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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.2016.010  Vol. 16 (2016) 165−187 

Submitted November 30, 2016  ISSN (online) 2450-7067 

Accepted December 28, 2016 ISSN (print) 1234-3862 

Elżbieta Szulc*, Dagna Wleklińska, Karolina Górna 

The Share of European Economies in the Process  
of Convergence of Long-term Interest Rates in the EU  

in the Period of 2006–2016 

A b s t r a c t. The paper refers to convergence of interest rates of ten-year government bonds 

emitted by EU countries. It is an attempt to assess the participation of particular economies in 

this process using the so-called vertical convergence. The primary tools of analysis were 

panel data models with fixed effects, including models that consider the links among econo-

mies, quantified by using a distance matrix between indicators of fiscal stability compre-

hended as the share of public debt in GDP. The analysis was conducted for the 27 members of 

the EU in the period between January 2006 and November 2016. 

K e y w o r d s: long term interest rates; process of convergence; fiscal stability; panel data 

models; matrix of economic connections. 

J E L Classification: C10; C23; C58; E43.  

Introduction 

According to the concept of fiscal stability, the value of public debt 

should be maintained at the level that will provide the opportunity to mini-

mise the negative effects of its occurrence and will not impede the govern-

ment’s tasks implemented to stabilise the economy (Buiter, 2006, p. 2; 

Marchewka-Bartkowiak, 2008, p. 55). Referring to the French economist, 

O. J. Blanchard, a stable fiscal policy occurs when it effectively aims to re-

                                                 
* Correspondence to: Elżbieta Szulc, Nicolaus Copernicus University, Faculty of Eco-

nomic Sciences and Management, 13A Gagarina Street, 87-100 Toruń, Poland, e-mail: 

eszulc@umk.pl 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

166 

duce the ratio of public debt to gross domestic product. Therefore, a long-

term debt growth rate should not exceed the height of interest rates, while 

the debt ratio, measured by the ratio of public debt to gross domestic prod-

uct, should not be higher than the difference between the rate of economic 

growth and long-term interest rates. Thus, a conclusion arises that an in-

crease in the government spending intended to support the economic situa-

tion in the country must be later offset by actions aimed at achieving fiscal 

stability by reducing the ratio of public debt to GDP (Blanchard, 1990,  

p. 13). 

Narrowing disparities among long-term interest rates that ensure fiscal 

stabilisation is identified with one of the most important dimensions of con-

vergence processes, which are considered in relation to European economies. 

The fundamental theory that should be used to explain the mechanism of 

European integration and also one of its components, the monetary integra-

tion, is the theory of optimum currency areas. According to it, one of the 

main factors that lead to fluctuations in the exchange rate is the interest rate 

differential. Regarding the fact that a monetary union operates in the terri-

tory of the Old Continent, those disparities may lead to asymmetric shocks. 

The monetary policy which is common to the member countries does not 

constitute an effective tool for leveling the risk and consequences of such 

shocks, and, therefore, their appearance may constitute a real threat to the 

stability of the monetary union. The theory of convergence points to key 

areas of similarity between particular economies and thus can help prevent 

an occurrence of the said shocks and ensure stabilisation of the exchange 

rate. It is particularly important that the process of proximity of interest rates 

is accompanied by fiscal convergence identified with the harmonisation of 

fiscal policies of the members of the EU, because an increase in the debt also 

leads to an escalation in the premium related to the growing risk of insol-

vency of the issuer, which is the main component of interest rates (Pełka, 

2008, p. 6371). 

Long-term interest rates as a macroeconomic aggregate not only reflect 

country-specific market factors which determine the nominal cost of the 

credit, for example, those connected with the rate of increase in prices or 

inflation expectations, but also indicate the manner of conducting and the 

quality of economic policy. 

The contemporary global financial crisis has revealed weaknesses in the 

functioning of the European Union, highlighting the need to reform its eco-

nomic and institutional mechanisms. The fiscal rules defined in the Maas-

tricht Treaty and the complementary Stability and Growth Pact have proven 

ineffective. However, the insufficient fiscal discipline in many economies 



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

167 

was not the reason but rather a consequence of the contemporary crisis with 

the underlying deep structural differences between European economies. The 

increasing integration of financial markets and a common monetary policy 

aimed at maintaining relatively low and often negative interest rates in pe-

ripheral European countries, led to a further deepening of the existing differ-

ences. 

