Regional convergence and divergence in the enlarged ...
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Regional convergence and divergence in the enlarged european union: a comparison of different spatial econometric approaches
Panagiotis Artelaris
Lecturer (under appointment), Harokopio University, Department of Geography,
Abstract
Three types of (βconvergence) econometric models have been used in the
growth and convergence literature; global, regime and local. The aim of this paper is
to present and compare, for the first, the estimation results obtained by all three
types of models. The convergence analysis is carried out for the regions of the
(enlarged) European Union (EU) over the period 19952005. Due to the fact that
spatial effects have been found to be significant for the EU regions, the paper
employs spatial, rather than nonspatial, models. As a result, the analysis includes
the estimation of the: (i) traditional OLS as well as the spatial error and spatial
autoregressive model (global approach) (ii) spatial error and spatial autoregressive
model with groupwise and/or structural instability (regime approach) (iii)
geographically weighted regression (GWR) model (local approach). The results show
that a regime model, and more particularly a spatial error model with groupwise
heteroskedasticity, seems to perform better than any other model used for examining
the regional convergence process in the EU. Global models seem to perform worse
than any other specification while the superiority of local (GWR) models usually
founded in the literature has not been confirmed. This might happen because,
normally, the estimates of local models are not compared with the estimates of
regime models but only with the estimates of global models.
Keywords: regional βconvergence and divergence, enlarged European Union, club
convergence, local convergence, spatial econometrics
JEL classification: C21, R11, R12, O52, O18
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1. Introduction Spatial or regional income inequality, a major dimension of overall inequality,
has been a matter of great academic and political concern for more than fifty years.
From the policy point of view, the investigation of the dynamics of regional inequality
can be seen as an evaluation of the effectiveness of regional policy measures. From
the theory point of view, regional inequality has been the subject of intensive
academic debate from different disciplines of social sciences, especially economics
and geography, since the 50’s (e.g. Myrdal, 1957; Hirschmann, 1958; Borts and
Stein, 1964; Williamson, 1965; Kaldor, 1970; Harvey, 1982). More recently, it has
gained much more attention mainly due to developments in the fields of economic
integration, economic geography and endogenous growth (e.g. Romer, 1986; Lucas,
1988; Krugman, 1991; Barro and SalaIMartin, 1995; Fujita et al., 1999).
Two main schools of thought have arisen in the ongoing debate concerning
regional inequalities: The spatial equilibrium (convergence) and spatial disequilibrium
(divergence) schools. The former, led by the neoclassical economic theory (Solow,
1956; Borts and Stein, 1964), predicts convergence among advanced and less
advanced regions (see Barro and SalaiMartin, 1995, for a review). In contrast, the
latter, led by the theory of cumulative causation (Myrdal, 1957; Hirschmann, 1958),
argues that regional inequality is likely to increase. For example, both the
endogenous (new) growth theories (see Aghion and Howitt 1998, for a review) and
the new economic geography (see Fujita et al. 1999, for a review) tend to agree that
growth is a spatially cumulative process, which is likely to increase inequalities.
In the midst of this theoretical spectrum, more recently growth models (e.g.
Azariadis and Drazen, 1990; Galor, 1996) generate multiple steady state (locally
stable) equilibria and convergence clubs (for a review see Azariadis, 1996). Club
convergence implies convergence to a common level only for economies that are
both identical in their structural characteristics and similar in their initial conditions. In
other words, these models transcend the “all or nothing” logic behind conventional
growth models and maintain that convergence may come about for different groups
of (regional) economies. Multiple equilibria and convergence clubs can emerge on
account of differences in, among others, human or physical capital, income
distribution, capital or market imperfections, local complementarities and
externalities.
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Because theories have an ambiguous prediction about the evolution of
regional disparities, a large number of empirical studies have recently been carried
out (see Magrini, 2004, for a review). The econometric models, usually used by these
studies, are known as βconvergence models and can be divided into three groups:
global models, regime models and local models. The first type of models (i.e. global
models), influenced by the neoclassical paradigm, produces parameter estimates
that represent an average type of economic behavior; that is, for each variable there
is one regression coefficient for the entire sample. In other words, estimated
parameters of these econometric models represent global averages of processes
assuming that the relationship of interest is stable over space. The second type of
models (i.e. regime models), based on recent theoretical models that yield multiple
steady state equilibria, requires the identification of a (preset) number of spatial
regimes (clubs) between which model coefficients and other parameters are allowed
to vary. In other words, this kind of analysis challenges the view that the estimated
parameters are identical across economies, suggesting that convergence is not a
widespread phenomenon and can occur only within small groups of economies.
