and a non-spatial innovation can be thought of as a continuous formulation of the Besag-York-Mollie model structure (Besag et al. 1991). In this model, the innovation structure is also decomposed into a spatial random effect and an independent component. A useful feature is that this specification can capture complex spatial models and allows the use of Bayesian estimation procedures proposed in Rue et al. (2009). Assuming a Bayesian perspective, the parameter vector has posterior distribution pðh; njyÞ / pðyjn; hÞpðhÞ:

ð9Þ

We assume independent priors for pðhÞ, and the elements yi being conditionally independent due to the Markov property, so we can write ! T Y pðh; njyÞ / pðyjn; hÞ ðpðnjhÞÞpðhÞ; ð10Þ t¼1

and using the Gaussian property of GMRF this posterior is given by

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A spatial error model with continuous random effects...

pðh; njyÞ ¼

ðr2e ÞdT=2

T 1X exp  2 ð y  zb  nÞ0 ð y  zb  nÞ re t¼1

! dimðhÞ Y

pðhi Þ

ð11Þ

i¼1

with T the number of time periods in the model. The hierarchical formulation of this class of processes enables to use the integrated nested Laplace approximations (INLA) proposed in Rue et al. (2009). This method is based on accurate analytical approximations to calculate the posterior distribution of parameters and latent factors in the general class of problems that can be written as GRMFs. This is the Bayesian methodology used in the estimation in this study, and for reasons of space we do not present the details of INLA; see Rue et al. (2009). To complete the model specification, we take a set of independent priors, using log-gamma distributions for the parameters of the spatial covariance function, Gaussian distributions for the parameters of the conditional mean and gamma distributions for the precision parameters. Due to the sample size, the results are robust to the choice of priors, and so are essentially invariant to the choice of alternative sets of priors. The hyperparameters used in the priors can be obtained from the author and are based on the default priors proposed in Lindgren et al. (2011) for the SPDE representation of spatial covariance functions. The most natural alternative to the approach presented in this paper is the spatial error structure in the proper Besag CAR (conditional autoregressive model) model proposed in Besag (1974). This model is based on a discrete Gaussian Markov random field structure and can be interpreted similarly to the hierarchical structure used in the continuous model, but relies on a lattice structure for the spatial random effects. In this model, the spatial effects nðsÞ are parameterized as a GMRF, but by the structure ! 1 X 1 ni jnj ; s; d  N xj ; ; i 6¼ j ð12Þ d þ ni i  j sðd þ ni Þ where the precision matrix is given by Q ¼ sðd I þ diagðN i Þ  RÞ

ð13Þ

where N i is the number of neighbors of region i, I is a identity matrix and R is a matrix with the structure 8 if i ¼ j > <0 if i  j ð14Þ Rij ¼ 1 > : 0 otherwhise with ni denoting the number of neighbors of node i, i  j indicates that the two nodes i and j are neighbors, s is a precision parameter for these spatial effects, and d is a parameter controlling the properness of the model, as discussed in Blangiardo and Cameletti (2015). The CAR model parameterizes the spatial effects by using the nearest neighbors, while the SPDE-Mate´rn model explores the continuous spatial

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Markov property. We perform a comparison of the continuous spatial error model with the proper Besag model in the next section. The main contribution of the present work is to show that the continuous formulation of spatial models, in the form proposed by Lindgren et al. (2011) and extended in several directions by Blangiardo et al. (2013), and Krainski et al. (2016), can also be applied in the analysis of spatial data observed in discrete structures such as areas and lattices. Other works, such as Cameletti et al. (2013) and Laurini (2016), apply this method in their original formulation for geostatistical (point referenced) data. The continuous formulation allows to approximate the true generator process, which evolves continuously in space, avoids the discontinuities generated by lattice models, and allows a clearer visualization of the spatial patterns of dependence in the data, by the projection of the spatial random effects in the spatial continuum. This methodology can also be a complement to local measures of spatial dependence, such as local indicators of spatial association (e.g., Anselin 1995, as discussed in the next section, as well.

