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© 2013 Nicolaus Copernicus University. All rights reserved.  
http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2013.007  Vol. 13 (2013) 127−143 

Submitted November 8, 2013  ISSN 
Accepted December 30, 2013 1234-3862 

Joanna Górna, Karolina Górna, Elżbieta Szulc* 

Analysis of β-Convergence. From Traditional  
Cross-Section Model to Dynamic Panel Model 

A b s t r a c t. The aim of the paper is to discuss the course of development of methodology of 
economic convergence analyses, which points up the necessity of taking into consideration 
spatial connections among regions in regional growth models. It presents empirical models of 
β-convergence concerning the economic growth of European regions using various methodo-
logical conceptions. In the paper the models offered by spatial econometrics are recommend-
ed. The empirical data refer to per capita GDP across the European Union regions at  
a NUTS-2 level over the period 1995–2009 (annual data). 

K e y w o r d s: economic convergence, spatial effects, connectivity matrix, spatial panel 
models.  

J E L Classification: C52. 

Introduction 
 The empirical analyses of β-convergence described in the growth litera-
ture can be divided into two parts. The first part contains the analyses which 
use cross-sectional data, while the second one refers to the analyses based on 
pooled time series and cross-sectional data. Among other things the works of 
Mankiw, Romer and Weil (1992), Barro and Sala-i-Martin (1995) can be 
found within the classic literature on cross-sectional data models of conver-
gence, while Islam (1995) is a representative for the second part of the 
growth literature. The theoretical framework for these analyses of growth 

                                                 
* Correspondence to: Elżbieta Szulc, Nicolaus Copernicus University, Department of 

Econometrics and Statistics, Gagarina 13A, 87-100 Toruń, Poland, e-mail: eszulc@umk.pl. 



Joanna Górna, Karolina Górna, Elżbieta Szulc 

DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

128

and convergence is the neo-classical Solow-Swan model (Solow, 1956; 
Swan, 1956). The Solow-Swan model also forms the base of many modern 
analyses in this domain. 
 The spatial and spatio-temporal econometrics points out that for more  
precise explanation of economic growth it is necessary to take into consider-
ation spatial connections among economics as the connected economies’ 
income levels are interdependent.  
 Elhorst, Piras and Arbia (2010) emphasize that the hypothesis that the 
relative location of an economy affects economic growth has been under-
pinned by theoretical extensions of the Solow-Swan model and confirmed by 
numerous and vast empirical analyses using mostly cross-sectional data. 
Exemplifying literature is as follows: Le Gallo, Ertur and Baumont (2003), 
López-Bazo, Vayá and Artis (2004), Abreu, de Groot and Florax (2005), 
Rey and Janikas (2005), Arbia (2006), Bode and Rey (2006), Fingleton and 
López-Bazo (2006), Ertur and Koch (2007), Rey and Le Gallo (2009). Up to 
now there have been fewer empirical analyses which support the hypothesis 
using panel data. The examples are: Badinger, Müller and Tondl (2004), 
Elhorst, Piras and Arbia (2010). 
 This paper presents a review of fundamental conceptions of verifying the 
hypothesis of β-convergence, starting with the traditional regional cross-
sectional data model, through the models which include the spatial connec-
tions among the regions and the panel data models without the spatial ef-
fects, concluding on the spatial panel data models. The considerations pre-
sented are a continuation of the previous works by Górna, Górna and Szulc 
(2013, 2014). 
 The contents of the successive sections of the paper are as follows: in 
Section 2 the subject and  range of the investigation are defined as well as 
the aim of the paper and the research hypothesis are formulated. Section 3 
characterizes the data used in the investigation. Section 4 presents the meth-
odology. In this section the theoretical models of β-convergence in formula-
tion of the cross-section regressions and the regressions for the pooled time 
series and cross-sectional data are presented. Moreover, in Section 4 the 
diagnostic tests for verification of the empirical models are pointed out. The 
results of the analysis are presented in Section 5. Recapitulation contains 
final conclusions and indicates further investigations. 