Finally, the third type of models (i.e. local models), based mostly on Tobler’s First
Law of Geography1, suggests that the relationship of interest can widely vary over
space, allowing the regression parameters to change across regions. As a result, a
separate regression equation is estimated for each observation uncovering possible
patterns neglected by all the other approaches.
Undoubtedly, in the regional growth and convergence literature, the majority of
studies have applied exclusively global models (e.g. Barro and Sallai Martin, 1995;
Armstrong, 1995; Persson, 1997; Paci, 1997; Baumont et al., 2000; Martin, 2001;
Carrington, 2003; Badinger et al., 2004; LopezBazo et al., 2004; Petrakos and
Artelaris, 2009). However, some studies have applied and compared the results of
global and regime models (e.g. Baumol and Wolff, 1988; Baumont et al., 2003;
Dall’erba, 2005; Ertur et al, 2006; Fischer and Stirböck, 2006; Le Gallo and Dall’erba,
2006; Monastiriotis, 2006; Dall’erba et al, 2008; Ramajo et al, 2008; Chapman et al.
2010; Mur at el., 2010; Petrakos et al., 2011; Tian, 2011), or global and local models
(e.g. Bivand and Brunstad, 2003; Bivand and Brunstad, 2005; Eckey et al., 2007;
1 This law states that “everything is related to everything else, but near things are more related than
distant things” (Tobler, 1970: 236).
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Ertur et al., 2007; Yıldırım et al., 2009; Sassi, 2010). Nevertheless, no study, to the
best of our knowledge, has applied and compared the results of all types of
econometric approaches.
In this context, the aim of this paper is to present and compare the estimation
results obtained by all three types of (βconvergence) econometric models  global,
regime and local. The convergence analysis is carried out for the regions of the
(enlarged) European Union (EU) over the period 19952005. This area has been
chosen, mainly, because much attention has been drawn to it recently2 and because
only few studies have examined regional convergence process in the enlarged EU3.
Due to the fact that spatial effects (i.e. spatial dependence and spatial heterogeneity)
have been found to be significant for the EU regions by several studies, the paper
employs spatial, rather than nonspatial, models. As a result, the following analysis
includes the estimation of the: (i) traditional OLS as well as the spatial error and
spatial autoregressive model (global approach) (ii) spatial error and spatial
autoregressive model with groupwise and/or structural instability (regime approach)
(iii) geographically weighted regression (GWR) model (local approach).
The results of the paper show that a regime model, and more particularly a
spatial error model with groupwise heteroskedasticity, seems to perform better than
any other model used for examining the regional convergence process in the
enlarged EU. The convergence parameter implies a very low convergence rate
(0.62%) and a very long halflife (111 years). Global models seem to perform worse
than any other specification while the superiority of local (GWR) models usually
founded in the literature has not been confirmed. This might happen because,
normally, the estimates of local models are not compared with the estimates of
regime models but only with the estimates of global models.
The rest of the paper is organized as follows. The next section briefly outlines
the different approaches of spatial βconvergence models. Section 3 gives a
description of the regional data and spatial weight matrices used in empirical analysis
as well as the method employed for the identification of regimes. Section 4 presents 2 Regional inequality scores high on the economic and political agenda of the European Union, mainly
due to the strong focus placed on achieving economic and social cohesion among its Member States
(European Commission, 2004). 3 See for example Fischer and Stirböck, (2006), Chapman et al., (2010) and Mur at el., (2010).
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the estimation results obtained by all three types of econometric models while a brief
summary and discussion of the results follow in the final section.
2. Different approaches of spatial βconvergence models Several conceptual frameworks have been developed in the growth literature
for examining convergence or divergence among economies (for a review see Islam,
2003). Undoubtedly, the most popular one is unconditional or absolute β
convergence that is derived from the neoclassical growth model (Solow, 1956),
following the seminal studies of Baumol (1986), Barro and SalaiMartin (1992) and
Mankiw et al. (1992). According to this, if economies are homogeneous, convergence
can occur in an absolute sense since they will converge towards the same steady
state. In other words, this concept implies that poor economies grow faster than rich
ones and therefore, over a long period of time, they converge to the same level of per
capita income.