3 Application to b-convergence analysis The analysis of b-convergence, popularized by Barro and Sala-i Martin (1995) and Sala-i Martin (1996a), is the main form of income convergence analysis and may be understood as the relationship between initial income and growth rate. A negative relationship between growth rate and initial income indicates a process known as beta-convergence, the richest economies grow more slowly, while poorer economies grow faster. A striking pattern in spatial statistics is that geographically close regions tend to have similar behavior, which in the context of economic growth, implies similar convergence patterns, as found by Dall’erba and Le Gallo (2008), Ramajo et al. (2008), Cravo et al. (2015) and Lim (2016). This mechanism is based on the neoclassical growth model (Solow 1956; Swan 1956), assuming an exogenous saving rate and a production function with decreasing productivity of capital and constant returns to scale. The model predicts that in the long run the growth rates show a process of convergence. The unconditional b-convergence analysis is based on the estimation of the following specification   1 yt;i log ¼ a þ by0;i þ ei ð15Þ t y0;i where yt;i denotes the per capita income in economy i at time t, a and b are parameters, and ei are idiosyncratic shocks. A detailed discussion of this specification can be found in Arbia (2006). Convergence processes in Europe have been studied by Arbia and Paelinck (2003), Fischer and Stirbo¨ck (2006), Mur et al. (2010), and in the USA by Rey and Montouri (1999), and Gonza´lez-Val (2015). The component ei in Eq. (15), which defines the idiosyncratic shocks, should capture all the unsystematic components that are not explained by initial income, or in another way, are orthogonal to the initial income component. When analyzing

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A spatial error model with continuous random effects...

regions within the same country or state, the spatial component is fundamental. Closer units are more subject to common shocks, such as climate, industrial agglomeration effects, similarities in institutions, human capital formation. A remarkable fact in spatial economics is that rich regions tend to be surrounded by rich regions and poor regions by poor regions. This spatial concentration of wealth effect violates the assumption of absence of correlation of specification of residuals in Eq. (15), necessary to the efficiency properties and for the usual calculation of the variance of parameters by least squares estimation, since in general the model specification does not include all the economic conditioning factors, causing a problem of omitted variable bias. But beyond the estimator’s properties, a correct analysis of the effects related to distances and locations is essential for economic analysis and the interpretation of b-convergence analysis. The existence of poverty traps or convergence clubs could dominate the effects of bconvergence, and thus, the tendency of poorer economies growing faster can be dominated by negative effects indicating spatial concentration. Previous analysis of income convergence in Brazil showed evidence for the existence of income convergence clubs (Mossi et al. 2003; Andrade et al. 2004; Laurini et al. 2005; Laurini and Valls Pereira 2009), indicating two convergence clubs, with a higher income club formed by municipalities of the Southeast, Center and South regions, and a low-income convergence club formed by the municipalities in the Northeast and North. In Andrade et al. (2004), Laurini et al. (2005), and Laurini and Valls Pereira (2009), the identification of convergence clubs, based on nonparametric methods and kernel estimators (see Fischer and Stumpner 2008; Monasterio 2010), neglect explicitly the spatial dynamics in convergence clubs. The importance of the spatial component in convergence clubs is shown in Fischer and Stirbo¨ck (2006), Fischer and Stumpner (2008), Fischer and LeSage (2015), among other studies. The approach proposed in this paper is a useful contribution to the literature by allowing to recover deviations from convergence by using continuous spatial random effects. This is important as it enables a direct spatial interpretation of the results of convergence club identified in other studies. The approach, however, provides a solid statistical foundation due to estimation of full posterior distribution of these random effects, summarized by the posterior mean and the credibility intervals, and a way of interpreting this phenomenon that does not depend on the choice of a spatial weight matrix. The model incorporates the information in the sample with regions with larger number of observed municipalities having tight posterior intervals, while regions with lower density of municipalities present lower accuracy in estimating the spatial random effects. 3.1 Data and results For the spatial convergence analysis of per capita income in Brazilian municipalities, we used the information on the 5,561 Brazilian municipalities in 1991 to 2010, built with the information extracted from the 1991, 2000, and 2010 Demographic Censuses by the United Nations Development Programme, Applied Economics Research Institute (IPEA) and the Joa˜o Pinheiro Foundation. Figure 1 displays the territorial division of municipalities in Brazil.