1. Subject and Range of the Investigation 
 The paper concerns the phenomenon of economic convergence. It pre-
sents changes of spatial differentiation of per capita incomes across the Eu-



Analysis of β-Convergence. From Traditional Cross-Section Model to… 

DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

129

ropean Union regions at a NUTS-2 level over the period 1995–2009. The 
question is about validity of economic convergence hypothesis in the area of 
the European countries in the investigated period. In the paper the methods 
of verification of the hypothesis are considered. 
 As a result of the analysis the empirical models of β-convergence con-
cerning the economic growth of the European regions were obtained. 
To achieve it various methodological conceptions, especially the models 
offered by spatial econometrics, have been used. The attention was paid to 
the so-called absolute β-convergence (Baumol, 1986; De Long, 1988; Arbia, 
2006). In the constructed models of convergence apart from per capita GDP 
in the initial (basic) period no additional variable explaining the state of 
economies was taken into consideration. On the so-called conditional β-
convergence in classic version, see e.g. Bal-Domańska (2010, 2011). On the 
contrary Elhorst, Piras and Arbia (2010) is an example of the spatial ap-
proach. 
 The aim of the investigation is to show that the empirical spatial models 
of convergence for cross-sectional data as well as the spatial panel data 
models have better statistical properties then the models which ignore the 
spatial and spatio-temporal connections among the regions. In addition, they 
allow more precise interpretation of the model parameters. The models pre-
sented are used to verify the hypothesis that relative location of a region 
affects the economic growth rate of the region. 

2. Data 
 In the investigation the data on per capita GDP for 261 regions of 27 
European countries were used. The data refer to the period of 1995–2009, 
i.e. to 15 years. They describe the per capita GDP spatial distributions and 
dynamics of incomes in the European Union and come from Eurostat data- 
-base (ec.europa.eu/eurostat/). All calculations were prepared with the use of 
R (versions 2.7.2 and 3.0.1). 
 Figure 1 presents the spatial distribution of per capita GDP values (ex-
pressed by log terms) at the beginning of 1995–2009 period, i.e. in the year 
1995 (Figure 1a) and the analogical distribution in the year 2009 (Figure 1b). 
It can be noticed that the spatial regularities of the distributions did not 
change in the period investigated. However, comparing the spatial trends, 
which have been fitted to the data and presented in Figure 2, it can be seen 
that the surface of the trend for 2009 is flatter than the surface for 1995. This 
fact seems to confirm the supposition that economic convergence of the Eu-
ropean regions occurs in the investigated period.  



Joanna Górna, Karolina Górna, Elżbieta Szulc 

DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

130

 
Figure 1.  Distribution of the per capita GDP in the European regions: (a) in the year 

1995, (b) in the year 2009 

 
Figure 2.  Trend surfaces of per capita GDP for the European regions: (a) in the year 

1995, (b) in the year 2009 

Then, Figure 3 compares the spatial distribution of per capita GDP in the 
year 1995 with the growth rates of GDP across the regions during the period 
1995–2009. This comparison shows that the poorer, at the beginning, regions 
have faster growth rates than the richer ones. Thus, the economic conver-
gence of the regions in the period considered is probable. This conclusion is 
conformable to the one formulated above. Moreover, the next Figure 4 
which presents the surfaces of regions’ per capita GDP (expressed by log 
terms) and of regions’ per capita GDP growth rates (also expressed by log 
terms), seems to substantiate the supposition that the convergence is possible 
as well. The tendencies in Figure 4 in parts a) and b) are inverted. Addition-
ally, Figure 5 presents the average annual growth rate of GDP across the 



Analysis of β-Convergence. From Traditional Cross-Section Model to… 

DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

131

regions of the EU over the investigated period (the map of spatial distribu-
tion of the growth rates and the trend surface of them respectively). 

