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© 2015 Nicolaus Copernicus University Press. 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.2015.001  Vol. 15 (2015) 5−26 

Submitted September 11, 2015  ISSN (online) 2450-7067 
Accepted December 15, 2015 ISSN (print) 1234-3862 

Elżbieta Szulc*, Dagna Wleklińska 

Spatio-temporal Analysis of Convergence  
of Development Level of Selected Stock Exchanges  

in the Period of 2004–2012 

A b s t r a c t. The paper concerns the convergence of selected stock exchanges from the point 
of view of their development. It presents the methodological approach which points up taking 
into account spatial and economic connections among stock markets in convergence analyses. 
In this analysis the need for division of the stock exchanges according to a spatial regimes is 
pointed up as well. The research includes 42 largest trading floors analyzed in the period of 
2004-2012. The empirical data refer to six diagnostic variables acknowledged as the im-
portant determinants of the development of stock markets. 

K e y w o r d s: stock exchanges, convergence, spatial regimes, physical and economic dis-
tance, connectivity matrix, spatial panel models.  

J E L Classification: C10, C12, C58, G15. 

Introduction 

 The paper concerns the convergence of selected stock exchanges, with 
European stock exchanges on the one hand and the Asian and American on 
the other, from the point of view of their development. The study is a con-
tinuation of our previous investigation, the results of which were published 
in Dynamic Econometric Models, 2014 (14) (Szulc et al., 2014, pp. 125– 
–144). The main findings of the quoted work were as follows: (1) Including 
the linkages that result from physical and/or economic distance between 

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

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



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

6

stock exchanges in their convergence models is justified and crucial for the 
analysis of this phenomenon. (2) As a result, it is possible to define the in-
fluence of the distance between exchanges on their economic development, 
the estimates of convergence parameter are more precise, and some statisti-
cal properties of the models are better. (3) Due to the heteroskedasticity, the 
empirical panel models for the exchanges investigated as a whole were not 
entirely satisfactory. It means that there are differentials in relationship  
between objects considered and their speed of convergence. (4) In some 
empirical models which we obtained there appeared the problem of autocor-
relation of residuals. 
 Motivated by the desire to improve the properties of the empirical mod-
els, firstly we decided to establish some spatial regimes and then repeat the 
research with the division of stock exchanges. The applied categorization 
involved placing European stock exchanges on one side, and the American 
and Asian stock markets on the other. The validity of this choice was con-
firmed by the results of the Chow test on spatial variability of the model 
parameters (Arbia 2006, p. 133) presented in Section 4 of this paper. 
 The discussion on the convergence of stock exchanges is associated with 
one of the directions of the analysis of the relationship between capital mar-
kets, which searches out the ever-increasing convergence of these markets 
from the point of view of their specific characteristics. This process, which 
can be referred to as convergence of stock exchanges is associated with an 
integration of the financial markets, and their growing interdependence, 
which in turn is associated with the liberalization of capital flows and tech-
nological progress. These processes are favorable for further development of 
stock markets, and thus the distinctions between them are becoming increas-
ingly blurred over time. 
 The problem of convergence of equity markets has already been consid-
ered in the literature on the subject, e.g. Aspergis, Christou, Miller (2014), 
Koralun-Bereźnicka (2008), Fraser, Helliar, Power (1994), Caporale, 
Erdogan, Kuzin (2009). In particular, the papers presenting the analysis of 
convergence of the stock exchanges with the prospect of space are the most 
interesting, e.g. Asgharian, Hess, Liu (2013), Suchecka, Łaszkiewicz (2011), 
Wójcik (2009). The literature indicates the validity of the analysis of the 
relationship between capital markets having regard on their location in the 
geographic space, and also takes into account the so-called economic dis-
tance between these markets. This paper refers to such a methodological 
approach. 
 The aim of the paper is to investigate whether, in the light of the current 
empirical analyses, one may observe the process of convergence of the main 



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

7

European as well as American and Asian stock markets. In addition, the 
importance of distance between the markets for the process is evaluated. 
Particularly, the role of economic distance is considered. The research is 
aimed to verify the hypothesis that the relative location of a stock market in 
the geographic as well as a particular economic space affects its growth rate.  
 The achievement of the objective and verification of the research hy-
pothesis was enabled through: defining a taxonomic measure of development 
of exchanges, then building of empirical dynamic models of stock exchang-
es’ convergence for pooled time series and cross-sectional data and for panel 
data in the traditional version, which ignore spatial and economic linkages 
between stock exchanges and dynamic spatial models (including spatial pan-
el models), i.e. models with regard to the described relationship, and finally 
the comparison of statistical properties and the interpretation values of mod-
el’ parameters in various versions. 