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scale approx 1:26,000,000 0

500

1,000 km

Fig. 1 Territorial division—Brazilian municipalities. Total of 5561 municipalities

We use per capita municipal income, defined as the average monthly income of the residents in each municipality, denominated in Brazilian currency (the Real, plural Reais, R$) of August 1, 2010. Table 1 shows the descriptive statistics of the sample, depicting the minimum and maximum values, the mean, median, and first and third quartiles of the distribution (.25 and .75 quantiles, denoted by q. in this in this and other tables), in Brazilian Real (R$) and US Dollar. The full sample consists of 5,561 municipalities in Brazil, but as we have two municipalities with missing data of initial income in 1991 (Pinto Bandeira and Alta Floresta D’Oeste), the sample size used in all estimations is 5,559 observations. We begin our analysis with the ordinary least squares (OLS) estimation bconvergence, with the results shown in Table 2. The results indicate the presence of b-convergence, by the estimated negative parameter associated with the log of initial income. To perform a spatia analysis of the b-convergence, Table 3 reports the results of the spatial error model, estimated by maximum likelihood, using a binary spatial weight matrix.1 In this matrix notation, the model in is given by Y ¼Xb þ e

1

ð16Þ

We also tested a specification of the SEM model using a distance-based weighting matrix. The results are similar, with an estimated intercept of 0.1652 and a parameter of -0.0234 to the initial income.

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A spatial error model with continuous random effects... Table 1 Descriptive statistics—growth rate and per capita income of Brazilian municipalities (5559 observations) Min.

0.25 q

Median

Mean

0.75 q

Max

Growth rate 1991–2010

0.03890

0.03256

0.04184

0.04204

0.05101

0.10470

Per capita income 1991 R$

19.00

68.67

114.75

134.25

179.88

677.56

Per capita income 1991 US$

33.24

120.10

200.70

234.80

314.60

1185.00

Per capita income 2010 R$

96.25

281.10

467.50

493.60

650.60

2044.00

Per capita income 2010 US$

55.03

160.72

267.31

282.23

372.00

1168.73

q stands for quantile

Table 2 Estimation results of the unconditional b-convergence, model without spatial effects, estimated by least squares (1991–2010)

Intercept

Parameter

sd

t stat.

pr([|t|)

0.1150

0.0014

79.37

\2e-16

0.0002

-50.69

\2e-16

p value

\2.2e-16

Log(income 1991)

-0.0138

Residual std. err.

0.0122

F stat

2569

R2

0.316

e ¼qWe þ g

ð17Þ

with Y and X denoting the dependent variable and the set of explanatory variables, in our case the growth rate and the initial income, respectively, e a vector of errors with spatial dependence and g the idiosyncratic errors. The b parameter represents the conditional effect of the explanatory variables and q the spatial dependency parameter. Note that the model parameters estimated by OLS and SEM models are significantly different, which may indicate a specification problem, such as omission of variables in the conditional mean or misspecification in the innovation structure, as discussed in (LeSage and Pace 2014, pp. 61–63). Using these model specifications, we do not have a direct way to analyze relevant issues, such as the importance of the spatial error component in each region of the country, or where the spatial effect is more relevant than the initial income in determining growth. They are also not informative about fundamental issues such as the presence of convergence clubs (Quah 1997) or poverty traps and low-income concentrations in specific regions, e.g., Azzoni (2001), and Fischer and Stirbo¨ck (2006). b-convergence analysis using continuous spatial random effects and the SPDEMate´rn approach allows a simpler interpretation of the effects of space on the growth process. The spatial random effects in the SPDE-Mate´rn estimate the direct effects of each location in the spatial continuum on the growth rate. To apply the SPDE-Mate´rn approach, we first define a triangle mesh from the shapefile with the territorial division of municipalities, shown in Fig. 1. For this, we use a Delauney triangulation algorithm that defines the maximum angle of 0.5 to 1.25 degrees for

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M. P. Laurini Table 3 Estimation results of the a spatial error model estimated by maximum likelihood estimation (1991–2010) Parameter

sd

z value

pr([|z|)

Intercept

0.1665

0.0019

85.967

\2.2e-16

Log(income 1991)

-0.0237

0.0003

-65.792

\2.2e-16

0.0104

LR 2688.7

\2.22e-16

q

0.7771

Variance

8.0256e-05

Log lik.