 
Figure 3.  Distributions of the per capita GDP: (a) in the year 1995, (b) growth rates 

during the period 1995–2009, in the European regions 

 
Figure. 4. Spatial distributions – trend surfaces: (a) of regions’ (log)  per capita GDP 

in the year 1995, (b) of regions’ (log) per capita GDP growth rates during 
the period 1995–2009  



Joanna Górna, Karolina Górna, Elżbieta Szulc 

DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

132

 
Figure 5.  Map and trend surface of average annual growth rate of GDP across the 

European regions 

3. Methodology 
 The values of per capita GDP in the area established are treated as  reali-
zations of spatial stochastic process ( )iZ s , where: [ ]iii yx ,=s – location 
coordinates on the plane, i = 1, 2, …, N – spatial units (regions) or of spatio-
temporal stochastic process ( )tZ i ,s , where i – as above and t = 1, 2, …, T – 
the successive years in the period considered. The cross-sectional data or the 
pooled time series and cross-sectional data are used. Each observation is 
connected with the established location on the plane within some structure of 
connections among the spatial units. The structure is quantified by spatial 
connectivity matrix W. The matrix W has as many rows and columns as 
there are the regions. Each row of the matrix contains non-zero elements in 
columns which correspond to the connected regions (the so-called 
neighbours), according to the established criterion (wij ≠ 0). Furthermore, the 
given region cannot be connected to itself, i.e. it cannot be a neighbour of 
itself, so wij = 0 for all i = j. Thus, the diagonal elements of W are zeros. 
 The successive specifications of β-convergence models are considered. 
The classical model of β-convergence using data in cross-section takes the 
form: 

[ ] .lnln 1
1

ii
i

iT GDP
GDP
GDP

εβα ++=⎥
⎦

⎤
⎢
⎣

⎡
 (1) 



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DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

133

The spatial cross-sectional data model of β-convergence which contains the 
spatially lagged dependent variable (Spatial Lag Model – SLM) can be writ-
ten as follows: 

[ ] .lnlnln
1

1
1

i
j

jT

ij
iji

i

iT

GDP
GDP

wGDP
GDP
GDP

ερβα +
⎥
⎥
⎦

⎤

⎢
⎢
⎣

⎡
++=⎥

⎦

⎤
⎢
⎣

⎡
∑
≠

 (2) 

Model (2) belongs to the class of the spatial autoregressive models (SAR). In 
turn, the cross-section model of β-convergence with the spatially autocorre-
lated error component (Spatial Error Model – SEM) takes the form: 

[ ] ,lnln 1
1

ii
i

iT GDP
GDP
GDP

ηβα ++=⎥
⎦

⎤
⎢
⎣

⎡
   ij

ij
iji w εηλη += ∑

≠

. (3) 

The use of cross-section regressions (1)–(3) for the investigation of econom-
ic convergence is connected with the loss of the information on variability of 
economies in time. This also means that in the analysis the individual econ-
omies’ features are omitted. Besides, the model (1) ignores the spatial con-
nections among the economies which appear important for describing the 
economic growth (see empirical characteristics of models (2)–(3)). 
 Below, the spatial models of β-convergence for pooled time series and 
cross-sectional data (TSCS) are considered. 
 Including into the model TSCS a spatial component leads to the follow-
ing specifications: 

1. The spatial autoregressive model (SAR_pooled) 

[ ] .lnlnln
1

1
1

it
jt

jt

ij
ijit

it

it

GDP
GDP

wGDP
GDP
GDP

ερβα +
⎥
⎥
⎦

⎤

⎢
⎢
⎣

⎡
++=⎥

⎦

⎤
⎢
⎣

⎡

−≠
−

−
∑  (4) 

2. The model with spatial autoregressive residuals (SE_pooled) 

[ ] ,lnln 1
1

itit
it

it GDP
GDP
GDP

ηβα ++=⎥
⎦

⎤
⎢
⎣

⎡
−

−

   .itjt
ij

ijit w εηλη += ∑
≠

 (5) 

 Pooled spatial and temporal data create the so-called panel data. Meth-
odology of the panel data analysis suggests including into the model individ-
ual or/and time effects which can be taken into consideration as fixed – and 
then they can be estimated – or as random becoming the part of the error 
component. 
  As much as in the cross-sectional data growth models also in the panel 
data models more and more frequently the spatial connections among the 



Joanna Górna, Karolina Górna, Elżbieta Szulc 

DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

134

regions are taken into account. It is caused by the opinion that the growth 
rate of any region is connected with the growth rates of its neighbours. 
 The spatial panel models for verification of β-convergence hypothesis 
are as follows: 