1. Subject and Range of the Investigation 

 The subject of the investigation contains the selected European, Asian 
and American stock exchanges, characterized in terms of their level of de-
velopment. The study included 42 largest trading floors in the period of 
2004–2012. The specification of the exchanges with the assignment to the 
relevant country is presented in Table 1. 
 The level of stock exchange development was defined by a synthetic 
measure based on six diagnostic variables, i.e. X1 − the capitalization of 
domestic shares, X2 − the capitalization of newly listed domestic shares,  
X3 − the total value of share trading, X4 − GDP per capita, X5 − the top 10 
most heavily capitalized domestic companies, X6 − the ratio of market capi-
talization to GDP. It was recognized that, in the light of theory and empirical 
analyses, the specified variables are important determinants of the develop-
ment of stock exchanges (see e.g. Demirgur-Kunt and Levine, 1996; Levine 
and Zelvos, 1996; Łuniewska and Tarczyński, 2006; Szulc et al., 2014; 
Wiśniewski, 2003). Taking into account the connections of the capital mar-
ket with the economy of the country of its location was also an important 
issue for the specification of the diagnostic variables. The range of infor-
mation provided by the World Federation of Exchanges (www.world-
exchanges.org), whence we get the data, played a significant role as well. 
 
 
 



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

8

Table 1. Specification of the stock exchanges considered 

North.South America and Asia 

Brazil BM&BOVESPA (BOV) Chile Santiago SE (SSE) 

Canada TMX Group (TMX) Colombia Colombia SE (CSE) 

Mexico 
Mexican Exchange 

(BMV) 
Bermuda Bermuda SE (BSX) 

Argentina 
Buenos Aires SE 

(BCBA) 
Peru Lima SE (BVL) 

United States 

Nasdaq OMX 
(NASDAQ) 

Nyse Euronext (US) 
(NYSE) 

Singapore Singapore SE (SE) 

Thailand Thailand SE (THSE) Philippines Phillippine SE (PSE) 

China 

Honk Kong SE (SEHK) 
Shanghai SE (SHSE) 
Shenzen SE (SZSE) 

Taiwan SE Corp. 
(TSEC) 

Japan 
Osaca SE (OSE) 
Tokyo SE Group 

(TSE) 

India 
National SE India 

(NSE) 
Indonesia Indonesia SE (ISE) 

South Korea Korea Exchange (KRX) Malaysia Bursa Malaysia (BM) 

Sri Lanka Colombo SE (CLSE)   

Europe 

Austrian Wiener Borse (AG) Cyprus Cyprus SE (CPSE) 

Egypt 
Cairo&Alexandria SE 

(EGX)* 
Greece 

Athens Exchange 
(ATHEX) 

Spain 
BME Spanish Ex-

change (BME) 
Netherlands 

Nyse Euronext 
(Europe) (NEE) 

Turkey Istanbul SE (ISSE) Ireland Irish SE (IRSE) 

Israel Tel Aviv SE (TASE)* Luxemburg 
Luxemburg SE 

(LXSE) 

Malta Malta SE (MSE) Hungary Budapest SE (BDSE) 

Germany Deutsche Borse (DB) Norway Oslo Bors (OBE) 

Poland Warsaw SE (WSE) Great Britain London SE (LSE) 

Switzerland 
SIX Swiss Exchange 

(SIX) 
Sweden 

Nasdaq OMX Nordic 
Exchange (NOMX) 

Note: Stock exchanges signed by * have been included in the European capital markets for the reason of 
their geographical proximity to the continent and their economic similarity as well. 

2. Methodology 

 The research was conducted in relation to the aggregate characteristic of 
the stock exchanges in the form of a taxonomic measure of development. 
This indicator is understood as a synthetic normalized formula expressed by 
(see Hellwig, 1968): 



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

9

,
2

1'

q

i
i Sq

q
q

+
−=  (1) 

where: 

iq  − the synthetic variable determining the level of development of the i-th 
exchange in relations to a development standard, 
q  − the average value of the synthetic variable, 

qS  − the standard deviation of the variable. 

In this approach the values of the synthetic variable iq  are calculated ac-
cording to the formula: 

( ) ,
1

2
0∑

=
−=

m

j
jiji zzq  (2) 

where: 

ijz  − the value of j-th diagnostic variable for i-th exchange standardized to 

0–1, 

jz0  − the value of j-th diagnostic variable for the standard of development 

standardized to 0–1.  
Thus, iq means a distance between i-th exchange and the development stand-
ard. 
 Through the use of the taxonomic measure of stock exchanges’ devel-
opment it is possible to present the rankings of exchanges and their changes 
in time, the evaluation of the correlation between stock exchanges in terms 
of development, the identification of linkages between markets in an eco-
nomic space, and finally, the analysis of the stock exchanges’ convergence, 
which is meant as equalizing their development levels. In this paper we fo-
cus on the examination of the concept of exchanges’ convergence, in the 
light of which the stock market with an initial lower level of development 
showed a faster growth rate in the considered measure of development.  
 The analysis of the stock exchanges' convergence was based on econo-
metric models of β-convergence, in particular, the spatial models for pooled 
time series and cross-sectional data (TSCS) as well as spatial panel models. 
The same classes of models were used previously (Szulc et al., 2014).The 
premises for the application of spatial models are as follows: 