17,927.6

the triangles inside and outside of the lattice regions. Note that we need to perform a triangulation outside the domain of analysis due to boundary effects in the representation of finite elements that can affect the calculation of the variance of representations. The triangulated Delauney mesh used in the estimation is shown in Fig. 2. In ‘‘Appendix’’, we present a robustness analysis with the results based on an alternative triangulation based only on the centroids of the municipalities. We estimate the SPDE-Mate´rn model from the following representation gr ðsÞ ¼ a þ y0 ðsÞb þ nðsÞ þ eðsÞ nðsÞ ¼ xðsÞ

ð18Þ

2

CovðxðsÞÞ ¼ r CðhÞ where gr(s) represents the average growth rate between 1991 and 2010 of the municipality in location s, measured by the longitude and latitude at the spatial centroid of the municipality in question, and y0 ðsÞ is the logarithm of monthly per capita income in 1991 in this same location s, and other components have the same meaning as defined in Sect. 2, with a denoting the fixed effect intercept and b the parameter associated with initial income. The results of this estimation are show in Table 4. This table presents the average values, standard deviation, quantiles .025, .5, .75, and the mode, summarizing the results of estimating posterior distribution. We point out that in this specification we also find the result of b-convergence, estimating a negative parameter with posterior mean of -0.0249, notably more negative than that found by the OLS estimation. In ‘‘Appendix’’, an alternative estimation can be found, using a grid constructed only using the centroids as nodes, representing a cruder approximation to the true random field. The results are similar, showing the robustness of the continuous approximation. The estimated value for the precision parameter e is high, indicating a relatively small impact of idiosyncratic effects, unrelated to the space component. The parameters s and j related to the spatial covariance function are presented in log scale, due to positivity constraints for the transformed parameters. As the interpretation is not natural for these parameters, we will focus our analysis on the spatial correlation function set by this model, presented in Fig. 3, that shows the spatial autocorrelation function obtained by the SPDE-Mate´rn model (solid line) compared with the sample autocorrelation as a function of distance measured in distance decimal degrees.

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A spatial error model with continuous random effects...

scale approx 1:26,000,000 0

500

1,000

1,500 km

Fig. 2 Triangulated Delauney mesh—Brazilian municipalities Table 4 Estimation results: unconditional b-convergence analysis, SPDE-Mate´rn model (1991–2010) Mean

sd

0.025 q

0.5 q

0.975 q

Mode

Intercept

0.1670

0.0037

0.1596

0.1670

0.1743

0.1670

Log(income 1991)

-0.0249

0.0003

-0.0257

-0.0249

-0.0242

-0.0249

Precision - e

22,432.7

788.5

21,077.1

22,358.7

24,154.0

22,147.0

Log - s

4.0208

0.0942

3.845

4.0169

4.2146

4.0055

Log - j

0.8965

0.1563

-1.218

-0.8902

-0.6052

-0.8711

Marginal lik.

18,358.5

q stands for quantile

The graph shows the theoretical correlation function for the Mate´rn covariance function constructed using the estimated posterior distribution of the parameters s and j, and transformed into a correlation matrix. The correlation function is calculated using the posterior mean and a 95% credibility interval using the .025 and .975 posterior quantiles for these parameters. We can observe the pattern of spatial

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1.0

M. P. Laurini

0.6 0.4 0.0

0.2

Correlation

0.8

Mean 2.5% Quantile 97.5% Quantile

0

2

4

6

8

Distance in Degrees Fig. 3 Estimated spatial autocorrelation: SPDE-Mate´rn model, distance in degrees

dependence for the spatial random effects in this model, and the close fit using the Mate´rn function, indicating that the proposed model can adequately approximate the spatial decay pattern observed in this process. This result is particularly important since the pattern of decay is estimated by the model, and not imposed by specifying a spatial weight matrix, possibly subject to ad hoc formulations and discontinuity effects. The tight credibility interval also indicates the high accuracy in the estimation made possible by the continuous structure of spatial random effects. The results are shown in Fig. 4, with the posterior mean of spatial continuous random effects and shown in Figs. 5 and 6, with the posterior standard deviation and the posterior 95% credibility interval of these random effects, are important contributions to the spatial analysis of income convergence. The Bayesian method allows estimating the full posterior distribution of the parameters and random effects, so we can build full credibility intervals for the spatial random effects estimated by the model, allowing a complete inferential analysis of the spatial convergence process. The posterior distribution of the continuous spatial random effects is based on the basis expansion discussed in Sect. 2 to represent the values of the Mate´rn process in space. Based on the posterior distribution of the parameters j and s, we use the basis expansion to estimate the posterior mean and credibility interval in each vertex of the triangulated mesh, and the results inside the triangles by interpolation. In Fig. 6, the credible interval is constructed by using the .025 and .975 quantiles of the posterior distribution of the spatial random effect for each point in the spatial continuum. The spatial random effects capture the impact of space on the convergence process and can be easily interpreted by analyzing the estimated posterior distribution, summarized by the posterior mean and quantiles. The great variability in the observed values of the spatial random effects is worth noting, ranging from 0.03 to 0.03; a range of variation similar to the observed growth rate. We also note that the random effects with negative values are concentrated in municipalities in the North region, while a clear concentration of positive random effects can be

