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© 2014 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.2014.007  Vol. 4 (2014) 125−144 

Submitted October 20, 2014  ISSN 
Accepted December 23, 2014 1234-3862 

Elżbieta Szulc*, Dagna Wleklińska,  
Karolina Górna, Joanna Górna 

The Significance of Distance Between Stock  
Exchanges Undergoing the Process of Convergence:  

an Analysis of Selected World Stock Exchanges  
During the Period of 2004–2012 

A b s t r a c t. The paper concerns the convergence of selected world stock exchanges from 
the point of view of their development in the context of geographical and economic distance 
between them. It presents the methodological approach which points up the necessity of 
taking into account spatial and economic connections among stock markets in convergence 
analyses. The research includes 46 largest trading floors analyzed in the period of 2004–2012. 
The empirical data refer to six diagnostic variables acknowledged as the important determi-
nants of the development of stock markets. 

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

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

Introduction 

 The paper concerns the convergence of selected world stock exchanges 
in the context of geographical and economic distance between them. The 
importance of the geographical and economic distance in revealing the link-
ages between stocks has already been considered in literature on the subject 

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

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



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

126

(e.g. Suchecka and Łaszkiewicz, 2011; Wójcik, 2009; Asgharianet et al., 
2013). In particular, the premises of the spatial perspective on capital mar-
kets’ analyses in two areas: geography of finance and capital, and behavioral 
finance were indicated (see e.g. Suchecka and Łaszkiewicz, 2011). 
 A spatial analysis of linkages between securities markets was carried 
among others by Asgharian et al. (2013). On the basis of their research, they 
stated that the similarity with regard to the economies’ components is the 
strongest source of linkages between stock markets but the connections that 
result from geographical neighborhood, bilateral FDI and stability of the 
bilateral exchange rate are important as well. 
 One of the directions of the analysis of the stock exchanges’ relation-
ships is considering the convergence of these markets from the point of view 
of their specific characteristics. This process is strictly connected with the 
integration and liberalization of stock markets and their growing interde-
pendence, which in turn is associated with the liberalization of capital flows 
and technological innovation. These processes are favorable for the devel-
opment of stock markets, and thus the distinctions between them are becom-
ing increasingly blurred over time.  
 The literature on the convergence of stock markets includes e.g. Fraser et 
al. (1994), Koralun-Bereznicka (2008) and Caparale et al. (2009). In the 
works the problem of linkages between the markets in geographical and 
economic spaces is particularly interesting. In the recent literature the hy-
potheses of convergence vs. divergence are formulated mainly in the context 
of the contemporary financial crisis of 2007–2010 (see e.g. Aspergis et al., 
2014). 
 The aim of the paper is to investigate whether, in the light of the current 
empirical analyses, one can observe the process of convergence of main 
stock markets in the world. In addition the importance of distance between 
the markets for the process is evaluated. Particularly the role of economic 
distance is considered. In the research the hypothesis that in the convergence 
of stock markets the spatial and economic connections among them are im-
portant, and so that relative location of a stock exchange affects the growth 
rate of the exchange, is verified. 
 The structure of the paper is as follows: in Section 1 the subject and 
range of the investigation are defined. It qualifies the investigated stock ex-
changes and characterizes the specified diagnostic variables. Section 2 pre-
sents the methodology. In this section a taxonomic measure of stock  
exchanges’ development is defined and the theoretical models of β-con-
vergence in formulation of the regressions for the pooled time series and 
cross-sectional data are presented. Moreover, in Section 2 the diagnostic 



The Significance of Distance Between Stock Exchanges … 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

 

127

tests for verification of the empirical models are pointed out. Section 3 con-
tains preliminary data analysis. The results of the research are presented in 
Section 4. Conclusion formulates final remarks and indicates further investi-
gations. 