The existing literature confirms an occurrence of convergence processes 

of long-term interest rates during the period of preparation of particular 

countries to adopt the common currency and introduce it into circulation 

(Adam et al., 2002; Pagano and von Thadden, 2004). As a consequence of 

maintaining stable exchange rates the possibility of devaluation of individual 

currencies in order to increase the competitiveness of the economy was elim-

inated, and the elimination of an exchange rate risk premium led to a reduc-

tion of discrepancies in terms of profits from bonds issued by countries be-

longing to the euro area. The vision of a common monetary policy required 

an introduction of preparatory mechanisms, resulting in an anchoring of 

inflation expectations. On the other hand, thanks to the common pursuit of 

the Maastricht criteria as well as restrictions on fiscal policy, credit risk was 

maintained at a relatively low level (Baele, 2004, p. 38; Lane, 2012, p. 1). 

However, in the recent period, the results of empirical analyses provide 

evidence that following the outbreak of the global financial crisis, bonds 

issued by European economies are no longer substitutes for each other. The 

search for the causes of the divergence of long-term interest rates was the 

subject of an empirical analysis conducted, inter alia, by Attinasi et al. 

(2009) and Klepsch, Wollmershäuser (2011). A partial justification for the 

differences in the spreads of the yield of government bonds in the European 

Union rests in the transfer of the credit risk from financial institutions to 

governments which were providing them with the financial aid (Afonso et 

al., 2015). Along with the intensification of the European debt crisis, in fear 

of a growing risk investors restructured their portfolios by deciding to invest 

in safer instruments that showed greater liquidity, thus leading to a growth in 

the divergence between the spreads of government bonds (Barrios et al., 

2009). 

Given the large diversity of European economies that leads us to believe 

that in the group of the countries that co-create the European Union there are 

economies that have both a destimulating and stimulating effect on the pro-

cess of convergence of long-term interest rates, the inference based on the 

classical model of β convergence, which is the most popular method of anal-

ysis of this phenomenon, may lead to erroneous conclusions (Nowak, 2007, 

p. 7778). Among the available empirical analyses regarding the phenome-



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

168 

non of convergence/divergence of long-term interest rates in the European 

Union there is an absence of studies which concern an assessment of the 

contribution of particular countries to this process. The conviction of the 

thus far insufficient exploration of the convergence issues, in particular with 

regard to the impact of individual countries on this process, was the main 

motivation of the authors to undertake this topic. 

The main objective of the empirical analysis conducted in this research is 

to examine the impact of individual EU member countries on the process of 

convergence of interest rates. In particular, we will be interested in the na-

ture of the said impact (positive or negative), and thus the answer to the 

question of which countries pose the greatest risk to the cohesion process of 

the UE as a whole. The conducted analyses will be accompanied by an as-

sumption of a considerable importance of the fiscal stability of a given econ-

omy in the process of convergence/divergence. Therefore, we hypothesised 

that the pace of changes occurring in long-term interest rates in a given 

country affects long-term interest rates in countries with a similar fiscal sta-

bility, measured by the ratio of public debt to GDP. 

1. Subject and Range of the Investigation 

The subject matter of the study concerns the process of convergence of 

long-term interest rates in the European Union in the period between January 

2016 and November 2016 with the consideration of the particular impact of 

individual countries on this process. Due to the unavailability of data, Esto-

nia was excluded from the analysis. 

The dependent variable in the models of convergence was an increase in 

the spread of yield of ten-year Treasury bonds. The analyses of the relation-

ships between the debt markets in Europe have been conducted using 

spreads understood as an excess in the yields expected by investors in con-

nection with the risks incurred by them.  

Due to the interest nature of debt instruments, using the rates of return 

from their prices could result in drawing the wrong conclusions. Yield 

spreads of ten-year Treasury bonds issued by the twenty-seven European 

economies have been calculated as the difference between the yields of gov-

ernment bonds issued by a given economy and the rate of return of another 

risk-free debt instrument, in this case the German ten-year treasury bond (see 

also Hull, 2011, p. 392). 

The empirical analysis concerning an occurrence of the process of con-

vergence of long-term interest rates in the European Union and establish-

ment of the impact of individual countries on this process were carried out 



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

169 

on the basis of monthly data. In order to take into account the impact of the 

global financial crisis and the debt crisis on the analysed phenomenon, the 

time range of the study was divided into two sub-periods, i.e. the period 

between 01.200612.2010 and 01.201111.2016. The data was obtained 

from the Eurostat and the World Bank databases. The calculations were per-

formed using the R-Cran (version 3.2.5), while the drawings were prepared 

with the use of MapViewer. 

2. Methodology 

 The verification of an occurrence of equalising levels of spreads of gov-

ernment bonds was based on the use of the panel data model, which takes 

into account individual effects (see e.g. Dańska-Borsiak, 2011; Suchecki 

(ed.), 2012). The said model takes the following form (FE_IND): 

,1

1

ititi

it

it s
s

s
 












 (1) 

where: 

i – individual characteristic of the issuer, 

its – value of the spread of interest rate of government bonds in country i 

in period t (benchmark – Germany), 

1its – value of the spread of interest rate of government bonds in country i 

in period t – 1. 