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Longitude Fig. 5 Standard deviation of the spatial random effects: SPDE-Mate´rn model

found in the Center, Southeast, and South regions. The estimated credibility interval shows that the effects are relevant to indicate the diverging regions (low posterior probability of values near zero) with consistently positive effects for South, Southeast, and Center-West regions and systematically negative effects in most of the North region.

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Fig. 6 Credibility interval of the spatial random effects: SPDE-Mate´rn model

This result has a very relevant economic interpretation. According to the hypothesis of b income convergence, poorer regions should grow faster, and regions with higher initial wealth should have relatively smaller growth. However, the results obtained with the inclusion of spatial random effects show that the effect of convergence can be canceled by the spatial effects, since the estimated posterior distribution of these effects shows that the random effects are statistically important. The highest growth rate expected for municipalities in the North region, with lower initial income observed in 1991, is affected by the negative spatial effect observed in this region, as observed by the posterior distribution of this process. Conversely, municipalities in the Center, Southeast, and South, with higher initial income, show the effect of convergence annulled by the presence of positive and statistically significant spatial random effects, leading to a higher growth rate. The municipalities in the Northeast region grew near the rates predicted by convergence effect, with spatial random effects close to zero. We can interpret this result as further evidence of the formation of convergence clubs, with a convergence club of rich regions formed by municipalities in the Center, Southeast, and South regions, a club formed by the municipalities of the North region, dominated by the negative spatial effects, and a third club formed by municipalities in the Northeast region, which has growth rate determined largely by the effect of income convergence. An alternative interpretation would be that the municipalities of the North region are caught in a poverty trap, canceling the effects of income convergence. This evidence is very similar to that obtained by the nonparametric estimations and stochastic kernel functions in Andrade et al. (2004), and Laurini et al. (2005), but obtained without the direct use of spatial dependence structures. Using the SPDE-Mate´rn model, we can estimate the deviations of the bconvergence process for each point in the spatial continuum. Note that the SPDEMate´rn model provides a full statistical analysis, and not only a visual or descriptive measure of divergence and club formation processes, since we have a complete estimation of the posterior distribution of spatial random effects and the parameters in the conditional mean. To facilitate the interpretation of these results, Fig. 7 shows the growth rate predicted by the SPDE-Mate´rn model of spatial random effects by combining the effect of convergence with the spatial random effects, and in Fig. 8 the expected

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Longitude Fig. 7 Posterior mean of the fitted growth rate: SPDE-Mate´rn model

growth between 1991 and 2010 projected by the OLS model (see Table 2). The SPDE-Mate´rn model generates a much more consistent result compared with the observed growth rate, especially in contrast to the growth projected by the OLS estimation. This result is depicted by using the dispersions between the fitted and observed growth for both methods, displayed in Fig. 9. The SPDE-Mate´rn model is more precisely adjusted compared to the OLS model without spatial effects. The correlation between the observed and fitted growth is 0.5622 in the OLS model, in contrast to the value of 0.9460 estimated by the SPDE-Mate´rn model. To perform a comparison with a discrete structure for the spatial random effects, we estimate the CAR-Besag model through the neighborhood structure determined by the territorial division of municipalities shown in Fig. 1, yielding the results shown in Table 5. It can be seen that the results are quite similar to those obtained by the SPDE-Mate´rn model for the intercept and the convergence parameter, supporting the evidence about the importance of spatial effects. Figs. 10 and 11 show the spatial random effects obtained by the CAR structure and also the growth rate predicted by this model, respectively. It is possible to note that the spatial patterns and growth effects are quite similar to those obtained by the SPDE-Mate´rn model, but without the continuous evolution of the spatial structure which can be obtained by the latter. The CAR model is an appropriate form of spatial adjustment, but without a continuous interpretation.