1. Subject and Range of the Investigation 

 The subject of the investigation contains the selected worldwide stock 
exchanges, characterized in terms of their level of development. The study 
included 46 largest trading floors in the period of 2004–2012. The specifica-
tion of the exchanges with the assignment to the relevant country are pre-
sented 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 capitaliza-
tion 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. Taking into account the connections of the capital 
market with the economy of the country of its location was also an important 
issue. The range of information provided by the World Federation of Ex-
changes played a significant role as well. 
 The capitalization of domestic shares is one of the most important pa-
rameters reflecting the situation in the securities market. It is calculated as 
the total number of shares issued by domestic companies. A high value of 
this indicator encourages large investors to invest their capital in a given 
market and shows its attractiveness compared to others. 
 From the point of view of the development of a given equity market, 
another important indicator is the capitalization of newly listed domestic 
shares. Contemporary capital market is very often treated as a short term 
mechanism where one can earn or lose money suddenly. In many cases, the 
financial performance of companies is ignored. With increasing stock quota-
tion and improving situation in the stock market, the interest of new compa-
nies investing in a given trading floor is also growing. In practice, this means 
that growing prices of shares allow investors to make a profit. However, the 
number of initial public offerings do not affect the conditions of a stock ex-
change. Only on the basis of their capitalization, the development potential 
of a given market can be assessed (Wiśniewski, 2003). 
  



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

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Table 1. Specification of the stock exchanges considered 

North/South America 
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) 

  

Asia/Pacific 
Australia Australian SE (ASX) 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) Thailand Thailand SE (THSE) 
Singapore Singapore SE (SSE)   

Europe/Middle East/Africa 
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 (Eu-
rope) (NEE) 

Iran Tehran SE (THRSE) Ireland Irish SE (IRSE) 

Israel Tel Aviv SE (TASE) Luxemburg 
Luxemburg SE 

(LXSE) 
Malta Malta SE (MSE) Mauritius Mauritius SE (SEM) 

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

Poland Warsaw SE (WSE) South Africa 
Johannesburg SE 

(JSE) 

Switzerland 
SIX Swiss Exchange 

(SIX) 
Sweden 

Nasdaq OMX Nordic 
Exchange (NOMX) 

Turkey Istanbul SE (ISSE) Hungary Budapest SE (BDSE) 
Great Britain London SE (LSE)   

 Other important parameter characterizing the capital market is the value 
of turnover. It is calculated as a total number of shares traded multiplied by 
their respective matching prices within a year. For a well-developed market 
the desired situation is that the annual turnover is higher than the total value 
of shares traded. 



The Significance of Distance Between Stock Exchanges … 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

 

129

 In order to connect the capital market with the country’s economy the 
GDP per capita indicator was used. It is one of the most popular parameters 
which reflects the level of citizens’ wealth. It has to be emphasized that most 
of the highly developed stock exchanges are located in developed countries. 
It is legitimate to use this indicator due to the fact that almost all exchanges, 
described as mature, are located in countries that have achieved a high level 
of development. 
 The top 10 most heavily capitalized domestic companies is one of the 
indicators reflecting market concentration. This parameter, as the only one 
out of the six variables taken into account, was treated as a destimulant. The 
concentration phenomenon takes place when a small number of large com-
panies has a significant share in the capitalization of a given stock exchange. 
A high value of this indicator is undesirable since it characterizes poorly 
developed markets. 
 The ratio of market capitalization to GDP reflects the relationship be-
tween economic development and maturity of an equity market. It was ob-
served that for well-developed stock markets, the value of this indicator is 
higher than 60 percent. It is believed that such participation allows operators 
to gather national and foreign capital. At the same time, the results of empir-
ical studies demonstrate that this ratio needs to be above 2 percent for the 
stock exchange to have significant influence on particular processes of 
a national economy. On the other hand, this variable should be treated with 
caution because of the fact that currently one company may be listed on 
more than one market (Łuniewska and Tarczyński, 2006, p. 45). 