 Model (1) was extended by introducing the connections described by the 

connectivity matrix based on an economic distance. The inclusion of these 

links modifies the formula (1) to take the following form (SAR_FE_IND): 

,
1

,1

1

it

ij it

it
tijiti

it

it

s

s
ws

s

s
 

 






















 (2) 

where: 

  – parameter constituting the average value of the effect of "neighbour-

ing" issuers for country   (the statistical significance means that there 

is a relation between the values of spread for a given country and the 

values of spread in countries with similar characteristics in terms of 

specific economic characteristics), 

ijw – element of the connectivity matrix which defines the connection be-

tween country i and j. 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

170 

For more details on the classes of models see e.g. Suchecki (ed.) (2012). The 

same classes of models, among others, were used in Szulc et al. (2014) and 

Szulc, Wleklińska (2015). 

 The matrix of an economic distance used in the analysis that comprises 

links that vary in time is a symmetric block diagonal matrix of the following 

form: 

  ,

*

*

*
1

*
,

*

NTNTT

tNTNTtijd





























D0

D

0D

D









 (3) 

 
NNtijt d


 *

,
*

D , for t = 1, 2, …, T. (4) 

The particular elements of matrix 
D which define the economic distance 

between pairs of objects, in general release, can be described by the formula: 

,
1

1

0

,,,,
*
, 



 



px

px

p

k

r

rtjprtip

p x

tij xx
k

d


 (5) 

where: 

rtipx ,,  – value of the variable which is the basis for the evaluation of the 

difference between units i and j in period ]),0[(
pxkrrt  , 

pxk  – value of the time lag of variable px , 

px – normalising parameter (along with an increase of the delta of the 

normalising parameter, the relationships between objects of 

a greater similarity grow). 

 In particular the determination of an economic distance between objects 

can be performed with the use of one variable. In the conducted study the 

distance was determined based on the 
GDP

debt
 ratio. The ratio of debt to gross 

domestic product is one of the most popular indicators of fiscal stability, 

which was the basis for the formulation of one of the assumptions contained 

in the Maastricht Treaty and the Stability and Growth Pact. The elements 


tijd ,  take the following form: 



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

171 

,*
,

jt

jt

it

it
tij

GDP

debt

GDP

debt
d   (6) 

where 
jt

jt

it

it

GDP

debt

GDP

debt
,  – public debt share in GDP in country i and j in the 

period t. 

Subsequently, the components of matrix 
D should be inversed, i.e.: 

.
1
*
,

,

tij

tij
d

d   (7) 

 The connectivity matrix must be row standardized which means that the 

sum of every row should be equal to 1. It can be obtained by the following 

transformation: 




j

tij

tij

tij
d

d
w

,

,

,  for i = 1, 2, …, N. (8) 

Then, the following assumption is true: 

  .1
1

,,  




NT

j

tijNTNTtij wiwW  (9)  

 The models described by the formulas (1) and (2) were used to evaluate 

the process of convergence/divergence of long-term interest rates in Europe. 

The impacts of individual states in the considered process were estimated 

using the theory of vertical convergence, which in its original form was 

complementary to the concept of income convergence. The classical models 

of absolute and conditional convergence did not provide the possibility to 

estimate individual effects of each of the objects in the process. Considering 

the fact that in large groups of objects, which are characterised by significant 

differences, both such objects exist for which the hypothesis of existence of 

convergence is justified and those that do not converge through the investi-

gated characteristics, false conclusions may be drawn with regard to whether 

the process of convergence between all analysed objects occurs or not. The 

idea of vertical convergence refers essentially to the marginal credit risk 

which is determined as the difference between the risk calculated with regard 

to full loan portfolio and the portfolio consisting of N – 1 components. 

 The values of vertical convergence for every issuer were obtained using 

the formula: 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

172 

,iNi bbconv   (10) 

where: 

b – speed of convergence, indicating the time necessary to equalise levels 

of the analysed phenomenon, wherein: 

 
,

1ln

T
b


  (11) 

iNb   – speed of convergence calculated for the researched group of ob-

jects with the exception of object i. 

 Value of iconv  tells us what is the direction and strength of the influence 

of particular states on the process of alignment of long-term interest rates. 