3.2 SPDE-Mate´rn model with covariates A relevant question is the interpretation of spatial random effects. Spatial random effects may be viewed as residuals not explained by initial income. These random

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Longitude Fig. 8 Fitted growth rate: OLS model Correlation 0.9460

0.08 0.06 0.04

SPDE Fit

0.088

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0.02

0.094 0.092 0.090

OLS fit

0.096

0.10

0.098

Correlation 0.5622

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Growth Rate

0.06

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0.10

Growth Rate

Fig. 9 Fitted versus observed growth rate: the OLS and the SPDE-Mate´rn models

effects capture spatial effects of all the variables that are omitted in the unconditional specification of the b-convergence model. Thus, an important question is the role of economic variables that have significant impact on the growth rate and may be spatially measured. With the inclusion of these variables, we can obtain a better estimation of the convergence pattern. To perform this analysis, we have chosen the following variables: expected years of schooling in 18 years of age (Educ), total fertility rate (Fer_tot), infant mortality under one year of age (Mort), gross attendance rate to higher education (R_higher), the Gini index (Gini), and the percentage of people in households with inadequate water supply and sewerage (Water_sewer), with all variables measured in 1991, reflecting the impact of initial conditions on the growth rate. The results of this estimation are reported in Table 6. We can see that education and the gross attendance rate to higher education have a positive impact on the

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A spatial error model with continuous random effects... Table 5 Estimation results: unconditional b-convergence for Brazilian municipalities, Besag proper model (1991–2010) Mean

sd

0.025 q

0.5 q

0.975 q

Mode

Intercept

0.1716

0.0053

0.1609

0.1715

0.1824

0.1715

Log(income 1991)

-0.0245

0.0004

-0.0252

-0.0245

-0.0238

-0.0245

Precision - e

36,678.4

2,725.1

31,706.84

36,537.5

42,412.2

36,221.8

Precision Besag

7361.37

498.056

6416.72

7350.63

8371.57

7335.11

d

0.0013

0.0009

0.0002

0.0011

0.0036

0.0006

Marginal Lik.

18,342.03

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0

q stands for quantile

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Longitude Fig. 10 Posterior mean of spatial random effects: CAR-Besag model

growth rate, while the Gini Index, measuring inequality, mortality rates, inadequate water, and sewerage and total fertility have negative effects. These results are consistent with the literature on endogenous growth, inequality, and poverty traps. The inclusion of these variables alters the effect of initial income, increasing the effect of b-convergence to a posterior mean of -0.0314, compared to the estimated value of -0.0249 obtained by the estimation without covariates. It is possible to observe that this effect is statistically relevant, since the intervals of credibility do not overlap in the two estimates. Figures 12 and 13 show the spatial random effects obtained by the SPDE-Mate´rn model with the inclusion of the covariates and the corresponding fit to growth rates, respectively. We note that the inclusion of these covariates reduces the magnitude of the spatial effects, especially the negative values observed for the North region, but the spatial effects are still relevant to the Center, Southeast, and South, capturing

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Longitude Fig. 11 Fitted growth rate: CAR-Besag model Table 6 Estimation results: conditional b-convergence analysis, SPDE-Mate´rn model (1991–2010) Mean

sd

0.025 q

0.5 q

0.975 q

Mode

Intercept

0.2224

0.0034

0.2157

0.2224

0.2290

0.2224

Log(income 1991)

-0.0314

0.0005

-0.0323

-0.0314

-0.0304

-0.0314

Educ

0.0009

0.0001

0.0007

0.0009

0.0011

0.0009

Fer_tot

-0.0029

0.0002

-0.0033

-0.0029

-0.0025

-0.0029 -0.0001

Mort

-0.0001

0.0000

-0.0002

-0.0001

-0.0001

R_higher

0.0005

0.0000

0.0004

0.0005

0.0006

0.0005

Gini

-0.0129

0.0019

-0.0165

-0.0129

-0.0092

-0.0129

Water_sewer

-0.0001

0.0000

-0.0001

-0.0001

-0.0001

-0.0001

Precision - e

27,725.5

917.72

25,967.8

27,710.1

29,573.3

27,679.5

Log - s

4.2661

0.1079

4.0602

4.2638

4.4837

4.2568

Log - j

-0.7471

0.1830

-1.1179

-0.7420

-0.4001

-0.7267

Marginal lik.

18,747.8

q stands for quantile

common characteristics of these regions that are not included in the adopted specification. We can speculate that these features are related to distinct patterns of immigration, institutional, and cultural factors regarding the North and Northeast regions, which are variables not included in this specification but may have effects on the growth rate. See, for example, Acemoglu et al. (2001), and Acemoglu and Robinson (2012).