− The result of including the spatial linkages between stock exchanges in 
the models of their convergence is better evaluation of the convergence 
phenomenon on the grounds of β parameter estimated. The estimate of 



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

10 

the parameter reflects more accurately the impact of the base level of de-
velopment of a given stock exchange on the growth rate of the stock  
exchange characteristics in question because such estimate is not influ-
enced by omitting spatial relationship. 

− The use of spatial models provides the opportunity to measure and inter-
pret the impact of connections of a given stock exchange with others on 
its development. 

The spatial models for pooled time series and cross-sectional data (TSCS) 
are presented by formulas 3 and 4, whereas the formulas 5 and 6 refer to the 
spatial panel models. 
 The model TSCS with spatial component takes the form of the spatial 
autoregressive model (SAR_pooled), i.e.: 

[ ] ,lnlnln
'

1

'
'

1'
1

'

it
jt

jt

ij
ijit

it

it

q

q
wq

q

q ερβα +











++=









−≠
−

−
∑   (3) 

or of the model with spatial autoregressive residuals (SE_pooled), i.e.: 

[ ] ,lnln '
1'

1

'

itit
it

it q
q

q ηβα ++=







−

−

 .itjt
ij

ijit w εηλη += ∑
≠

  (4) 

The spatial panel models used in the investigation were as follows: 

[ ] ,lnlnln
'

1

'
'

1'
1

'

it
jt

jt

ij
ijiti

it

it

q

q
wq

q

q ερβα +











++=









−≠
−

−
∑   (5) 

i.e. the spatial autoregressive panel model with individual fixed effects (the 
spatial autoregressive fixed-effect model) (SAR_FE_IND) and 

[ ] ,lnln '
1'

1

'

ititi
it

it q
q

q ηβα ++=







−

−

 ,itjt
ij

ijit w εηλη += ∑
≠

  (6) 

i.e. the spatial error panel model with individual fixed effects (SE_FE_IND). 
 Elements wij in the formulas (3)–(6) come from connectivity matrix W 
which refers to the linkages between exchanges considered. Various types of 
weights wij may be pointed out according to the established criteria (see e.g. 
Haining, 2005, pp. 83–84). 
 In this paper the linkages between stock exchanges will be defined with 
the use of two approaches. The first uses a matrix of connections with 
weights established on the basis of the physical distance between the centers 
of the countries where the stock exchanges are located. The second consists 



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

11

in the consideration of the economic distance in the matrix of connections. 
The essence of the second approach is to establish similarity of the exchang-
es on the basis of the value of the taxonomic measure of exchanges’ devel-
opment. 
 The quantification of the spatial linkages between stock exchanges on 
the basis of the geographical distance was carried out according to the fol-
lowing scheme: 
1. Determining the spatial relationships using the linkages matrix S, with 

elements: 








=

≠
=

,if0,

if,
1

ki

ki
ds

ikik  (7) 

where: ikd − the physical distance between capitals of the countries where 
the i-th and the k-th stock exchanges are located. 

2. Row standardization of the connectivity matrix to one, i.e.: 

.

1
∑

=

=
N

k
ik

ik
ik

s

s
w  (8) 

3. Construction of the block matrix of connections, i.e.: 

,

9

2

1



















=

W00

0W0

00W

W

L

MOMM

L

L

 (9) 

where: 921 ... WWW ===  – matrixes of the spatial connections based on 
the physical distance, the same for all the considered years. 
 In the second approach, i.e. with the use of an economic distance be-
tween stock exchanges, the following scheme was used: 

1. Determining the linkages between stock exchanges with the use of an 
economic distance, expressed by the formula: 

( ) ,
1

2∑
=

−=
m

j
kjijik zzd  (10) 

where: ikd − the economic distance between i-th and k-th stock exchange, 



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

12 

kjij zz ,  − the values of standardized diagnostic variables for each i-th and  

k-th stock exchange, 
j = 1, 2, …, 6 − the number of the diagnostic variable. 

2. Construction of the matrix of linkages between stock exchanges, with 
elements: 








=

≠
=

.if0,

if,
1

ki

ki
dw

ikik      (11)    

3. Row standardization of the connectivity matrix to one, i.e.: 

.