3. Preliminary Data Analysis 

 There is an obvious connection between the size of the public debt and 

deficit and the yields on long-term government bonds. The height of an in-

terest rate of debt securities issued by the Treasury affects the pace of an 

increase of the debt to GDP ratio, thereby impacting the solvency of the 

government. The rising level of the public debt may lead to an expectation of 

a higher risk premium by the investors reflected in the increase in bond 

yields, which in turn may accelerate the pace of debt accumulation. The 

positive relationship between the rate of debt accumulation and its level is 

often referred to the economic phenomena known as "the snowball effect". 

 The problem of an increasing debt-to-GDP ratio and therefore the declin-

ing confidence in the countries issuing debt securities has been taken up 

again by the theorists of the market due to the outbreak of the global finan-

cial crisis. However, the doubts concerned with the solvency of governments 

grew since the crisis of the 1930s, when the fear of the consequences of 

many developed countries caused them to implement a highly expansionary 

fiscal policy leading in consequence to a sharp increase in their debt. In par-

ticular, countries belonging to what was conventionally called "the southern 

Europe" began to grapple with growing internal and external imbalances, 

which was soon reflected in the decline in demand for the bonds issued by 

the Treasury. 

 Figure 1 shows bar graphs concerning the values of the public debt of the 

member countries  of the European Union in the period between 20062015. 

The graphs have been performed with the use of annual data. The horizontal 

line indicates an excess of the rate of one hundred percent of the gross do-

mestic product. It may be noticed that at the end of the analysed period, the 



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

173 

countries from the southern Europe were most affected by the debt crisis. 

Nevertheless, the problem of debt affected not only the countries considered 

to be poorer, such as Greece, Portugal, Lithuania, Latvia and Estonia, but 

also Ireland, whose economic indicators of the last decade were to set an 

example for the countries of the central and eastern Europe. Among  the 

countries whose value of debt in the recent years has exceeded one hundred 

percent of the gross domestic product we have: Portugal, Belgium, Cyprus, 

Greece and Italy. Until 2014, Iceland belonged to this group as well, howev-

er, in 2015 the debt ratio did not exceed seventy-eight percent of GDP. 

 

Figure 1. Bar charts of debt to GDP ratio in the years 2006–2015 

 In turn, Figure 2 shows bar graphs reflecting the ratio of deficit to the 

gross value of gross domestic product.  

 The rapid increase in debt is one of the main factors having a destabiliz-

ing effect on the economic system. However, along with an increase of na-

tional prosperity the level of the public debt acceptable for investors increas-

es, though what is also not without significance is the credit history of a giv-

en country. Among the economies that faced the necessity to apply for inter-

national assistance are Romania and Latvia, which in 2010 were at risk of 

bankruptcy.  

 According to the estimates of the European Commission, Romania's 

public debt ratio amounted to 30.8 percent of gross domestic product, 

whereas budget deficit to 6.4 percent of GDP. In the case of Latvia, which 

has been on the map of Europe for less than twenty-five years, these indica-

Ireland

Great Britain

SpainPortugal

Luxemb.

Belgium

Netherl.
Germany

Denmark
Sweden

Finland

Estonia

Latvia

Lithuania
Poland

Cyprus
Malta

Czech Rep.

Romania

Bulgaria

Greece

Italy

Austria

Slovenia

Slovakia

Croatia

Hungary

2006 2007 2008 2009 2010 2013 20142011 2012 2015

France



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

174 

tors were 44.7 percent of GDP and 6.4 percent of GDP, respectively. In the 

same period, the United Kingdom, which is one of the most developed coun-

tries of the world, made use of external financing to cover the budget deficit 

of ten percent of the gross domestic product, with the debt level reaching 

82.5 percent.  

 It should be noted, however, that the level of debt which is safe from the 

point of view of an investor and creates an incentive to invest in the bonds of 

a given country may suddenly change. An example would be the situation of 

Greece before the outbreak of the financial crisis, whose bonds were very 

popular among investors despite the fact that the public debt was oscillating 

within the range of one hundred percent of gross domestic product. The es-

calation of the crisis raised uncertainty among investors and as a conse-

quence the deficit and the value of the public debt of Greece rose sharply 

thus discouraging investors to make further investments and bringing the 

country to the brink of bankruptcy. 

 

Figure 2. Bar charts of deficit to GDP ratio in the years 2006–2015 

 It can be concluded that if the level of indebtedness of a country exceeds 

one hundred percent of the gross domestic product with the ratio of the 

budget deficit oscillating within the range of ten percent, there is a likelihood 

of defaults on receivables from loans. It should be emphasized that if, on 

a yearly basis, a country emits bonds with a value equal to or exceeding the 

level of GDP and then repays them along with the interest through the issu-

ance of new securities, as long as the rate of economic growth is higher than 

Ireland Great Britain

Spain

Portugal

Luxemb.