Unconditional βconvergence hypothesis is normally tested by the following
crosssectional OLS regression model4, in matrix form:
εβ ++= 0yaSgT ),0(~ 2 Iσε Ν (1)
where Tg is the (nx1) vector of average growth rates of log per capita GDP in the [0, T]
period for the n regions; S is the sum vector; 0y is the vector of log per capita GDP
levels at date 0; εi is the error term ; and α and β are parameters to be estimated. A
negative and significant β coefficient indicates unconditional βconvergence across
the territorial units of analysis, in a given time period, while a positive and significant
β coefficient indicates unconditional βdivergence. The convergence process is
traditionally characterized by its convergence speed and its halflife5. The convergence
speed can be estimated by the formula b=ln(1+Tβ)/T (where T is the length of the time
interval) and the half life by the formula τ=ln(2)/ln(1+β).
As many scholars have shown, the standard unconditional βconvergence
model might be misspecified because of omitted spatial effects, namely, spatial 4 Durlauf and Quah (1999) and Temple (1999) present an overview of the weaknesses of this type of
analysis. 5 The halflife is the time necessary for the economies to fill half of the variation that separates them
from their steady state.
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dependence and spatial heterogeneity (Abreu et al., 2005, Fingleton and Lopez
Bazo, 2006). The former can be defined as the coincidence of value similarity with
locational similarity (Anselin, 1988:11) and can be caused by several factors, such as
trade between regions, labor and capital mobility, and technology and knowledge
diffusion. The latter is the presence of variation or instability over space of the
relationships under study (Anselin, 1988:13) and it can arise, mainly, because of
presence of strong geographic patterns or spatial regimes (e.g. coreperiphery
pattern). The presence of spatial effects may lead to serious bias and/or inefficiency
in the estimates of the coefficients. Conventional econometric methods, such as
OLS, totally neglect the role played by space and geographical location in regional
convergence processes. However, recent advances in spatial data analysis, such as
spatial econometrics, can take into account both spatial heterogeneity and spatial
dependence, facilitating the consideration of spatial inequalities and enhancing the
reliability of the empirical work (Rey and Janikas, 2005; Wei and Ye, 2009). Spatial
effects have been found to be significant for the EU regions by many studies (e.g.
Fingleton, 1999; Baumont et al., 2003; Carrington, 2003; Badinger et al., 2004; Ertur
et al., 2006; Fischer and Stirböck, 2006; Basile, 2008; Dall’erba et al., 2008;
Chapman et al., 2010).
Spatial effects among regions are typically taken into account via a spatial (n x
n) weight matrix (W) (Anselin and Bera, 1998). This matrix is a formal expression of
spatial adjacency between regions and defines the structure and the intensity of
spatial effects. Each row of the matrix contains nonzero elements for the columns
corresponding to “neighbors”. By convention, the diagonal elements of W are set to
zero, implying that each location is not a neighbor of itself. Usually, spatial weights
matrices—for ease of interpretation—are used in rowstandardized form so that the
sum of the weights for each observation equals one. Several forms of spatial weight
matrices have been suggested and used in the literature (see for example, Anselin
and Bera, 1998; Haining, 2003; Abreu et al., 2005). The specification of weight matrix
is commonly based on three criteria6: simple contiguity, decreasing functions of
distance, and knearest neighbors. The selection is of paramount importance
because the choice might have a substantive effect on the obtained results.
6 Spatial weights must be exogenous to the model (such as geographybased measures, i.e., based
on contiguity or distance) to avoid identification problems (Anselin, 2001).
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Spatial dependence can be normally incorporated into a regression model through a spatial lag (spatial autocorrelation pertains to the dependent variable) or a
spatial error term (spatial autocorrelation is considered in the error term)7 .
Formally, a spatial lag unconditional βconvergence model can be expressed
in matrix notation as:
εβρ +++= 0yWgaSg TT (2)
),0(~ 2 Iσε Ν
where ρ is the coefficient of the spatial lag term Wg measuring how neighboring
observations affect the dependent variable (a spatial autoregressive coefficient), W
defines the (nxn) spatial weight matrix that captures the spatial interaction between
economies and the rest of the notations are the same as before. The spatially lagged
dependent variable is calculated as the weighted average of the values of the
dependent variable of neighboring regions.