1
∑

=

∗ =
N

k
ik

ik
ik

w

w
w  (12) 

4. Construction of the block matrix of cross-sectional and time connec-
tions which may be described in the following form: 

,

9

2

1





















=

∗

∗

∗

∗

W00

0

0W0

00W

W

L

MOM

L

L

 (13) 

where: 921 ... ∗∗∗ ≠≠≠ WWW  – matrixes of connections, taking into account 
the economic distance between exchanges, different for successive years. 
 In order to evaluate the quality of the empirical models in the investiga-
tion the following tools were used: the Moran test for verifying spatial inde-
pendence of the residuals, the Lagrange Multiplier tests (LMlag, LMerr) and 
their robust versions (RLMlag, RLMerr) as spatial dependence diagnostics, 
the Likelihood Ratio test (LR) for testing the significance of the spatial de-
pendence, the Breusch-Pagan heteroskedasticity test, the Chow test for veri-
fying the spatial changeability of β parameters and the need for including 
fixed effects in the spatial panel models (on the tools see e.g. Arbia, 2006; 
Millo and Piras, 2012; Mutl and Pfaffermayr, 2011; Baltagi et al., 2003; 
Suchecki (ed.), 2012). 
 All calculations were performed with R (version 3.0.1) and the graphical 
illustrations – with the use of MapViever and Corel. 



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

13

3. Preliminary Data Analysis 

 Figure 1 shows locations of the investigated exchanges on the world map 
and bar charts of taxonomic measure of development (TMD) in the years 
2004–2012. This presentation allows us to observe changes in the level of 
development of the individual stock exchanges and a comparison of the dy-
namics of changes by their spatial location as well. It is worth noting that 
most of the developing economies’ stock exchanges, both on the European 
continent, as well as American and Asian, are characterized by a relatively 
stable level of the taxonomic measure of development throughout the whole 
adopted time horizon, even during a sharp slowdown in the economic condi-
tions caused by the global financial crisis. This finding is particularly evident 
in relation to such exchanges as e.g. BCBA, BVL, SSE, BDSE, EEZ, CLSE, 
ISE, PSE. 

 
 Figure 1. Bar charts of TMD for the investigated stock exchanges in the years 

2004–2012 

 Figures 2 and 3 show the value of the taxonomic measure of develop-
ment (surface of the wheel) for each stock exchange included in the study for 
the year 2004 and 2012, respectively. This graphical presentation is useful 
for a preliminary assessment of changes in the global capital market over the 
considered period. In 2004, two dominant financial centers are clearly visi-
ble. In the west, it is NYSE and NASDAQ, while in central Europe, the 
London Stock Exchange and NYSE Euronext Europe stand out in particular. 
In turn, in 2012 a slight strengthening of the position of the two largest US 
stock exchanges: NYSE and NASDAQ may be observed. However, the most 
spectacular changes can be seen in the case of the NOMX Central European 
stock exchange. NOMX has strengthened at the expense of two neighboring 



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

14 

stock exchanges LSE and NEE, gaining a leading position in 2012 and clear-
ly outperforming their level of development. With regard to the second 
group of the analyzed exchanges, there were no significant changes in the 
values of the synthetic measure of development. Therefore NYSE and 
NASDAQ are again placed in the dominant position of the ranking. 

 
Figure 2. The taxonomic measure of stock exchanges’ development in 2004 

 
Figure 3. The taxonomic measure of stock exchanges’ development in 2012 

4. Results of the Econometric Analysis 

In order to justify the division of the considered stock exchanges into two 
groups the Chow test of spatial changeability of β parameters was  applied. 
The results are presented in Table 2. The hypothesis that parameters in  
β-convergence models estimated in the investigation are constant should be 
rejected. This leads to the identification of spatial regimes and means that the 



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

15

convergence of the European and Asian/American stock markets should be 
investigated separately. 

Table 2. Results of the tests for spatial invariance of the β-convergence parameters 

Models Linear regression 

Spatial autoregressive 
model 

Spatial error 
model 

Variant I Variant II Variant I Variant II 

Values of Chow test 210.709 113.517 117.036 113.557 121.783 
p-value 0.0000 0.0000 0.0000 0.0000 0.0000 

 The successive tables presented below contain information on the use-
fulness of various methodological concepts expressed by the spatial models 
presented in Section 2, in comparison with the linear regression model, i.e. 
the traditional model not including the spatial effects. Tables 3–6 refer to the 
empirical models obtained for the European stock exchanges, and tables 7– 
–10 for the Asian and American stock exchanges. 
 In Tables 3 and 4 there are presented the results of the estimation and 
verification of the three models for pooled time series and cross-sectional 
data: the linear regression model (TSCS), the spatial autoregressive model 
(SAR_pooled) and the spatial error model (SE_pooled). Table 3 contains the 
results obtained in the case when, for the purpose of quantification of the 
connections among the investigated exchanges, the matrix W of the physical 
distance between them was used (variant I). Table 4 presents the analogical 
results, but in the spatial models there was used the connectivity matrix W* 
of the economic distance between the exchanges (variant II). 
 The classical model estimated with the use of the pooled time series and 
cross-sectional data does not satisfy the fundamental criteria of statistical 
verification. The main drawback of this model is autocorrelation of residu-
als, which is confirmed by the result of the Moran test (see Tables 3 and 4). 
 In order to propose an alternative opposed to the classical model the 
Lagrange Multiplier tests (LM) were used (see Tables 3 and 4). The LM 
tests for the linear model for the pooled time series and cross-sectional data 
used consider the spatial lag model (spatial autoregressive) and the spatial 
error model as alternatives (LMlag and LMerr, respectively). Tables 3 and 4 
report 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 assump-
tion that ρ ≠ 0) as well. 
 