1. The spatial autoregressive panel model with individual fixed effects (the 
spatial autoregressive fixed-effects model) (SAR_FE_IND) 

[ ] .lnlnln
1

1
1

it
jt

jt

ij
ijiti

it

it

GDP
GDP

wGDP
GDP
GDP

ερβα +
⎥
⎥
⎦

⎤

⎢
⎢
⎣

⎡
++=⎥

⎦

⎤
⎢
⎣

⎡

−≠
−

−
∑  (6) 

2. The spatial autoregressive panel model with individual and time fixed 
effects (SAR_FE_TWO-WAY) 

[ ] .lnlnln
1

1
1

it
jt

jt

ij
ijitti

it

it

GDP
GDP

wGDP
GDP
GDP

ερβγα +
⎥
⎥
⎦

⎤

⎢
⎢
⎣

⎡
+++=⎥

⎦

⎤
⎢
⎣

⎡

−≠
−

−
∑ (7) 

3. The spatial error panel model with individual fixed effects 
(SE_FE_IND) 

[ ] ,lnln 1
1

ititi
it

it GDP
GDP
GDP

ηβα ++=⎥
⎦

⎤
⎢
⎣

⎡
−

−

   .itjt
ij

ijit w εηλη += ∑
≠

 (8) 

4. The spatial error panel model with individual and time fixed effects 
(SE_FE_TWO-WAY) 

[ ] ,lnln 1
1

ititti
it

it GDP
GDP
GDP

ηβγα +++=⎥
⎦

⎤
⎢
⎣

⎡
−

−

.itjt
ij

ijit w εηλη += ∑
≠

 (9) 

5. The spatial autoregressive panel model with individual random effects 
(SAR_RE_IND) 

[ ] ,lnlnln
1

10
1

it
jt

jt

ij
ijit

it

it

GDP
GDP

wGDP
GDP
GDP

ζρβα +
⎥
⎥
⎦

⎤

⎢
⎢
⎣

⎡
++=⎥

⎦

⎤
⎢
⎣

⎡

−≠
−

−
∑  (10) 

itiit εαζ += , or ittiit εγαζ ++=  (in the case SE_RE_TWO-WAY). 
6. The spatial error panel model with individual random effects 

(SE_RE_IND) 

[ ] ,lnln 10
1

itit
it

it GDP
GDP
GDP

ζβα ++=⎥
⎦

⎤
⎢
⎣

⎡
−

−

 (11) 



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DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

135

itiit ηαζ += , itjt
ij

ijit w εηλη += ∑
≠

, and in the model with individual 

and time random effects (SE_RE_TWO-WAY), ittiit ηγαζ ++= . 

 Economic convergence is said to be confirmed by the data if the esti-
mates of β coefficients in models (1)–(11) are negative and statistically sig-
nificant. Furthermore, if parameters ρ in models: (2), (4), (6), (7), (10) and 
parameters λ in models: (3), (5), (8), (9), (11) are significantly different from 
zero, then in the convergence process the spatial connections among econo-
mies are important, and the hypothesis that the rate of growth of any econo-
my is related to that of its neighbours is confirmed. 
 As a result of explicit including the components of spatial dependence 
into the economic growth model is the evaluation of the convergence phe-
nomena on the ground of β parameter estimated better than in the traditional 
approach. Then the estimate of the parameter will not be influenced by omit-
ting the dependence and it will more precisely reflect the influence of the per 
capita GDP in the basic period on the growth rate of incomes. 
 In order to evaluate the quality of the empirical models in the investiga-
tion the following tools were used: the Moran test (Moran’s I) for spatial 
independence of the residuals, the Lagrange Multiplier tests (LMlag, LMerr) 
and their robust versions (RLMlag, RLMerr) as spatial dependence diagnos-
tics, the Likelihood Ratio test (LR) for testing the significance of the spatial 
dependence, the Breusch-Pagan heteroskedasticity test, the Chow test for 
spatial invariance of β-convergence parameters and for verifying the need of 
including into the spatial panel models the fixed effects, the Lagrange Multi-
plier tests to verify the spatial interactions and the random effects, the 
Hausman test for choosing between the fixed-effects (FE) model and the 
random-effects (RE) model (on the tools see e. g. Arbia, 2006; Millo and 
Piras, 2012; Mutl and Pfaffermayr, 2011; Baltagi et al., 2003; Suchecki (ed.), 
2012). 