A spatial error unconditional βconvergence model can be expressed in matrix
notation as follows:
uWySgT
+=++=
ελεεβα 0 ),0(~ 2INu σ (3)
where λ is a coefficient indicating the extent of spatial correlation between the
residuals and the rest of the notations are the same as before. If appropriate, these
models are estimated by means of maximum likelihood or method of moments
techniques (Anselin and Bera, 1998).
However, in case of spatial heterogeneity (i.e. in case of presence of variation
or instability over space), the results of the aforementioned models might lead to
inappropriate, erroneous and misleading conclusions about convergence process8.
Clearly, spatial heterogeneity is consistent with many geographical processes such
as uneven spatial development (Wei and Ye, 2009). These processes suggest that
economic behaviours might not be stable over space complicating the formulation of
a general statement (Anselin, 1990). In an econometric model, this type of
heterogeneity can be reflected by varying coefficients (structural instability) or by
varying error variances across observations (groupwise heteroskedasticity) or both
(Le Gallo and Dall'erba, 2006).
7 For a taxonomy of spatial econometric models, see Anselin (2003). 8 For an excellent overview of spatial heterogeneity see Ertur and Le Gallo (2009).
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The most known form of structural instability into regional science is the
situation where the estimation parameters of an econometric model correspond to
different spatial “regimes” (Anselin, 1990). In this case, spatial regimes (clubs) are
used to model spatial instability of the parameters and each regime is characterized
by different parameter set. For example, assuming two regimes (clubs), the form of a
spatial error convergence model with structural instability is as follows9:
εββ ++++= 0222201111 yDDayDDagT (4)
uW += ελε and ),0(~ 2Iu uσΝ
where 1D and 2D are dummy variables corresponding to spatial regimes. In this case,
each regime is characterized by a different value for the coefficient.
Another form of spatial heterogeneity, closely related to convergence club, is
known as groupwise heteroskedasticity. This type of heterogeneity means that the
variance across error terms in one regime is different from the variance across error
terms in the other regime. In other words, the assumption of homoskedasticity may
hold within each regime, but not between regimes. Assuming two regimes, the form
of a spatial error model with groupwise heteroskedasticity is shown below:
uWySgT
+=++=
ελεεβα 0 (5)
)0
0,0(~
22,
12,
⎥⎥⎦
⎤
⎢⎢⎣
⎡
Ι
Ι
B
ANuε
ε
σ
σ
where 2εσ represents the variance within each regime.
Finally, a spatial error model with both structural instability and groupwise
heteroskedasticity can be expressed as follows:
εββ ++++= 0222201111 yDDayDDagT (6)
uW += ελε and )0
0,0(~
22,
12,
⎥⎥⎦
⎤
⎢⎢⎣
⎡
Ι
Ι
B
ANuε
ε
σ
σ
In the extreme case, however, of complete spatial heterogeneity – or just in
the case of significant spatial heterogeneity both global and regime (club)
convergence models would be inappropriate because each region might be 9 In order to save space the form of a spatial lag model with structural instability and/or groupwise
heteroskedasticity is not presented here. For an extensive presentation of spatial βconvergence
models see Baumont et al. (2003), Ertur et al. (2006), and Le Gallo and Dall’erba (2005).
8
considered to be completely or highly unique10. In this case, no general statements
can be formulated and an alternative approach is required. The GWR approach
(Brundson et al., 1996, Fotheringham et al., 2002) can provide a reliable basis for
dealing with this issue11. In essence, GWR extends the traditional regression
framework by allowing local rather than global parameters to be estimated. In other
words, instead of calibrating a single regression equation, GWR generates a
separate regression equation for each observation, allowing the estimation of
different convergence rates for each region. Thus, the traditional global regression
model is a special case of the GWR.
Consider a global regression model written as:
∑ ++= ιεkikki xaay 0 GWR extends this conventional regression framework so that the model is
expressed as follows:
∑ ++= ιεkikikii xaay 0
where kia represents the value of ka at point i.
In this model it is assumed that observed data near to point i have more of an
influence in the estimation of the coefficient values of than do data that are further
away from i. Hence, data from observations close to i are weighted more than data
from observations farther away. Each equation is calibrated using a different
weighting of the observations contained in the data set. Due to the fact that there are
more unknown parameters than degrees of freedom, the local regression coefficients
are estimated by weighted least squares (WLS). The GWR estimator is
ywxxwxa it
it 1)( −
∧
= where iw is an n by n matrix whose diagonal elements denote the geographical
weighting of observed data for point i.