2. Methodology 

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

,
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. 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

130

In this approach the values of the synthetic variable iq are calculated accord-
ing 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 stan-
dard. 
 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 the linkages between markets in eco-
nomic space, and finally the analysis of the stock exchanges’ convergence, 
which is meant as equalizing their development levels.  
 In the analyses of the stock exchanges' convergence the econometric 
models of β-convergence, in particular the spatial models for pooled time 
series and cross-sectional data (TSCS) and spatial panel models, were used. 
 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
 


























   (3) 

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

  ,lnln '
1'

1

'

itit
it

it q
q

q
 












 .itjt
ij

ijit w   


  (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
 


























   (5) 

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



The Significance of Distance Between Stock Exchanges … 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

 

131

  ,lnln '
1'

1

'

ititi
it

it q
q

q
 












 ,itjt
ij

ijit w   


  (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. Assuming that 
there are N stock exchanges, the matrix has as many rows and columns as 
there are exchanges, i.e. N by N matrix W is considered. Each row of the 
matrix contains non-zero elements in columns which correspond to the con-
nected objects, according to the received criterion. Furthermore, the given 
object cannot be connected to itself, so wij = 0 for all i = j. Thus, the diagonal 
elements of W are zeros. 
 In the majority of the spatial analyses the starting point in establishing 
the spatial connections is the binary matrix of neighborhood. The neighbors 
are usually established according to the common border criterion. Then, the 
rows in the connectivity matrix are normalized, so that the row sums are 
equal to 1, as a result of dividing each entry on a row by the sum of the row 
values (the so-called row standardization to one). The weights wij that are 
established in this way signify that each j-th neighbour of the i-th spatial 
units is treated identically, and the greater the strength of its interactions with 
the neighbours is, the fewer neighbours it has. 
 A different situation occurs when the weights wij are functions of some 
properties of the space, e.g. of the length of the common border, of the dis-
tance between the centers of the regions or of other measures of similarity 
between the regions, e.g. of the so-called economic distance between them. 
Various types of weights wij may be pointed out according to the established 
criteria (see e.g. Haining, 2005, p. 83–84 ). 
 In this paper the linkages between stock exchanges will be defined with 
the use of two approaches. The first one 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 one con-
sists in that in the matrix of connections the economic distance (the essence 
of which is to establish similarity of the exchanges on the basis of the value 
of the taxonomic measure of exchanges’ development) is taken into consid-
eration. 
 The economic distance was expressed as: 

  ,
1

2



m

j
kjijik zzd  (7) 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

132

where:  

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. 

In both approaches the elements of the linkages’ matrix are equal to: 












ki

ki
dw

ikik

if0,

if,
1

   (8)   

Then, as a result of row standardization to one, the matrixes of the connec-
tions based on the physical or economic distance are obtained. 
 Since the models (3)–(6) refer to the pooled time series and cross-
sectional data, the block matrixes of connections were used, i.e. 

,

9

2

1





















W00

0W0

00W

W







 (9) 

where: 921 ... WWW   – matrixes of the spatial connections based on 
the physical distance, the same for all the considered years, and 

,

9

2

1































W00

0

0W0

00W

W







 (10) 

where: 921 ...   WWW  – matrixes of connections, taking into account 
the economic distance between exchanges, different for successive years. 
 The convergence of the exchanges is confirmed by the data, if the pa-
rameter estimates β in models (3)–(6) are negative and statistically signifi-
cant. In addition, if parameters  in models (3) and (5) and parameters  in 
models (4) and (6) are significantly different from zero, then in the conver-
gence process the spatial connections among stock exchanges are important 
and the hypothesis that the rate of growth of any stock exchange is related to 
that of its neighbors is confirmed. 
 The inclusion of the spatial elements in the analysis of the convergence 
of stock exchanges allows us to identify the relationships between them in 



The Significance of Distance Between Stock Exchanges … 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

 

133

a geographical and economic space. Furthermore, the spatial models of con-
vergence have better statistical properties, and thus allow for wider econom-
ic interpretation. 
 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 need for including fixed effects into 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. 