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

16 

Table 3. Results of the estimation and verification of β-convergence models for 
pooled time series and cross-sectional data, obtained for European stock 
exchanges – variant I 

 Linear regression Spatial autoregressive 
model 

Spatial error 
model 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–0.2087 
(0.0055) 
–0.1356 
(0.0007) 

– 
 
– 

 
–0.1316 
(0.0275) 
–0.0789 
(0.0128) 
0.6779 

(0.0000) 
– 

 
–0.0533 
(0.4338) 
–0.0527 
(0.0704) 

– 
 

0.6964 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.0711 

–60.5170 

 
– 

–109.7800 

 
– 

–107.0800 

Heteroskedasticity 
Breuch-Pagan test 

 
3.2874 

(0.0698) 

 
2.2977 

(0.1296) 

 
2.7537 

(0.0970) 

Autocorrelation of 
residuals 

Moran test 

 
11.9203 
(0.0000) 

 
–1.1350 
(0.1282) 

 
–0.6776 
(0.2490) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

145.0560 
(0.0000) 
116.1595 
(0.0000) 

– 
 

 – 

 
 

51.2580 
(0.0000) 

– 
 
– 
 

35.9081 
(0.0000) 

– 

 
 

48.5600 
(0.0000) 

– 
 
– 
 
– 
 

7.0116 
(0.0081) 

Speed of convergence 
Half-life 

0.0182 
38,05 

0.0103 
67.47 

0.0068 
102.42 

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

Since the LMlag tests are more significant than the LMerr, and the RLMlag 
tests are more significant than the RLMerr, the spatial lag models should be 
preferred. Subsequently, the significance of the spatial effects in the SAR 
and SE models using the Likelihood Ratio test (LR) was confirmed (see 
Tables 3 and 4). The results show that the statistical properties of the ob-
tained empirical models are the same (the spatial autocorrelation of residuals 
of linear regression model, significant LM statistics and significant spatial 



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

17

effects confirmed by the LR test), irrespective of which connectivity matrix 
(of physical or of economic distance) was used in the spatial models. 

Table 4. Results of the estimation and verification of β-convergence models for 
pooled time series and cross-sectional data, obtained for European stock 
exchanges – variant II 

 Linear regression Spatial autoregressive 
model 

Spatial error 
model 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–0.2087 
(0.0055) 
–0.1356 
(0.0007) 

– 
 
– 

 
–0.1219 
(0.0336) 
–0.0747 
(0.0144) 
0.7442 

(0.0000) 
– 

 
–0.0638 
(0.4084) 
–0.0800 

(0.02534) 
– 
 

0.7920 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.0711 

–60.5170 

 
– 

–118.4300 

 
– 

–117.5600 

Heteroskedasticity 
Breuch-Pagan test 

 
3.2874 

(0.0698) 

 
2.2312 

(0.1353) 

 
2.1093 

(0.1464) 

Autocorrelation of 
residuals 

Moran test 

 
12.9587 
(0.0000) 

 
1.4475 

(0.0739) 

 
1.7664 

(0.0387) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

156.2542 
(0.0000) 
137.3223 
(0.0000) 

− 
 

− 
 

 
 

59.9100 
(0.0000) 

– 
 
– 
 

27.8846 
(0.0000) 

– 

 
 

59.0380 
(0.0000) 

– 
 
– 
 
– 
 

8.9527 
(0.0028) 

Speed of convergence 
Half-life 

0.0182 
38.05 

0.0097 
71.42 

0.0104 
66.50 

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

 Similarly, irrespective of which the connectivity matrix was applied in 
the spatial models, parameters ρ and λ are statistically significant. It is worth 
noting that the fact of including the connectivity matrixes in the considered 
models has a crucial impact on convergence parameters (β). Absolute values 



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

18 

of the parameters for the SAR and SE models are lower than for the tradi-
tional model which does not take into account the connections across the 
investigated stock exchanges. 