Belgium

Netherl.
Germany

Denmark
Sweden

Finland

Estonia

Latvia

Lithuania
Poland

Cyprus

Malta

Czech Rep.

Romania

Bulgaria

Greece

Italy

Austria
Slovenia

Slovakia

Croatia

Hungary

2006 2007 2008 2009 2010 2013 20142011 2012 2015

France



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

175 

the cost of debt operation, i.e. the interest rates on the issued government 

bonds, this mechanism will be repeated periodically whenever such a neces-

sity arises. This practice was used by many European economies, some of 

which implemented it at a significantly higher level of debt, such as Italy, 

Greece, and Portugal. 

 Figures 3 and 4 show the development of ten-year yield spreads of gov-

ernment bonds of the EU member states in the period between January 2006 

and November 2016. As it was mentioned above, due to the unavailability of 

data the analysis does not include Estonia. In order to maintain sufficient 

transparency of the presented data included in the analysis the states are 

divided into two groups. The vertical line on both graphs corresponds to the 

temporal division into two sub-periods adopted in this study. 

 The behaviour of the European government bonds market in the period 

of 20062016 was determined to a large extent by an increase in the overall 

risk level, which occurred alongside the outbreak of the international finan-

cial crisis. The increase in the profitability of treasury bonds issued by the 

member countries was accompanied by an increase in their diversity. It 

should be noted however, that in the period preceding the outbreak of the 

financial crisis, the European government bonds market was seen as relative-

ly stable. For a long time, investors did not see the signs indicating anything 

in the near future that would shake the foundations of the unified Europe 

thus leading to the debt crisis. 

 Starting with the collapse of the Lehman Brothers bank in September 

2008, we may observe a rapid and progressive decompression of the yield 

spreads of ten-year bonds issued by the European economies. The solvency 

of each country was verified with the result of a lowering of their ratings, 

both several times in the period considered, as well as one-offs however by 

a few levels. The increase in the budget deficit in the most indebted euro 

zone countries such as Portugal, Italy, Ireland, Greece and Spain, and the 

growing borrowing needs of the public sector in these countries were reflect-

ed in the declining creditworthiness measured by a rating, as well as in the 

growth of profitability of bonds issued by them. 

 The situation in the European debt markets improved slightly in the 

summer of 2010, when the prices of Greek bonds slightly rose, which meant 

a reduction in their spreads in relation to German securities. It was expected 

that the system procedures of the European Union and the global initiatives 

taken by the G-20 are starting to deliver the expected results and have 

a calming effect on the international financial market. Since the beginning of 

2011, however, the crisis has entered a new phase leading to a continuous 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

176 

increase in the spreads of Irish and Portuguese bonds. Similarly, the profit-

ability of Greek ten-year bonds has clearly increased. 

 
Figure 3. Yield spread of ten-year government bond of selected European Union 

member states in the period between 01.200611.2016  

 
Figure 4. Yield spread of ten-year government bond of selected European Union 

member states in the period between 01.200611.2016 (cont.) 

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30 

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b

 

20
16

Ju
n

 

20
16

O
ct

 

Latvia Lithuania Luxembourg Malta 

Netherlands Poland Portugal Romania 

Slovakia Slovenia Spain Sweden 

United Kingdom 



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

177 

4. Results of the Econometric Analysis 

 The models given by the formulas (1) and (2) have been estimated and 

verified for two separate periods: the period covering the outbreak of the 

global financial crisis, i.e. 01.200612.2010 and the "post-crisis" period 

between 01.201111.2016. The results obtained for the first period are 

shown in Table 1, while Table 2 contains results for the second period. The 

value of β parameter for the panel data model with individual effects, which 

was estimated for the first period, points that the levels of spreads of ten-year 

treasury bonds issued by the European economies equalise through time. 

Since this parameter is statistically significant, it can be concluded that the 

member states of the European Union have implemented one of the most 

important principles contained in the Treaty of Maastricht that concern the 

convergence of economies. 

Table 1. Results of the estimation and verification of FE_IND and SAR_FE_IND 

models for all analysed issuers in the period between 01.2006–12.2010 

Parameters 
Models 

FE_IND SAR_FE_IND 

 
 

 

–0.0876 
(0.0068) 

 
 

–0.0057 
(0.8126) 
0.0697 

(0.0630) 

Autocorrelation of residuals 
Moran test 

–0.0236 
(0.9386) 

–0.0077 
(0.6807) 

Speed of convergence 0.0019 0.0001 

Note: Numbers in brackets refer to the p-values. All calculations were performed with R – version 3.2.5. 