3. Preliminary Data Analysis 

 Figure 1 shows locations of 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 change in the arrangement of their spatial location as well. It is 
interesting that some of the Asian trading floors, e.g. OSE, SZSE, KRX, 
PSE, did not record any decrease in the value of taxonomic measure of de-
velopment after the beginning of the financial crisis in 2007. In the worst 
case, these stock exchanges reacted with slowdown or stagnation of growth. 
 Obviously, the beginning of a crisis has caused a decline in the US stock 
exchanges but also stimulated the process of making up the development 
imbalance between emerging and well-developed markets. Interestingly, the 
same exchanges, for which synthetic variable already showed lower values 
in 2008 (NOMX, SE, SHSE, BM), recorded an increase in the value of this 
variable the following year (as opposed to all other stock exchanges). 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

134

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

2004–2012 

 Figures 2 and 3 show the value of taxonomic measure of development 
(surface of the wheel) for each stock exchange included in the study for 2004 
and 2012 respectively. This graphical presentation is useful for a preliminary 
assessment of changes in the global capital market over the considered peri-
od. In 2004, two dominant financial centers are clearly visible. In the west, it 
is NYSE and Nasdaq, while in central Europe, the London Stock Exchange 
and NYSE Euronext Europe stand out in particular. The reason for achieving 
such good results by the latter is certainly the fact that NYSE Euronext Eu-
rope is an example of a trading platform created by the consolidation of the 
stock exchanges of Paris, Amsterdam, Brussels and Lisbon. Against the 
background the Tokyo Stock Exchange stands out of the Asian stock ex-
changes in 2004. In 2012, by contrast, 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. 
 Comparing Figures 2 and 3 allows to observe that the Asian stock mar-
kets have strengthened themselves at the expense of the US and European 
stock exchanges within the nine years of the research period. Asia is current-
ly the largest region of emerging markets in the world and is a cradle of the 
fastest-growing economies. The advantage of these markets arises not only 
from the fact that this area is inhabited by more than 60 percent of the 
world's population, but also from the reasonable, in comparison to other 
countries, fiscal and monetary policy. Economic liberalization and increas-
ing competitiveness of these markets still attracts many foreign investors. 



The Significance of Distance Between Stock Exchanges … 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

 

135

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

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

 For the purpose of a preliminary assessment of the relationship between 
analyzed stock markets in the context of physical distance, for all pairs of 
stock exchanges the values of Pearson correlation coefficient for the TDM in 
the period of 2004–2012 were calculated. 
 Figure 4 is a graphic illustration of the relationships, for which the values 
of the correlation coefficient are greater than 0.9. It may be noticed that most 
of these connections are located on the Old Continent. On one hand, it 
demonstrates a high integration of European stock exchanges but also carries 
the risk of transmitting negative pulses occurring within a trading floor, for 
further linked to it causing the contagion effect. For comparison, the links 
identified on the basis of the value of the correlation coefficient for the peri-
od 2007–2010 crisis, are shown in Figure 5. According to some beliefs (see 
e.g. Login and Solnik, 2001) the strength of the relationship between the 
stock exchanges during the downturn increases and decreases with the im-
provement of the general economic situation in the world. This hypothesis is 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

136

confirmed by the number of links between stock exchanges marked in Figure 
5, for which the value of the correlation coefficient exceeds 0.9. Therefore, it 
seems that along with deteriorating sentiments in the global capital market, 
an increase of the correlation between securities markets might be expected, 
as long as the trend will not be reversed. 

 
Figure 4. The significant linkages between investigated stock markets according to 

the Pearson correlation coefficient in the period of 2004–2012 

 
Figure 5. The significant linkages between investigated stock markets according to 

the Pearson correlation coefficient in the period of 2007–2010 

4. Results of the Econometric Analysis 

 The successive tables presented below contain the information on the 
usefulness of various methodological conceptions expressed by the spatial 
models, presented in Section 3, in comparison with the linear regression 
model, i.e. the traditional model without the spatial effects. 
  