5. Results 
 The successive tables presented below contain the information on the 
usefulness of various methodological conceptions expressed  by the model 
specifications presented in Section 4. The part of the information which re-
fers to the models estimated on the ground of cross-sectional data is also 
presented in the works: Górna, Górna and Szulc (2013, 2014), which unlike 
this paper are limited to the β-convergence analysis by using cross-section 
regressions. 



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DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

136

 Table 1 contains the results of estimation and verification of three mod-
els: the linear regression model, i.e. the traditional model without the spatial 
effects, the spatial autoregressive model (SAR) and the spatial error model 
(SE). 

Table 1.  The results of estimation and verification of the cross-sectional data mod-
els of β-convergence 

 Linear regression 
 

Spatial autoregressive 
model 

Spatial error  
model 

Parameters 
α 
 
β 
 
ρ 
 
λ 

 
3.8739 

(0.0000) 
–0.3535 
(0.0000) 

− 
 
− 

 
2.8602 

(0.0000) 
–0.2618 
(0.0000) 
0.2860 

(0.0000) 
− 

 
3.6383 

(0.0000) 
–0.3269 
(0.0000) 

− 
 

0.4548 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.7642 

–166.71 

 
− 

–191.86 

 
− 

–195.33 
Heteroskedasticity 

Breusch-Pagan test 
 

11.0877 
(0.0009) 

 
13.1741 
(0.0003) 

 
4.0292 

(0.0450) 
Autocorrelation of 

residuals 
Moran test 

 
 

5.4531 
(0.0000)

 
 

2.0380 
(0.0416) 

 
 

–0.0946 
(0.0753) 

Spatial  
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
− 
 

26.3535 
(0.0000) 
29.7572 
(0.0000) 

− 
 
− 

 
 

27.1540 
(0.0000) 

− 
 
− 
 

4.0923 
(0.0431) 

− 

 
 

30.6230 
(0.0000) 

− 
 
− 
 
− 
 

7.4940 
(0.02654) 

Speed of convergence 
Half-life 

0.0291 
23.8477 

0.0202 
34.2531 

0.0264 
26.2654 

Note:  numbers in brackets refer to the p-values. 

 Diagnostics for the models considered suggest that the classical model is 
the worst of them. This result is conformable to our anticipation because the 



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DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

137

assumptions of the model, especially the same variance in the space and 
independence across residuals for all regions, are usually unrealistic in prac-
tice. In this case, the Breusch-Pagan statistic is significant, leading to reject-
ing the model assumption of homoskedasticity. In addition, on the basis of 
the Moran’s I test it is necessary to state that the hypothesis of independence 
of the traditional model residuals should be rejected. 
 As the Moran test does not admit an explicit alternative hypothesis op-
posed to the null, the Lagrange Multiplier tests (LM) were used (see Ta-
ble 1). The LM tests for the linear model used consider the spatial lag model 
(spatial autoregressive) and the spatial error model as alternatives (LMlag 
and LMerr, respectively). Table 1 reports the results of using the robust tests 
(RLMlag, in which H0: ρ = 0 under the assumption that λ ≠ 0 and RLMerr, 
where H0: λ = 0 under the assumption that ρ ≠ 0) as well. Subsequently, the 
significance of the spatial effects in SAR and SE models using the Likeli-
hood Ratio test (LR) was confirmed. 
 Taking into account the spatial connections across the European regions 
at a NUTS-2 level over the period 1995–2009, in the β-convergence models 
has removed the problem of autocorrelation of the residuals (at the level of 
significance γ = 0.01). However, the problem of variance heteroskedasticity 
has remained, especially in the case of the spatial autoregressive model. The 
spatial heteroskedasticity can be caused by omitting the factor responsible 
for systematic spatial variability. In this connection in Górna, Górna and 
Szulc (2014) the additional analysis searching for the spatial regimes was 
performed. For this purpose the considered area of the regions has been di-
vided into two sub-areas (see Figure 6). To justify the division the Chow test 
for verifying the spatial changeability of β parameters was used (see also 
Arbia, 2006, p. 133). Table 2 contains the results of the test. 