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Fig. 12 Posterior mean of the spatial random effects of the conditional convergence model: SPDEMate´rn model

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Longitude Fig. 13 Posterior mean of fitted growth of the conditional model: SPDE-Mate´rn model

3.3 Spatiotemporal model with AR(1) dynamics Another relevant question is about the pattern of time dependence in the spatial random effects. If these effects are persistent, changes in spatial factors may have

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quite lasting effects on the growth rate. To address this issue, we modified the analysis of income convergence using a spatiotemporal model, which captures the dynamics of the spatial random effects in time. One possible formulation is through a specification of a first-order effect autoregressive process, AR(1), for the dynamics of the spatial random effects, similar to the modeling proposed in Cameletti et al. (2013). In this specification, we modified the hierarchical formulation to capture the dynamics of observed variables and latent factors in space and time, using the following representation yðs; tÞ ¼ zðs; tÞb þ nðs; tÞ þ eðs; tÞ nðs; tÞ ¼ anðs; t  1Þ þ xðs; tÞ:

ð19Þ

In this model, the set of indices ðs; tÞ refer to a stochastic process observed in location s at time t. The process is assumed to depend on several variables located in space-time, an idiosyncratic process eðs; tÞ and a spatial random effect in time nðs; tÞ evolving as a first-order autoregressive process with common parameter a, which captures the persistence of spatial random effects in time. We implemented the space-time model given by Eq. (19) for the data on municipal income, using per capita income in the year 2000. The sample is split into two periods, similar to the analysis in Le Gallo and Dalle´rba (2006), stacking the first period with the growth between 1991 and 2000 and the initial income measured in 1991, and a second period with the growth between 2000 and 2010, with initial income in 2000. Note that the models estimate the same b parameter for all samples, but impose a spatiotemporal process for the random effects. The results of this model specification are given in Table 7. We can see that we get an initial income parameter close to -0.0354, closer to that obtained in Table 6. In this specification, we can observe the random effects for the periods 1991–2000 and 2000–2010, shown in Fig. 14. The patterns of spatial random effects change between the two periods analyzed. It is noteworthy that between 1991–2000 the random effects are most striking for the North and Northeast and part of the Center, and in the next time period these effects appear to be softened in significant parts of these regions. We can also note that the positive effect remains for the Southeast and South regions in both periods. The value of the autoregressive parameters is Table 7 Estimation results: unconditional b-convergence for Brazilian municipalities, SPDE-Mate´rn model with AR(1) dynamics (1991–2010) Mean

sd

0.025 q

0.5 q

0.975 q

Mode

Intercept

0.2280

5e-03

0.2183 0

.2280

0.2380

0.2279

Log(initial income)

-0.0354

6e-04

-0.0366

-0.0354

-0.0342

-0.0354

Precision - e

3582.8

57.7595

3472.0

3581.8

3698.7

3579.3

Log - s

3.6372

0.0797

3.4755

3.6394

3.7885

3.6460

Log - j

-0.9068

0.1195

-1.1313

-0.9111

-0.6623

-0.9240

AR(1)

0.2568

0.1049

0.0442

0.2595

0.4543

0.2647

Marginal lik.

28,678.4

q stands for quantile

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Fig. 14 Posterior mean of spatial random effects of the model with AR(1) dynamics: SPDE-Mate´rn model

estimated with a posterior mean of 0.2568, indicating that although there is significant time persistence in dynamic spatial effects, it is possible to observe important changes in this pattern, even for these relatively short periods of time.

4 Closing remarks In this paper, we analyzed some spatial and spatiotemporal models with continuous spatial random effects, based on the use of a spatial covariance function of the Mate´rn class. This approach is based on the computational properties derived from the equivalence between the solution of a stochastic partial differential equation and the continuous covariance function of a random field. This representation allows representing continuous spatial random effects using a hierarchical formulation for a Gaussian Markov random field that enables to use of Bayesian methods for the estimation of complex spatial models. The continuous spatial estimation approach used in this study provides a direct interpretation of spatial effects, by using a formulation consistent with the original ideas of the spatial models proposed in Whittle (1954). The approach also avoids some of the problems associated with the use of spatial weight matrices, especially the discontinuities generated by the lattice structures in discrete spatial models. The use of a Mate´rn covariance matrix, with parameters estimated using the observed sample, also avoids the use of arbitrary decay structures, allowing accurately to recover spillovers and spatial persistence patterns. The Bayesian approach is also important because it allows the estimation of the full posterior distribution of parameters and spatial random effects. This property is important for building credibility intervals for the spatial random effects throughout the continuous space and allows to interpret the impact of these effects on the models under review. The spatial random effects can be interpreted as mechanisms of deviation from the convergence hypothesis, and continuous estimation allows to visualize how this mechanism occurs in each region of space. The spatial random effects estimated in the distinct model specifications indicate that the spatial components are very important and can dominate the convergence effects. Our results support previous findings about the importance of spatial effects