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DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

 

137

Table 2. Results of the estimation and verification of β-convergence models for 
pooled time series and cross-sectional data – variant I 

 Linear regression Spatial autoregressive 
model 

Spatial error 
model 

Parameters 
 
 
 
 
 
 
 

 
–0.2276 
(0.0000) 
–0.1252 
(0.0000) 

– 
 
– 

 
–0.1704 
(0.0000 ) 
–0.0965 
(0.0000) 
0.7082 

(0.0000) 
– 

 
–0.1702 
(0.0004) 
–0.0922 
(0.0000) 

– 
 

0.7255 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.0644 

–161.0100 

 
– 

–264.4000 

 
– 

–261.9500 
Heteroskedasticity 
Breuch-Pagan test 

 
1.8260 

(0.1766) 

 
1.1870 

(0.2759) 

 
1.5371 

(0.2151) 
Autocorrelation of 

residuals 
Moran test 

 
17.7228 
(0.0000) 

 
–0.8701 
(0.1923) 

 
–0.7309 
(0.2312) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

316.6723 
(0.0000) 
291.9936 
(0.0000) 

– 
 

 – 

 
 

105.3900 
(0.0000) 

– 
 
– 
 

26.0514 
(0.0000) 

– 

 
 

102.9300 
(0.0000) 

– 
 
– 
 
– 
 

1.3727 
(0.2414) 

Speed of convergence 
Half-life 

0.0167 
41.47 

0.0127 
54.63 

0.0121 
57.31 

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

 Table 2 contains the results of estimation and verification of three mod-
els for pooled time series and cross-sectional data: the linear regression 
model (TSCS), the spatial autoregressive model (SAR_pooled) and the spa-
tial error model (SE_pooled). In the spatial models for the purpose of quanti-
fication of the connections among exchanges investigated the matrix W was 
used, taking into account the physical distance between them (variant I). 
Table 3 presents the results for the three analogical models, but in the spatial 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

138

models the connectivity matrix W* of the economic distance between the 
exchanges was used (variant II). 

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

 Linear regression Spatial autoregressive 
model 

Spatial error 
model 

Parameters 
 
 
 
 
 
 
 

 
–0.2276 
(0.0000) 
–0.1252 
(0.0000) 

– 
 
– 

 
–0.1670 
(0.0000) 
–0.0981 
(0.0202) 
0.8030 

(0.0000) 
– 

 
–0.1477 
(0.0110) 
–0.1221 
(0.0000) 

– 
 

0.8311 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.0644 

–161.0100 

 
– 

–279.9800 

 
– 

–285.4900 
Heteroskedasticity 
Breuch-Pagan test 

 
1.8260 

(0.1766) 

 
0.4405 

(0.5069) 

 
0.2427 

(0.6223) 
Autocorrelation of 

residuals 
Moran test 

 
25.8554 
(0.0000) 

 
3.0317 

(0.0011) 

 
3.3425 

(0.0004) 
Spatial 

dependence 
LR 

 
LMlag 

 
LMerr 

 
RLMlag 

 
RLMerr 

 
 
– 
 

597.6765 
(0.0000) 
590.0261 
(0.0000) 

 
 
 
 

 
 

120.9600 
(0.0000) 

– 
 
– 
 

9.6901 
(0.0019) 

– 

 
 

126.4800 
(0.0000) 

– 
 
– 
 
– 
 

2.0397 
(0.1532) 

Speed of convergence 
Half-life 

0.0167 
41.47 

0.0129 
53.69 

0.0163 
42.57 

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

 The classical model estimated using the pooled time series and cross-
sectional data does not satisfy the fundamental criterions of statistical verifi-
cation (see Tables 2 and 3). This result is consistent with our prediction be-
cause the assumptions of the model, especially the same variance in the 
space and independence across residuals for all singled out objects, are usu-