 The inclusion of the connectivity matrix, which was defined based on 

differences between countries in terms of the ratio 
GDP

debt
, causes a reduction 

in value of  parameter and at the same time causes it to lose its statistical 

significance. Therefore, the study, which takes into account the spatial rela-

tions between economies which are quantified on the basis of the debt ratio 

would lead to the conclusion of an absence of a process of convergence of 

interest rates of government bonds in period I. Interestingly, however, pa-

rameter ρ in the SAR_FE_IND model proved to be statistically insignificant. 

This demonstrates a lack of dependence between the pace of changes in the 

interest rates of government bonds in one country and their paces in coun-

tries with a similar financial stability, i.e. in countries with a similar share of 

debt in gross domestic product. If so, the preferred model in this analysis, 

should be the FE_IND model without spatial effects. 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

178 

 During the period between 01.2011–11.2016 the β parameter in the 

FE_IND model remains negative, however, it is statistically insignificant, 

thus it should be concluded that the European Union faces a process of de-

compression of long-term interest rates. Changes in the profitability of gov-

ernment bond yields during this period were caused by a differentiated risk 

of insolvency of issuers, which was basically caused by ineffective, with 

regard to stability and growth, national fiscal and structural policies in many 

countries and also, in some way, by investors’ speculations. The said period 

reveals heterogeneity of the European Union, which departs from the criteria 

of an optimal currency area. In the conditions of an escalating debt crisis in 

Europe, the common monetary policy for entire territory has not been able to 

equally impact on the economic situation of all individual countries. 

 The introduction of spatial dependencies to the model applied for the 

second period turns out to be an appropriate approach as the parameter ρ in 

this case is statistically significant. It can be concluded that during the eco-

nomic downturn the differences in the debt play a greater role in explaining 

the formation of the levels of spreads in government bond yields. Interest-

ingly, in the SAR_FE_IND model the parameter β is positive. It indicates 

a process opposite to that equalising the levels of spreads in the second peri-

od. In this case we should rather speak of a divergence and a deepening di-

versification of spreads of government bonds in the EU countries. 

Table 2. Results of the estimation and verification of FE_IND and SAR_FE_IND 

models for all analysed issuers in the period between 01.2011–11.2016 

Parameters 
Models 

FE_IND SAR_FE_IND 

 
 

 

–0.0087 
(0.1072) 

 
 

0.0084 
(0.0143) 
0.2231 

(0.0000) 

Autocorrelation of residuals 
Moran test 

–0.0682 
(1.0000) 

–0.0162 
(0.8948) 

Speed of convergence 0.0001 –0.0001 

Note: Numbers in brackets refer to the p-values. All calculations were performed with R – version 3.2.5. 

 The approach presented in the methodology as the vertical convergence 

enables an assessment of the impact of particular issuers on the analysed 

phenomenon. The numerical values contained in Table 3 indicate an individ-

ual effect of each of the considered countries in the process of conver-

gence/divergence in models (1) and (2) in both studied periods. 



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

179 

Table 2. The values of vertical convergence for FE_IND and SAR_FE_IND models 

in the two analysed periods between 01.2006–12.2010 (period I) and 

01.2011–11.2016 (period II) 