Table 5. Results of the estimation and verification of panel models with fixed effects 
obtained for the European stock exchanges – variant I 

 FE_IND SAR_FE_IND SE_FE_IND 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–1.6250 
(0.0000) 
–0.8927 
(0.0000) 

– 
 
– 

 
–1.2397 
(0.0000) 
–0.6785 
(0.0000) 
0.3543 

(0.0006) 
– 

 
–1.5821 
(0.0000) 
–0.8696 
(0.0000) 

– 
 

0.5218 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.4375 

–117.1200 

 
– 

–127.0400 

 
– 

–132.3600 

Heteroskedasticity 
Breuch-Pagan test 

 
30.4129 
(0.0336) 

 
27.5709 
(0.0689) 

 
28.4229 
(0.0559) 

Autocorrelation of 
residuals 

Moran test 

 
7.0928 

(0.0000) 

 
2.0868 

(0.0185) 

 
–0.4411 
(0.3296) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

16.7366 
(0.0000) 
38.3239 
(0.0000) 

− 
 

− 
 

 
 

11.9190 
(0.0006) 

– 
 
– 
 

0.1360 
(0.7123) 

– 

 
 

17.2380 
(0.0000) 

– 
 
– 
 
– 
 

21.7233 
(0.0000) 

Chow test F – 
 

65.9038 
(0.0000) 

83.4606 
(0.0000) 

Speed of convergence 
Half-life 

0.2790 
2.48 

0.1418 
4.89 

0.2546 
2.72 

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

 Tables 5 and 6 contain the results of the estimation and verification of 
exemplary panel models used in the investigation, i.e. the panel model with 
fixed effects without the spatial component (FE_IND), the spatial auto-



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

19

regressive panel model with fixed effects (SAR_FE_IND), and the spatial 
error panel model with fixed effects (SE_FE_IND). Just as in the pooled 
time and cross-sectional data models also in the panel data models the con-
nections among the stock exchanges in two variants (connections according 
to physical/economic distance) were taken into account. Fixed effects are 
significant in the considered models. It means that individual characteristics 
of every exchange are valid for their convergence. 

Table 6. Results of the estimation and verification of panel models with fixed effects 
obtained for the European stock exchanges – variant II 

 FE_IND SAR_FE_IND SE_FE_IND 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–1.6250 
(0.0000) 
–0.8927 
(0.0000) 

– 
 
– 

 
–1.1442 
(0.0000) 
–0.6248 
(0.0000) 
0.4542 

(0.0000) 
– 

 
–1.6095 
(0.0000) 
–0.8894 
(0.0000) 

– 
 

0.6309 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.4375 

–117.1200 

 
– 

–132.0000 

 
– 

–138.2700 

Heteroskedasticity 
Breuch-Pagan test 

 
30.4129 
(0.0336) 

 
27.3471 
(0.0727) 

 
27.4311 
(0.0713) 

Autocorrelation of 
residuals 

Moran test 

 
7.4093 

(0.0000) 

 
3.0568 

(0.0011) 

 
1.2405 

(0.1074) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

22.4971 
(0.0000) 
41.8167 
(0.0000) 

– 
 
– 
 

 
 

16.8810 
(0.0000) 

– 
 
– 
 

0.4012 
(0.5265) 

– 

 
 

23.1530 
(0.0000) 

– 
 
– 
 
– 
 

19.7207 
(0.0000) 

Chow test F – 
 

22.0179 
(0.0000) 

28.7812 
(0.0000) 

Speed of convergence 
Half-life 

0.2790 
2.48 

0.1225 
5.66 

0.2752 
2.52 

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



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

20 

 The diagnostics of the considered models suggests that the classical pan-
el model is the worst among them. In this case, the Breusch-Pagan statistic is 
significant (at the level of significance γ = 0.05), leading to the rejection of 
the model assumption of homoskedasticity. In addition, on the basis of the 
Moran test the hypothesis of the independence of the model residuals should 
be rejected (comp. the TSCS model). 

 Table 7. Results of the estimation and verification of β-convergence models for 
pooled time series and cross-sectional data, obtained for Asian and Ameri-
can stock exchanges – variant I 

 Linear regression Spatial autoregressive 
model 

Spatial error 
model 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–0.4678 
(0.0000) 
–0.2084 
(0.0000) 

– 
 
– 

 
–0.4181 
(0.0000) 
–0.1845 
(0.0000) 
0.3065 

(0.0000) 
– 

 
–0.3749 
(0.0000) 
–0.1751 
(0.0000) 

– 
 

0.3188 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.0987 

–31.8650 

 
– 

–51.2710 

 
– 

–49.7970 

Heteroskedasticity 
Breuch-Pagan test 

 
18.7771 
(0.0000) 

 
19.1601 
(0.0000) 

 
18.4701 
(0.0000) 

Autocorrelation of 
residuals 

Moran test 

 
6.1474 

(0.0000) 