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139

ally unrealistic in practice. Though in this case the Breusch-Pagan statistic is 
insignificant, on the basis of the Moran test 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-
bles 2 and 3). The LM tests for the linear model for the pooled time series 
and cross-sectional data used consider the spatial lag model (spatial auto-
regressive) and the spatial error model as alternatives (LMlag and LMerr, 
respectively). Tables 2 and 3 report the results of using the robust tests 
(RLMlag, in which H0:  = 0 under the assumption that   0 and RLMerr, 
where H0:  = 0 under the assumption that   0) as well. Since the LMlag 
tests are more significant than the LMerr, and the RLMlag are significant 
while the RLMerr are insignificant, the spatial lag models should be pre-
ferred. Subsequently, the significance of the spatial effects in SAR and SE 
models using the Likelihood Ratio test (LR) were confirmed. 
 Likewise, irrespective of which connectivity matrix (of physical or of 
economic distance) in the spatial models has been used, parameters ρ and λ 
are statistically significant. It is worth noting that the fact of including the 
connectivity matrixes in the considered models has crucial impact on con-
vergence parameters (). Absolute values of the parameters for the models 
SAR and SE are lower than for the traditional model which does not take 
into account the connections across investigated stock exchanges. In turn, 
comparing the  parameters in the spatial models which contain the matrix 
of physical distance with the parameters of the models which contain the 
matrix of economic distance one can see that the convergence parameters are 
higher in the second case. It can be supposed that geographical distance has 
less impact on the process of equalizing the differentiation of stock markets. 
 Evaluation of statistical properties of the received empirical models re-
veals that in the models constructed with the use of economic distance be-
tween the stock exchanges the problem of autocorrelation of the residuals 
has not been eliminated. It is a significant drawback of these models. Solv-
ing the problem requires further investigation towards an appropriate modi-
fication of the connectivity matrix. 
 Tables 4 and 5 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 spatial component, the spatial autoregressive panel 
model with fixed effects, and the spatial error panel model with fixed effects. 
Just as in the pooled time and cross-sectional data models also in the panel 
data models the connections among the stock exchanges in two variants 
(connections according to geographical/economic distance) were taken into 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

140

account. Fixed effects are significant in the considered models. It means that 
individual characteristics of every exchange are valid for their convergence. 

Table 4. Results of the estimation and verification of panel models with fixed effects 
– variant I 

 FE_IND SAR_FE_IND SE_FE_IND 
Parameters 

 
 
 
 
 
 
 

 
–1.5352 
(0.0000) 
–0.8535 
(0.0000) 

– 
 
– 

 
–1.2475 
(0.1027) 
–0.6956 
(0.0510) 
0.5141 

(0.0000) 
– 

 
–1.3614 
(0.0000) 
–0.7559 
(0.0000) 

– 
 

0.6723 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.3629 

–260.7000 

 
– 

–327.3200 

 
– 

–328.9800 
Heteroskedasticity 
Breuch-Pagan test 

 
98.3157 
(0.0000) 

 
99.3100 
(0.0000) 

 
96.7010 
(0.0000) 

Autocorrelation of 
residuals 

Moran test 

 
13.5916 
(0.0000) 

 
13.5916 
(0.0374) 

 
–0.4039 
(0.3432) 

Spatial 
dependence 

LR 
 

LMlag 
 

LMerr 
 

RLMlag 
 

RLMerr 

 
 
– 
 

122.7834 
(0.0000) 
171.3005 
(0.0000) 

 
 
 
 

 
 

68.6220 
(0.0000) 

– 
 
– 
 

62.2343 
(0.0000) 

– 

 
 

70.2730 
(0.0000) 

– 
 
– 
 
– 
 

13.7172 
(0.0002) 

Chow test F – 
 

87.4795 
(0.0000) 

482.3193 
(0.0000) 

Speed of convergence 
Half-life 

0.2400 
2.89 

0.1489 
4.65 

0.1763 
3.93 

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

 Diagnostics for the considered models suggest that the classical panel 
model is the worst of them. In this case, the Breusch-Pagan statistic is signif-
icant, leading to rejecting the model assumption of homoskedasticity. In 



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141

addition, on the basis of the Moran test the hypothesis of independence of 
the model residuals should be rejected. 

Table 5. Results of the estimation and verification of panel models with fixed effects 
– variant II 

 FE_IND SAR_FE_IND SE_FE_IND 
Parameters 

 
 
 
 
 
 
 

 
–1.5352 
(0.0000) 
–0.8535 
(0.0000) 

– 
 
– 

 
–1.2113 
(0.0000) 
–0.6740 
(0.0000) 
0.6129 

(0.0000) 
– 

 
–1.3165 
(0.0000) 
–0.7465 
(0.0000) 

– 
 

0.8174 
(0.0000) 