Table 2. Results of the test for spatial invariance of the β-convergence parameters 
Chow test Linear regression Spatial lag model Spatial error model 

Values of test 8.2701 12.1347 18.9393 
p-value 0.0407 0.0069 0.0003 

 The hypothesis of constancy of parameters in β-convergence models 
estimated in the cited investigation should be rejected. This leads to identifi-
cation of spatial regimes and means that in the considered area as a whole 
there are differentials in relationships between regional growth rate and ini-
tial per capita GDP. The models of β-convergence according to the sub-areas 
considered were estimated and verified and the differentiation of the parame-



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DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

138

ters of β-convergence as well as of properties of the empirical models ob-
tained were stated (for the details, see Górna, Górna and Szulc, 2014). 

 
Figure 6. Classification of the EU regions within the two spatial regimes 

 The classical model estimated with the use of the pooled time series and 
cross-sectional data like the classical cross-section regression does not satis-
fy the fundamental criteria of statistical verification. 
 For the purpose of spatial effects verification in the models for pooled 
time series and cross-sectional data the Lagrange Multiplier tests analogical 
to those applied in the case of the cross-section regressions are used. The 
results of them have confirmed the need of re-specifications towards the 
models with the spatial effects (see Table 3). Moreover, the significance of 
the effects with the aid of the LR test was confirmed. 
 According to the methodology of panel data modeling in the investiga-
tion the reasonableness of including into the spatial models the fixed or/and 
the random effects was considered. For this purpose the Chow test (the spa-
tial model for pooled TSCS data vs. the spatial panel model with fixed ef-
fects) and the LM tests (to verify spatial interactions and random effects) 
were used. The results of the Chow test  have pointed out the statistical sig-
nificance of the fixed effects in the spatial autoregressive panel model but 
not in the panel spatial error model. On the basis of the joint LM test, which 
tested the hypothesis: H0: λ = 2

ασ = 0 under the alternative that at least one 
component was not zero, it was stated that the null should be rejected. 



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139

Table 3.  Results of estimation and verification of β-convergence models for pooled 
time series and cross-sectional data 

 Linear regression Spatial autoregressive 
model 

Spatial error  
model 

Parameters 
α 
 
β 
 
ρ 
 
λ 

 
0.3558 

(0.0000) 
–0.0326 
(0.0000) 

− 
 
− 

 
0.3132 

(0.0000) 
–0.0299 
(0.0000) 
0.4044 

(0.0000) 
− 

 
0.3334 

(0.0000) 
–0.0303 
(0.0000) 

− 
 

0.4250 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.1439 
–9442 

 
− 

–9974.6 

 
− 

–9981.7 
Heteroskedasticity 
Breuch-Pagan test 

 
184.5378 
(0.0000) 

 
243.7477 
(0.0000) 

 
250.5849 
(0.0000) 

Autocorrelation of 
residuals 

 Moran test 

 
 

26.7685 
(0.0000)

 
 

–2.1504 
(0.0158) 

 
 

–3.333 
(0.0304) 

Spatial  
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
− 
 

702.5570 
(0.0000) 
713.3021 
(0.0000) 

− 
 
− 

 
 

534.61 
(0.0000) 

− 
 
− 
 

12.5044 
(0.0004) 

− 

 
 

541.73 
(0.0000) 

− 
 
− 
 
− 
 

23.2494 
(0.0000) 

Speed of convergence 
Half-life 

0.0331 
20.82 

0.0304 
22.73 

0.0308 
22.43 

Note:  numbers in brackets refer to the p-values. 

 Next, in the investigation the so-called marginal LM tests for verification 
of the hypotheses: H0: λ = 0 (under the assumption that 2

ασ = 0) and H0: 2
ασ =0 

(assuming λ = 0), were used respectively. In both of the cases the result of 
the test has pointed out the lack of the bases for rejecting the null. Therefore, 
the conditional LM tests, which are more useful tests in this framework, 
because they test for one effect, and are robust against the other (H0: λ = 0 



Joanna Górna, Karolina Górna, Elżbieta Szulc 

DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

140

assuming 2
ασ = 0 or 2

ασ ≠ 0; H0: 2
ασ = 0 with λ = 0 or λ ≠ 0), confirmed that 

the interactions of spatial and random effects were possible. 
 The Hausman test was used to choose between the FE and RE models. It 
has suggested the choice of the spatial panel model with the fixed effects. 