Usually, the weighting function selected is a Gaussian one and its form is:
)( 2
2
h
d
ij
ij
eW =
10 For an overview of the possible reasons of spatial variation of parameter estimates see
Fotheringham et al. (1997: 6061). 11 For strengths and weaknesses of GWR see Ali et al., (2007) and Yildirim et. al. (2009).
9
where ijd is the distance between regions and h is a bandwidth that indicates the
extent to which the distances are weighted. Usually, the crossvalidation score or
Akaike information criterion (AIC) test is used to determine the optimal bandwidth
distance or the optimal number of neighboring spatial units 12.
3. Data, Spatial Weight Matrices and Spatial Regimes 3.1 Data
The empirical analysis presented below is based on, disaggregated at the
Nomenclature of Territorial Units for Statistics (NUTS) II spatial level, data derived
from European Regional Database (elaborated by Cambridge Econometrics) and
covers the period 19952005. This dataset is based mainly on information supplied
by REGIO, the Eurostat regional database. The sample includes 257 regions
belonging to 25 European countries (Belgium, Denmark, Germany, Greece, Spain,
France, Ireland, Italy, the Netherlands, Luxemburg, Austria, Portugal, Finland,
Sweden, the United Kingdom, the Czech Republic, Hungary, Poland, Slovakia,
Estonia, Lithuania, Latvia and Slovenia, Bulgaria, and Romania). Cyprus and Malta
are excluded from the sample for statistical reasons, whereas some regions such as
Canary Islands, French Overseas Departments, the Azores Islands, and Madeira are
excluded because of data limitations.
3.2 Spatial Weight Matrices Given the characteristics of European regions sample, the use of different k
nearest neighbor matrices, calculated from the distance between region centroids, is
advisable. These matrices are more appropriate because the islands are connected
to the continent avoiding rows and columns with only zero values. Furthermore,
major methodological problems, arising from the varying number of neighbors, are
avoided (see, for example, Anselin, 2002; Le Gallo and Ertur, 2003; BouayadAgha
et al., 2011). In this matrix form, the critical cut off for each region is determined so
that each region has the same number of neighbors (Haining, 2003:80). This kind of
matrix also has been used for EU regions by, among others, Le Gallo and Ertur
(2003), Le Gallo and Dall’erba (2006), Basile (2008), Battisti and Di Vaio (2008),
12 For a more thorough discussion of these issues, see Brunsdon et al., (1996) and Fotheringham et
al., (2002).
10
Dall’erba et al. (2008), Chapman et al. (2010), Mohl and Hagen (2010), Bouayad
Agha et al. (2011), and Miguélez and Moreno (2011). As a check of robustness of
results, five different weight matrices are defined by setting k to 5, 10, 15, 20, and 25.
However, because results are similar, leading to the same conclusions, only the
results of k10 are presented.
3.3 Spatial Regimes
As mentioned above, the idea of club (regime) convergence is based on
recent models that yield multiple steady state (locally stable) equilibria and classify
economies into different groups with different convergence characteristics. Because
economic theory does not offer much guidance, empirical studies—at both national
and regional levels—use a wide variety of criteria and methods to identify the number
and characteristics of groups or regimes (see, for example, Baumol and Wolff, 1988;
Chatterji 1992; Quah 1993; Durlauf and Johnson 1995; Desdoigts, 1999; Liu and
Stengos, 1999; Hansen, 2000; Corrado et al., 2005; Bartkowska and Riedl, 2012). All
these studies challenge the view that the parameters describing economic growth are
identical across economies, concluding that convergence is not a widespread
phenomenon and has occurred only within small groups of economies. In other
words, these studies have confirmed the club convergence hypothesis suggesting
that the assumption of a single linear growth model that applies to all economies is
simplistic, misleading and invalid.
However, in the context of regional economies where space play a prominent
role (e.g. coreperiphery pattern), clubs, normally, are determined by geographic
criteria using Exploratory Spatial Data Analysis, a useful tool for uncovering spatial
effects and spillovers among spatial entities (Haining 1990; Anselin, 1998). This
approach has been used by numerous studies such as Baumont et al. (2003), Le
Gallo and Dall’erba (2004), Ertur et al. (2006), Fischer and Stirböck (2006), Dall’erba
et al. (2008), Ramajo et al. (2008), and Chapman et al. (2010). All these studies have
highlighted the role of space in regional convergence process and have shown the
presence of spatial regimes among European regions.