Goodness of fit 
Adjusted R2 

AIC 

 
0.3629 

–260.7000 

 
– 

–336.8200 

 
– 

–350.8200 
Heteroskedasticity 
Breuch-Pagan test 

 
98.3157 
(0.0000) 

 
98.5343 
(0.0000) 

 
95.8613 
(0.0000) 

Autocorrelation of 
residuals 

Moran test 

 
18.8266 
(0.0000) 

 
5.6920 

(0.0000) 

 
3.9209 

(0.0000) 
Spatial 

dependence 
LR 

 
LMlag 

 
LMerr 

 
RLMlag 

 
RLMerr 

 
 
– 
 

184.0365 
(0.0000) 
311.6529 
(0.0000) 

 
 
 
 

 
 

78.1170 
(0.0000) 

– 
 
– 
 

17.7730 
(0.0000) 

– 

 
 

92.1220 
(0.0000) 

– 
 
– 
 
– 
 

145.3894 
(0.0000) 

Chow test F – 
 

83.2527 
(0.0000) 

95.0155 
(0.0000) 

Speed of convergence 
Half-life 

0.2400 
2.89 

0.1401 
4.95 

0.1715 
4.04 

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

 The necessity of model re-specifications towards the spatial panel mod-
els was also confirmed by the Lagrange Multiplier tests. All the tests are 
statistically significant and unfortunately robust versions of the tests do not 
provide unambiguous conclusions on what kind of the spatial connections, 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna, Joanna Górna 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 125–144 

142

autoregressive or error, should be applied to the models. Moreover, the sig-
nificance of the spatial effects with the aid of the LR test has been con-
firmed. For investigating the reasonableness of including the fixed effects in 
the spatial models the Chow test (the spatial model for pooled TSCS data vs. 
the spatial panel model with fixed effects) was used. 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 4 and 5). 
 Taking into account the geographical connections (variant I) among the 
investigated stock exchanges in the panel convergence models has removed 
the problem of autocorrelation of the residuals (at the level of significance  
=0.01). However, in the case of using the matrix of economic distance (var-
iant II) the autocorrelation of model residuals has not been eliminated. In 
turn, the problem of heteroskedasticity has remained in both cases. There-
fore, in further investigation searching for the spatial regimes will be per-
formed. 

Conclusions 

 The paper's findings show that including the linkages that result from 
physical and/or economic distances between stock exchanges in the models 
of their convergence is justified and very important for the analyses of the 
phenomenon. In other words, the results of the investigation provide evi-
dence of spatial effects in the empirical models of stock exchanges' conver-
gence. As a result, it is possible to define the influence of the distance be-
tween exchanges on their economic development, the estimates of conver-
gence parameter are more precise, and some statistical properties of the 
models are better. 
 During the investigation it was observed that geographical distance has 
less impact on the process of equalizing differentiation of stock markets then 
the economic distance between them. 
 Due to the heteroskedasticity, the empirical panel models for the ex-
changes investigated as a whole were not entirely satisfactory. It means that 
there are differentials in relationships between objects considered and their 
speed of convergence. Thus, in further investigation the spatial regimes will 
be searched for. For example, we will investigate the convergence of the 
European, Asian and American stock markets, separately. In addition, we 
will continue the work on improving other properties of the empirical mod-
els, e.g. on removing the problem of autocorrelation of the residuals. 



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Znaczenie odległości między giełdami papierów wartościowych 
w procesie ich konwergencji. Analiza wybranych giełd światowych 

w okresie 2004–2012 

Z a r y s  t r e ś c i. Artykuł dotyczy analizy konwergencji wybranych giełd światowych 
z punktu widzenia poziomu ich rozwoju, w kontekście geograficznej i ekonomicznej odległo-
ści między nimi. Przedstawia podejście, które wskazuje na potrzebę uwzględniania prze-
strzennych i ekonomicznych powiązań między rynkami giełdowymi w analizach ich konwer-
gencji. Badanie obejmuje 46 największych parkietów, analizowanych w okresie 2004–2012. 
Dane empiryczne odnoszą się do 6 zmiennych diagnostycznych, uznanych jako ważne deter-
minanty rozwoju rynków giełdowych. 

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