Country FE_IND I SAR_FE_IND I FE_IND II SAR_FE_IND II 
Austria 

Belgium 

Bulgaria 

Croatia 

Cyprus 

Czech Republic 

Denmark 

Finland 

France 

Germany 

Greece 

Hungary 

Ireland 

Italy 

Latvia 

Lithuania 

Luxembourg 

Malta 

Netherlands 

Poland 

Portugal 

Romania 

Slovakia 

Slovenia 

Spain 

Sweden 

United Kingdom 
 

–1.978E-04 

–1.966E-05 

–9.050E-05 

–1.092E-05 

–7.016E-05 

–1.058E-04 

–4.643E-05 

2.581E-05 

–5.015E-05 

–2.993E-05 

–1.959E-06 

–5.007E-05 

–1.055E-04 

–1.994E-04 

1.278E-05 

–5.970E-05 

–1.156E-04 

5.281E-05 

–1.004E-05 

1.101E-06 

–2.333E-04 

1.332E-04 

2.929E-04 

1.139E-03 

–7.314E-05 

–2.921E-05 

–1.308E-04 
 

–2.931E-05 

–1.373E-05 

2.743E-05 

2.093E-05 

4.153E-05 

4.999E-05 

4.003E-05 

4.348E-05 

1.183E-05 

4.352E-05 

4.175E-05 

4.342E-05 

4.446E-05 

3.473E-05 

1.625E-05 

4.099E-05 

4.155E-05 

4.570E-05 

4.564E-05 

4.460E-05 

6.394E-05 

7.529E-05 

6.739E-05 

2.103E-04 

2.755E-05 

3.759E-05 

3.421E-05 
 

1.837E-06 

–6.786E-06 

7.330E-06 

7.062E-06 

1.372E-06 

4.723E-06 

–2.431E-06 

–3.075E-06 

1.137E-06 

–7.202E-06 

–4.328E-06 

–1.099E-06 

7.484E-06 

4.399E-06 

–4.330E-06 

–2.743E-06 

2.912E-06 

–1.051E-05 

6.585E-07 

5.809E-06 

–4.054E-06 

–8.785E-06 

–1.359E-05 

4.256E-06 

2.164E-06 

1.405E-05 

4.641E-06 
 

–1.162E-05 

–2.258E-05 

–6.546E-06 

–6.544E-06 

–9.359E-06 

–8.369E-06 

–1.194E-05 

–1.295E-05 

–1.392E-05 

–1.506E-05 

–1.269E-05 

–2.193E-05 

–7.521E-06 

–1.430E-05 

–1.898E-05 

–2.149E-05 

–1.528E-05 

–1.770E-05 

–9.869E-06 

–4.258E-06 

–1.903E-05 

–1.276E-05 

–1.192E-05 

–1.574E-05 

–6.340E-06 

–1.355E-05 

–1.006E-05 
 

 Figure 5 shows the influence of particular countries issuing ten-year 

bonds on the convergence of long-term interest rates resulting from the mod-

el (1) in period I. The positive effects are visible only in several countries, 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

180 

mainly those located in Eastern Europe (Finland, Latvia, Malta, Poland, 

Romania, Slovakia, Slovenia). 

 
Figure 5. The impact of individual countries on the process of equalisation of 

spreads of government bonds in the period between 01.2006–12.2010 

(FE_IND) 

 
Figure 6. The distribution of values of vertical convergence of spreads of govern-

ment bonds in the period between 01.2006–12.2010 (FE_IND) 

positive impact negative impact

strong negative impact

weak negative impact

strong positive impact

weak positive impact



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

181 

 Figure 6 is a graphic illustration of the distribution of vertical conver-

gence indicators divided into four classes. The areas marked in black and the 

brightest grey are characterized by high (above average) negative and posi-

tive values of the index, respectively. Similarly, areas marked in a darker 

grey colour and a slightly lighter grey are characterised by low (not exceed-

ing the average) negative and positive values. The corresponding pairs of 

maps show results for subsequent analysed regimes differentiated by model 

form and time (see Fig. 712). 

 The strongest negative impact on the convergence process of long-term 

interest rates in the period between 01.200612.2010 is observed in such 

countries as Portugal, the United Kingdom, Ireland, the Czech Republic, 

Italy, Austria and Bulgaria. With the exception of the Czech Republic and 

Bulgaria, these are the countries where the debt ratio considerably exceeded 

60 percent of GDP. The massively activated automatic stabilisers and the 

introduction of packages whose purpose was to stimulate the economic 

growth in order to maintain, at all cost, the current level of global demand, 

ultimately led to an abrupt increase in debt and budget deficits. The biggest 

surprise is the presence of Ireland in this group, which in the years between 

20002007 could pass as a role model for other European economies. 

 Figures 7 and 8 are concerned with the SAR_FE_IND model for the 

period between 01.200612.2010. In the case of such a model approach, we 

may speak of a positive impact of the majority of countries on the analysed 

process. However, due to the poor quality of the current model, which was 

mentioned above, these results should be considered with caution. 

 In turn, Figures 9 and 10 show the results obtained on the basis of a pan-

el data model with individual effects for period II. In this case, it can be not-

ed that in the period between 01.201111.2016 there is a decrease in the 

number of countries adversely affecting the process of convergence of long-

term interest rates. Paradoxically, the β parameter in the considered model 

turns out to be statistically insignificant, and, thereby, does not confirm an 

occurrence of the analysed phenomenon of convergence. 

 The results of analysis of vertical convergence for the model which takes 

into account the dependencies based on the share of debt in GDP for the 

second researched period are shown in Figures 11 and 12. It should be noted 

that in this case all the countries have a negative (some greater, other smaller 

 see Figure 12) effect on the alignment process of bond yields. As it was 

mentioned earlier, it is not surprising that instead of convergence we may in 

fact be looking at the divergence process. 