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on the income convergence process, similar to other studies, like Dall’erba and Le Gallo (2008), Ramajo et al. (2008), Cravo et al. (2015) and Lim (2016). The results achieved are consistent with the existence of three convergence clubs, or spatial regimes, a first regime with positive spatial effects, a second regime with negative effects and a third one with validity of the hypothesis of b-convergence. We could interpret this result according to theories on convergence clubs and poverty traps, but using an approach that directly permits identifying the observation units in each club. We also studied the spatial random effects in the presence of additional explanatory variables and the persistence of spatial effects using a spatiotemporal convergence model, analyzing the impact of additional explanatory variables and the persistence of these effects in time. Although the formulation used in this study is a spatial error model, note that it is possible to use more complex spatial specifications in the modeling approach. The hierarchical design and the Bayesian estimation allow to incorporate complex structures such as autoregressive models for the conditional mean, especially useful for modeling continuous spillover effects and incorporating model misalignment (dependent and explanatory variables are observed at different points in the space), non-Gaussian processes, measurement errors (e.g., Laurini (2016)), and coregionalization models. Details of these specifications can be found in Krainski et al. (2016). The Bayesian estimation procedures used also permit to implement model selection procedures, since they allow the numerical evaluation of the marginal likelihood of each model, and the construction of model selection measures as Bayes factors. We believe that this class of models may play an important role for the econometric analysis of spatial data. We hope that the analysis carried out in this work indicates the advantages of this methodology and will inspire the development of new methods based on continuous spatial models.

Appendix This appendix briefly presents a robustness analysis using a SPDE-Mate´rn model specification relying on an alternative triangulation, based on a convex grid generated by the geographic centroid of each municipality. This specification can be thought of as the most basic triangulation that preserves the existing information in the spatial distribution of municipalities, with a unique focal point for each observed region. Figure 15 shows this triangulation, constructed using only nodes generated by the geographic centroid of each municipality. The posterior distribution of the estimated parameters is shown in Table 8. We can note that the results are quite similar to those observed with the use of finer triangulation (see Table 2), showing that the results are robust to the choice of an alternative mesh. Figure 16 shows the posterior mean for spatial random effects estimated based on this alternative triangulation. In general, the results are quite similar to those observed in the original formulation. The model captures the existing precision in the distribution of points in space, with more precise confidence intervals for the regions with the highest density of points, and intervals

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A spatial error model with continuous random effects... Constrained Delaunay triangulation

Fig. 15 Triangulated Delauney mesh: Brazilian municipalities using centroids

Table 8 Estimation results: b-convergence for Brazilian municipalities, SPDE-Mate´rn model with alternative triangulation (1991–2010) Mean

sd

0.025 q

0.5 q

0.975 q

Mode

Intercept

0.1656

0.0032

0.1594

0.1656

0.1718

0.1655

Log(Income 1991)

-0.0247

0.0004

-0.0255

-0.0247

-0.0240

-0.0247

Precision - e

31,442.8

1545.8

28,519.5

31,404.0

34,591.5

31,326.2

Log - s

3.7637

0.0744

3.6195

3.7628

3.9119

3.760

Log - j

-0.6458

0.1088

-0.8639

-0.6438

-0.4368

-0.638

Marginal Lik.

18,320.5

q stands for quantile

more open in regions with lower density of points, especially in the northern region of the country. Note that this result is due to the interpolation mechanism used in the representation of the solution of the partial differential stochastic equation used in the representation of the spatial random field (see Eq. (7)), which is also used in determining the posterior distribution of spatial random effects.This result also indicates that continuous representation of space can properly handle the presence of an irregular lattice structure, a major problem in model specifications with areal data, as discussed in Wall (2004). The posterior distribution of spatial random effects captures the amount of data available for density regions on the map,

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avoiding the problems associated with the use of spatial weights where this information is not used optimally.

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