 
–0.5729 
(0.2834) 

 
–0.3759 
(0.3535) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

39.7261 
(0.0000) 
32.3990 
(0.0000) 

– 
 
– 

 
 

21.4060 
(0.0000) 

– 
 
– 
 

12.2581 
(0.0005) 

− 

 
 

19.9320 
(0.0000) 

– 
 
– 
 
– 
 

4.9310 
(0.0264) 

Speed of convergence 
Half-life 

0.0292 
23.73 

0.0255 
27.19 

0.0241 
28.81 

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



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

21

 The necessity of model re-specifications towards the spatial panel mod-
els was also confirmed with the Lagrange Multiplier tests. All the tests are 
statistically significant, except for the robust version (RLMlag), which sug-
gests that the spatial error panel model should be preferred. Moreover, the 
significance of the spatial effects with the aid of the LR test has been con-
firmed (see Tables 5 and 6). 

Table 8. Results of the estimation and verification of β-convergence models for 
pooled time series and cross-sectional data, obtained for Asian and Ameri-
can stock exchanges – variant II 

 Linear regression Spatial autoregressive 
model 

Spatial error 
model 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–0.4678 
(0.0000) 
–0.2084 
(0.0000) 

– 
 
– 

 
–0.3915 
(0.0000) 
–0.1751 
(0.0000) 
0.6866 

(0.0000) 
– 

 
–0.3812 
(0.0002) 
–0.2014 
(0.0000) 

– 
 

0.7268 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.0987 

–31.8650 

 
– 

–73.7960 

 
– 

–76.4300 

Heteroskedasticity 
Breuch-Pagan test 

 
18.7771 
(0.0000) 

 
18.4305 
(0.0000) 

 
17.8364 
(0.0000) 

Autocorrelation of 
residuals 

Moran test 

 
11.8902 
(0.0000) 

 
1.7776 

(0.0377) 

 
1.8976 

(0.0289) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
– 
 

120.0009 
(0.0000) 
117.0748 
(0.0000) 

− 
 

− 
 

 
43.9310 
(0.0000) 

– 
 
– 
 

3.2764 
(0.0703) 

– 

 
46.5650 
(0.0000) 

– 
 
– 
 
– 
 

0.3503 
(0.5540) 

Speed of convergence 
Half-life 

0.0292 
23.73 

0.0241 
28.81 

0.0281 
24.66 

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



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

22 

For the purpose of investigating the reasonableness of including the fixed 
effects in the spatial models there was applied the Chow test which considers 
the spatial model for pooled TSCS data vs. the spatial panel model with 
fixed effects. The results of the Chow test  have pointed out the statistical 
significance of the fixed effects in the spatial autoregressive panel model, as 
well as in the panel spatial error model (see Tables 5 and 6). 

Table 9. Results of the estimation and verification of panel models with fixed effects 
obtained for Asian and American stock exchanges – variant I 

 FE_IND SAR_FE_IND SE_FE_IND 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–1.5180 
(0.0000) 
–0.9213 
(0.0000) 

– 
 
– 

 
–1.4074 
(0.0000) 
–0.8537 
(0.0000) 
0.2192 

(0.0000) 
– 

 
–1.4489 
(0.0000) 
–0.8805 
(0.0000) 

– 
 

0.2941 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.4167 

–94.2000 

 
– 

–107.3700 

 
– 

–107.9300 

Heteroskedasticity 
Breuch-Pagan test 

 
43.7765 
(0.0081) 

 
47.0493 
(0.0033) 

 
46.0142 
(0.0044) 

Autocorrelation of 
residuals 

Moran test 

 
5.4701 

(0.0000) 

 
0.7071 

(0.2397) 

 
–0.0850 
(0.4661) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

23.3580 
(0.0000) 
24.4627 
(0.0000) 

− 
 

− 
 

 
 

15.1720 
(0.0000) 

– 
 
– 
 

3.2420 
(0.0718) 

– 

 
 

15.7270 
(0.0000) 

– 
 
– 
 
– 
 

4.3467 
(0.0371) 

Chow test F – 
 

123.4199 
(0.0000) 

129.7126 
(0.0000) 

Speed of convergence 
Half-life 

0.3178 
2.18 

0.2403 
2.88 

0.2656 
2.61 

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



Spatio-temporal Analysis of Convergence of Development Level… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

23

 Taking into account the geographical connections (variant I) among the 
European stock exchanges investigated, in the panel convergence models 
there has been removed the problem of autocorrelation of the residuals  
(in the spatial autoregressive panel model at the level of significance  
γ = 0.01). However, in the case of using the matrix of economic distance 
(variant II) the residual autocorrelation has been eliminated only from the 
spatial error panel model. 