 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

182 

 
Figure 7. The impact of individual countries on the process of equalisation of   

spreads of government bonds in the period between 01.2006–12.2010 

(SAR_FE_IND) 

 
Figure 8. The distribution of values of vertical convergence of spreads of govern-

ment bonds in the period between 01.2006–12.2010 (SAR_FE_IND) 

positive impact negative impact

strong negative impact

weak negative impact

strong positive impact

weak positiwe impact



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

183 

 
Figure 9. The impact of individual countries on the process of equalisation of 

spreads of government bonds in the period between 01.2011–11.2016 

(FE_IND) 

 

 
Figure 10. The distribution of values of vertical convergence of spreads of govern-

ment bonds in the period between 01.2011–11.2016 (FE_IND) 

positive impact negative impact

strong negative impact

weak negative impact

strong positive impact

weak positiwe impact



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

184 

 
Figure 11. The impact of individual countries on the process of equalisation of 

spreads of government bonds in the period between 01.2011–11.2016 

(SAR_FE_IND) 

 
Figure 12. The distribution of values of vertical convergence of spreads of govern-

ment bonds in the period between 01.2011–11.2016 (SAR_FE_IND) 

positive impact negative impact

strong negative impact weak negative impact



 The share of European economies in the process of convergence… 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

185 

Conclusions 

 The global financial crisis and the debt crisis in the European Union has 

had a severe impact on many economic processes taking place in the world. 

The recession tensions increasing in the global and European economy sig-

nificantly affected the convergence of long-term interest rates, which is re-

flected by the state of public finances and the business cycle. At the root of 

the increasing divergence observed in the analysed period there are different 

approaches to fiscal policy and significant discrepancies concerning the state 

of the public finances of European economies. 

 The aim of the study was to identify the impact of individual EU mem-

ber countries on the processes of convergence/divergence of long-term inter-

est rates. Empirical analyses provide a clear evidence that in the first ana-

lysed sub-period i.e. between 01.200612.2010 the divergence in yield 

spreads of ten-year Treasury bonds issued by the European economies were 

reduced, as evidenced by statistically significant and negative β parameter. 

However, given the fact that during this period a vast majority of the EU 

member states was characterised by a negative impact on the process of con-

vergence of long-term interest rates, it is likely that the justification of the 

significance and the negative value of β parameter, may be the fact of ful-

fillment of the convergence criteria during the introduction of euro into the 

circulation, i.e. in 2002 as well as in the first years of its operation. Probably 

the high rate of convergence in the period preceding the time span of the 

analysis carried out in this article enabled maintenance of the desired proper-

ties of the convergence coefficient, despite the fact of an occurrence of 

a destimulating effect of most economies on the process of convergence of 

interest rates. 

 The properties of the spatial model of convergence of ten-year Treasury 

bonds estimated for the first period do not allow confirmation of the hypoth-

esis concerning the fact that the pace of changes of long-term interest rates in 

a given country affect long-term interest rates in countries with a similar 

fiscal stability, measured by the ratio of public debt to GDP. The same mod-

el estimated for the second period leads to a different conclusion, namely 

such that confirms the hypothesis adopted in the paper. 

 In the period between 1.201111.2016 the differences in debt played 

a significant role in explaining the formation of the levels of spreads in gov-

ernment bond yields. The interest rates dictated by the European Central 

Bank exerted a varying impact on the economic situation of particular econ-

omies, thus leading only to an increase in the heterogeneity of the European 



Elżbieta Szulc, Dagna Wleklińska, Karolina Górna 

DYNAMIC ECONOMETRIC MODELS 16 (2016) 165–187 

186 

Union, with the result that in relation to that period it is appropriate to speak 

of a move towards divergence of interest rates in the Community. 

 In order to confirm these suppositions the research should be continued 

with consideration of an extended time span of the analysis to take into ac-

count the period of adoption of the euro. 

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Udział gospodarek europejskich w procesie zbieżności  
długoterminowych stóp procentowych w UE w okresie 2006–2016 

Z a r y s  t r e ś c i. Artykuł dotyczy procesu zbieżności stóp procentowych dziesięcioletnich 

obligacji skarbowych krajów UE. Stanowi on próbę oceny udziału poszczególnych gospoda-

rek europejskich w tym procesie z wykorzystaniem koncepcji tzw. konwergencji pionowej. 

Podstawowym narzędziem analizy były modele panelowe z ustalonymi efektami, w tym 

modele uwzględniające powiązania między gospodarkami, kwantyfikowane za pomocą ma-

cierzy odległości między wskaźnikami stabilności fiskalnej, jako udział długu publicznego 

w PKB. Analizę przeprowadzono dla 27 krajów członkowskich UE w okresie styczeń 2006–

listopad 2016. 

S ł o w a  k l u c z o w e: długoterminowe stopy procentowe, proces zbieżności, stabilność 

fiskalna, modele panelowe, macierz powiązań ekonomicznych. 

http://dx.doi.org/10.1257/jep.26.3.49