As a result, in this paper, the identification of regimes is performed by
employing one type of ESDA tool, the GetisOrd statistic13 (Ord and Getis, 1995).
13 The results of the analysis are not presented here but are available from the authors upon request.
11
Based on this statistic, the sample is divided into two groups of regions depending on
both their geographical location and their level of per capita income. A positive value
denotes a spatial cluster of highvalue observations, whereas a negative one
suggests a spatial cluster of lowvalue observations14. According to this statistic, 157
regions are found to be in the first regime (more advanced regions/core) and 100
regions in the second regime (less advanced regions/periphery). The first regime
includes primarily regions of Central and East Europe as well as Greece, Portugal
and Spain while the second one includes regions from the rest countries.
4. Estimation Results 4.1. Global Convergence The (preliminary) OLS estimation results15 of the global regression model for the
period 19952005 are presented in Table 1. The results confirm unconditional β
convergence hypothesis implying a slow convergence speed (0.80% per annum) and
a long halflife (87 years). However, although the diagnostic tests do not reveal
multicollinearity, they indicate problems with the assumptions of normality,
homoscedasticity and autocorrelation. More specifically, the value of the
multicollinearity condition number is far below the critical range (2030), indicating
that multicollinearity is not a problem. On the other hand, the JarqueBera test is
highly significant, suggesting that OLS errors are not normally distributed16 while
significant heterogeneity is identified by diagnostic tests for heteroskedasticity (i.e.
White test and BreuschPagan/KoenkerBassett test). Furthermore, all diagnostic
tests for spatial autocorrelation17 (i.e., Moran I, LM lag and LM error) reject the null
hypothesis of absence of spatial autocorrelation in the residuals showing evidence of
spatial misspecification18.
14 For further details of this approach see Le Gallo and Dall’erba, (2006). 15 All calculations in this and the following section are performed using Spacestat 1.91. 16 Due to the large sample, this is not an important problem (Fischer and Stirböck, 2006). 17 LMERR is the Lagrange multiplier test for residual spatial autocorrelation, and RLMERR is its
robust version. Similarly, LMLAG is the Lagrange multiplier test lagged for spatially endogenous
variable, and RLMLAG is its robust version. 18 The presence of spatial effects has also been by the Exploratory Spatial Data Analysis (ESDA). The
estimations are available upon request.
12
Table 1. Estimation Results for the global OLS model (19952005)
0.0413 Constant (0.000) 0.0076
Ln of Initial GDP per capita (0.000)
Convergence speed 0.80% Half life 87 Coefficient of Determination 0.210 Akaike (AIC) 1546.2 Schwarz (SC) 1539.5 Multicollinearity (condition number) 6.562
67.21 JarqueBera (Normality)
(0.000) 22.92 BreuschPagan/ KoenkerBassett (heteroskedasticity)
(0.000) 28.06 White (heteroskedasticity)
(0.000) Diagnostics for spatial dependence
8.964 Moran
(0.000) 73.96
LMERR (0.000) 4.371
R LMERR (0.036) 69.66
LMLAG (0.000) 0.066
R LMLAG (0.796)
Note: p values presented in parentheses
As a result, alternative and more appropriate spatial econometric models are
required because, under such conditions, the estimates based on OLS are either
inefficient, or biased, or both (Anselin and Bera, 1998). Following the decision rule19,
suggested by Anselin and Florax (1995), the spatial error model is considered as the
most appropriate specification20.
The regression estimates of spatial error model21 are displayed in the second
column of Table 2. Similar to the OLS estimates, the results confirm unconditional β
convergence hypothesis implying, however, a slower convergence speed (0.68% per
19 The decision rule states that if LMLag is more significant than LMError, and Robust LMLag is
significant, but Robust LMError is not, then the appropriate model is the spatial lag model.
Conversely, if LMError is more significant than LMLag, and Robust LMError is significant, but
Robust LMLag is not, then the appropriate model is the spatial error model. 20 It is worth noting that Moran I statistic cannot discriminate the form of the spatial dependence. 21 The models also are estimated by GMM, but the results are similar.