Table 10. Results of the estimation and verification of panel models with fixed ef-
fects obtained for Asian and American stock exchanges – variant II 

 FE_IND SAR_FE_IND SE_FE_IND 

Parameters 

α 
 

β 
 

ρ 
 

λ 

 
–1.5180 
(0.0000) 
–0.9213 
(0.0000) 

– 
 
– 

 
–1.3046 
(0.0000) 
–0.7933 
(0.0000) 
0.5110 

(0.0000) 
– 

 
–1.3980 
(0.0000) 
–0.8588 
(0.0000) 

– 
 

0.6899 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.4167 

–94.2000 

 
– 

–121.4400 

 
– 

–127.2300 

Heteroskedasticity 
Breuch-Pagan test 

 
43.7765 
(0.0081) 

 
47.7040 
(0.0027) 

 
44.2658 
(0.0071) 

Autocorrelation of 
residuals 

Moran test 

 
9.8500 

(0.0000) 

 
3.3626 

(0.0004) 

 
1.4042 

(0.0801) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

50.3592 
(0.0000) 
76.4993 
(0.0000) 

– 
 
– 
 

 
 

29.2450 
(0.0000) 

– 
 
– 
 

3.6813 
(0.0550) 

– 

 
 

35.0310 
(0.0000) 

– 
 
– 
 
– 
 

29.8214 
(0.0000) 

Chow test F – 
 

41.4627 
(0.0000) 

45.2704 
(0.0000) 

Speed of convergence 
Half-life 

0.3178 
2.18 

0.1971 
3.52 

0.2447 
2.83 

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



Elżbieta Szulc, Dagna Wleklińska 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

24 

 Generally, we may say that the statistical properties of the convergence 
models of the European stock exchanges are better than of the models ob-
tained in our first study mentioned earlier, in which exchanges from different 
parts of the world were taken into account. 
 The other tables show the results of the estimation and verification of the 
convergence models obtained for Asian and American stock exchanges. 
Table 7 shows the characteristics of the models obtained for the pooled time 
series and cross-sectional data in the classical and spatial version (the spatial 
autoregressive model and the spatial error model), respectively. The spatial 
components in the spatial models are included through the matrix quantify-
ing the physical distance between the stock exchanges. The characteristics of 
the spatial models using the matrix of the economic distance in comparison 
with the characteristics of the model without the spatial connections are pre-
sented in Table 8. 
 In both variants of quantification of relationships between the stock ex-
changes spatial models are better than the models which do not take into 
account the connections, in terms of the autocorrelation of residuals. Unfor-
tunately, all the models obtained are characterized by heteroskedasticity of 
variance. The panel models obtained for Asian and American stock ex-
changes have the analogical fault (see Tables 9 and 10). 

Conclusions 

 The analysis confirms the earlier findings that the inclusion of the link-
ages which result from physical and/or economic distance between the stock 
exchanges in the models of their convergence is justified. In other words, the 
results of the investigation provide another evidence for the existence of 
spatial effects in the empirical models of stock exchanges' convergence. 
 The earlier study (Szulc et al., 2014) found that the geographical distance 
has less impact on the process of equalizing differentiation of stock markets 
then the economic distance between them. In this study the finding was not 
revealed as clearly. The empirical models of convergence obtained for the 
European stock exchanges satisfy the basic criteria of statistical verification. 
Unfortunately, this is not the case of the models obtained for the Asian and 
American stock exchanges. 
 The analyses of the process of convergence of stock exchanges should be 
further continued in terms of methodology as well as for the purpose of 
searching of properly established spatial regimes. 
 



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DYNAMIC ECONOMETRIC MODELS 15 (2015) 5–26 

25

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Przestrzenno-czasowa analiza konwergencji poziomu rozwoju  
wybranych giełd papierów wartościowych w okresie 2004–2012 

Z a r y s  t r e ś c i. Artykuł dotyczy analizy konwergencji wybranych giełd papierów warto-
ściowych z punktu widzenia poziomu ich rozwoju. Przedstawia podejście, które wskazuje na 
potrzebę uwzględniania przestrzennych i ekonomicznych powiązań między rynkami giełdo-
wymi w analizach ich konwergencji. Przeprowadzone badanie pokazuje także, że analiza 
konwergencji giełd w ustalonym zakresie przestrzennym wymaga podziału rozważanych 
giełd zgodnie z ustalonymi reżimami przestrzennymi  Badanie obejmuje 42 wybrane parkiety, 
analizowane w okresie 2004–2012. Dane empiryczne odnoszą się do 6 zmiennych diagno-
stycznych, uznanych jako ważne determinanty rozwoju rynków giełdowych. 

S ł o w a  k l u c z o w e: giełda papierów wartościowych, konwergencja, reżimy przestrzenne, 
odległość fizyczna, odległość ekonomiczna, macierz sąsiedztwa, przestrzenne modele pane-
lowe.  
  