Table 4. Selected characteristics of spatial panel models 

Parameter 
SAR_FE_IND SAR_RE_IND SE_FE_IND SE_RE_IND 

Estimate of 
parameter 

Statistic 
t 

Estimate of 
parameter 

Statistic 
t 

Estimate of 
parameter 

Statistic 
t 

Estimate of 
parameter 

Statistic 
t 

α 
β 
ρ 
λ 

− 
–0.064 

 0.340 
− 

− 
–16.08 
19.42 
− 

0.319 
–0.030 
0.344 
− 

26.23 
–24.10 
20.24 
− 

− 
–0.083 
− 

0.349 

− 
–16.13 
− 

19.62 

0.335 
–0.030 
− 

0.359 

26.57 
–23.50 
− 

19.60 
Speed of 

convergence 
Half-life 

 
0.0658 
10.4832 

 
0.0308 
22.4255 

 
0.0865 
7.9733 

 
0.0309 
22.3506 

Chow test F 121.4659 
(0.0000) 

 
 

–331.9926 
(1.0000)  

Hausman test 74.5562 
(0.0000) 

LM tests 
Joint LM (Random effects and spatial autocorrelation as alternative hypothesis): 460.0374; 
p-value=0.0000 
Marginal LM: 
LM1 (Random effects as alternative hypothesis):  –0.0005; p-value =1 
LM2 (Spatial autocorrelation as alternative hypothesis): 0.0064; p-value=0.9949 
Conditional LM (assuming 02 ≥ασ ; Spatial autocorrelation as alternative hypothesis): 24.5265; 
p-value=0.0000 
Note:  numbers in brackets refer to the p-values. 

Conclusions 
 On the ground of the analysis the following conclusions can be formulat-
ed: 
1. The fact of including the components of spatial dependence (the spatially 

lagged dependent variable or the spatial autoregressive error component) 
in the economic convergence model is justified and valid for the analyses 
of per capita incomes across the regions investigated and of income 
changes in time. 

2. As a result of that it is possible to define the influence of the neighbour 
connections on the economic growth, the estimate of parameter β is more 
precise, and properties of the model are better. 



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DYNAMIC ECONOMETRIC MODELS 3 (2013) 127–143 

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3. Like in the growth models for the cross-sectional data in the panel data  
models it is necessary to take into consideration the spatial connections 
among the regions. 

4. The dynamic panel models with the spatial effects are the natural exten-
sion of cross-section regressions as the tools of verifying hypothesis of 
economic convergence. 
In further investigations the models of convergence with the additional 

explanatory variables and also with these variables spatially lagged are to be 
considered. Additionally, more diagnostic tests for the spatial panel models 
are expected to be used.  

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Analiza β-konwergencji. Od tradycyjnego modelu przekrojowego do 
dynamicznego modelu panelowego 

Z a r y s  t r e ś c i. Celem artykułu jest omówienie kierunku rozwoju metodologii badania 
konwergencji gospodarczej, wskazującego na potrzebę uwzględniania w modelach wzrostu 
regionów powiązań przestrzennych między nimi. Artykuł prezentuje empiryczne modele  
β-konwergencji dotyczące wzrostu gospodarczego regionów Europy, otrzymane przy wyko-
rzystaniu różnych koncepcji metodologicznych. W artykule rekomenduje się modele z zakre-
su ekonometrii przestrzennej. Dane empiryczne dotyczą PKB per capita w regionach NUTS-2 
27 państw europejskich, będących członkami Unii Europejskiej. Zakres czasowy analizy 
obejmuje lata 1995–2009 (dane roczne).  

S ł o w a  k l u c z o w e: konwergencja gospodarcza, efekty przestrzenne, macierz sąsiedztwa, 
przestrzenny model panelowy.  