13
annum) and a longer halflife (102 years). The coefficient λ for the spatial error
variable is positive and highly significant, implying that spillover effects exist between
adjacent regions. Moreover, the significant decrease in the values of information
criteria (i.e Akaike and Schwarz) give evidence of a better fit of this spatial
specification with respect to OLS22. The other diagnostics tests, such as the LR and
Wald test on common factor hypothesis, the LR test on spatial error dependence and
the LMtest on residual spatial lag dependence, do not show signs of
misspecification. However, both BreuschPagan and spatial BreuschPagan tests are
significant indicating that there is still some remaining heteroscedasticity.
4.2. Club Convergence The presence of heteroskedasticity found above can be reflected by varying
coefficients (structural instability) or by varying error variances across observations
(groupwise heteroskedasticity) or both. As a result, firstly, a spatial error model with
structural instability is estimated.
22 The coefficient of determination is inappropriate and meaningless in spatial econometrics. As a
result, Akaike (AIC) and Schwarz (SC) information criteria are computed (Anselin, 1988:246).
14
Table 2. Estimation Results for the ML spatial error model, ML spatial error model with structural instability and ML spatial error model with groupwise heteroskedasticity, (19952005)
Spatial Error Model
Spatial Error Model with structural instability
Spatial Error Model with groupwise heteroskedasticity
Periphery Core
constant 0.0385 0.0375 0.0292 0.0364
(0.000) (0.000) (0.024) (0.000)
Ln of initial GDP per capita 0.0065 0.0047 0.0038 0.0060
(0.000) (0.033) (0.356) (0.000)
Spatial coefficient 0.5539 0.5500 0.5520
(0.000) (0.000) (0.000)
Convergence speed 0.68% 0.48%  0.62%
Halflife 102 143  111
Akaike (AIC) 1614.4 1612.7 1664.6
Schwarz (SC) 1607.6 1598.5 1657.5
BreuschPagan (heteroskedasticity) 59.365 45.56
(0.000) (0.000)
spatial BreuschPagan(heteroskedasticity) 59.365 45.56
(0.000) (0.000)
LR test on spatial error dependence 68.163 67.53
(0.000) (0.000)
LM test on spatial lag dependence 0.0057 1.064
(0.939) (0.302)
15
Spatial Error Model
Spatial Error Model with structural instability
Spatial Error Model with groupwise heteroskedasticity
LR test on common factor hypothesis 15.024 15.348
(1.000) (1.000)
Wald test on common factor hypothesis 0.8639 0.286
(0.352) (0.866)
Chow – Wald test 2.333
(0.311)
Individual stability test (constant) 0.367
(0.544)
Individual stability test (GDP per capita) 0.029
(0.862) 2ˆεσ core 7.005
(0.000) 2ˆεσ periphery 8.743
(0.000)
LR test for groupwise heteroskedasticity 50.00
(0.000)
Note: p values presented in parentheses
16
The estimates of the model are presented in the third column of Table 2.
The results show that although convergence parameters are relatively similar
between regimes, β coefficient is statistically significant only for the regime
relating to the less developed regions. The convergence speed for the regions
belonging to this regime is very slow (0.48%) and the halflife is almost 150
years. Concerning the diagnostics, it is evident that this specification suffers
from severe problems. Firstly, the information criteria (i.e. Akaike and Schwarz)
clearly indicate a worse fit of this spatial model with respect to the spatial error
model. Secondly, heteroskedasticity tests are significant showing evidence of
remaining heteroscedasticity. Finally, the ChowWald test for overall structural
instability does not reject the joint null hypothesis while the individual coefficient
stability tests do not reject the corresponding null hypotheses. All of these
diagnostics reveal major concerns about the appropriateness of this
specification.
As a result, a spatial error model with groupwise heteroskedasticity
should be considered. The estimates of this model are presented in the fourth
column of Table 2. The results show a significant (at the 1% level) and negative
(−0.006) β coefficient, corresponding to a low convergence rate (0.62%) and a
long halflife (111 years). The model performs better than any of the previous
models in terms of information criteria (i.e. Akaike and Schwarz criterion)23.
Furthermore, both the LRtest on groupwise heteroskedasticity and the
estimated variances for two regimes are significant, implying the
appropriateness of this specification.
4.3. Local Convergence
The presence of spatial heterogeneity found in most of the previous
econometric models suggests a more explicit focus on this issue. As a result, a
GWR convergence model is estimated24. Table 3 presents the estimation
23 This model also performs better than any other regression model employed (e.g. spatial error
model with structural instability and groupwise heteroskedasticity, Durbin model, cross
regressive model). The estimates of these models are not presented here but are available from
the authors upon request. 24 All computations in this section are performed by using the GWR 3.0 software package.
17
results of this model25. GWR parameter estimates are described by their
median, minimum, and maximum value as well as interquartile range. The
convergence parameters seem to vary over the entire study area. The extent of
variation is remarkable, as it is evident from the significant difference between
the minimum and the maximum value. The GWR parameters of the initial per
capita GDP lie in the range between 0.058 and 0.228, instead of 0.007 for the
OLS model and 0.006 for the spatial error model with groupwise
heteroskedasticity. In other words, the (local) βcoefficients change signs
implying convergence for some regions and divergence for others.
Clearly, Akaike information criterion gives evidence of a better fit of this
spatial specification with respect to traditional OLS or (global) spatial error
model. However, according to this criterion, the superiority of GWR models,
usually founded in the literature, is not confirmed because the GWR model
performs worst compared to the spatial error model with groupwise
heteroskedasticity26. As a result, the latter specification seems to be the most
appropriate one.
Table 3. Estimation Results for the GWR model (19952005)
Constant (c) GDP per capita Convergence Speed
Minimum 0.0584 0.0677 11.3%
Lower quartile 0.0175 0.0129 1.39%
Median 0.0329 0.0049 0.5%
Upper quartile 0.0554 0.0024 
Maximum 0.2286
0.0455

Global OLS 0.04135 0.0076 0.80%
Akaike OLS 1544 Akaike GWR 1630
R2 OLS 0.20 R2 GWR 0.59
N nearest neighbours: 19 Global test of nonstationarity F: 5.28**
** significant at the 1 % level
25 The Akaike information criterion (AIC) test is used to determine the optimal number of
neighboring spatial units. 26 As it evident from Table 1 and Table 3, the computation of AIC is similar between
Spacestat 1.91 and GWR 3.0.
18
5. Discussions and Conclusions The aim of the paper was to present and compare the estimation results
obtained by using all types of (spatial) βconvergence models that is global,
regime and local. The results of the paper showed that a regime model and
more particularly a spatial error model with groupwise heteroskedasticity seems
to perform better than any other model used for examining the regional
convergence process in the enlarged EU over the period 19952005. Global
models seem to perform worst than any other specification while the superiority
of local (GWR) models, usually founded in the literature, is not confirmed. This
might happen because, typically, the estimates of GWR models are not
compared with the estimates of regime models but only with the estimates of
global models.
As far as regional convergence is concerned, the estimation results
showed a negative and significant β coefficient corresponding to a very low
convergence rate (0.62%) and a very long halflife (111 years). Some of the
results obtained in the paper confirm several wellknown findings. For example,
spatial effects still matter among European regions, and as a result traditional
nonspatial econometric models are inadequate and erroneous. Moreover, the
spatial error specification seems to be the most appropriate for the regions of
the EU (see for example Fingleton and LopezBazo, 2006). Finally, the speed of
convergence is very slow and varies widely among regions implying
convergence for some regions and divergence for others. On the other hand,
however, some of the results found in this study are in variance with findings
obtained by previous similar studies. For example, the speed of convergence is
significantly lower than the speed estimated by other similar studies (e.g.
Fischer and Stirböck, 2006; Ramajo et al., 2008; Chapman et al., 2010).
Furthermore, the spatial error model with structural instability, usually
recognized as the most adequate specification (e.g. Fischer and Stirböck, 2006;
Chapman et al., 2010), seems to perform very poorly.
Undoubtedly, these three types of analysis should not be seen as
competitive, but rather as complementary. Each approach focuses on a
different aspect of convergence process and results in different findings and
conclusions. The choice of the regression type should depend on the aim of the
study (Ali et al., 2007). Sometimes all types of models might be necessary to be
19
performed and compared. In general terms, global (convergence) models are
more popular among economists since they usually try to find general rules and
“laws”. In contrast, local (convergence) models are more popular among
geographers since they typically assume different processes for each area
rather than general rules and global regularities”. In many instances, however,
regime models might be superior to any other type of model, reconciling, in this
way, the two abovementioned diametrically opposing approaches.
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