The Convergence in the Sustainability of the Economies of the European Union Countries between 2006 and 2016

Abstract

On the background of the exponential growth of the world’s population, doubled by the decrease of natural resources and the continuous, accentuated degradation of the quality of the environment, with global warming as its main effect, ensuring the sustainability of economic and social processes is becoming a growing concern. At the European Union level, it is important that all member countries adhere to and implement common measures on sustainable development, which involve, inter alia, ensuring the convergence of policies and their effects at EU level. The EU through detailed SDGs presents the structure of a system of indicators structured on 17 objectives, indicators taken over, implemented, and calculated by EUROSTAT. The study proposes, based on a Composite Index of Sustainable Development of EU Countries’ Economies (ISDE-EU), the analysis of the convergence of the sustainability of EU states’ economies, not so much at individual level, but at cluster level, each cluster containing EU countries with similar/close ISDE-EU levels and dynamics. The results of the analysis confirm the partial existence of the beta and sigma convergence of the sustainability of EU countries’ economies. Please note that, at the time when we processed data, the UK was an EU state, which is why it was included in the analysis.

1. Introduction

On the background of the exponential growth of the world’s population, doubled by the decrease of natural resources and the continuous, accentuated degradation of the quality of the environment, with global warming as its main effect, ensuring the sustainability of economic and social processes becomes an increasingly important issue of concern. At EU level, it is important that all member countries adhere to and implement common measures on sustainable development, which involves, among other things, ensuring the convergence of policies and their effects at EU level [1].

The EU (European Union) through SDGs (Sustainable Development Goals) [2] presents a system of indicators structured on 17 objectives, indicators taken over, implemented, and calculated by EUROSTAT. Based on these, Turturean et al. [3] proposed the construction of the Composite Index of Sustainable Development of EU Countries’ Economies (ISDE-EU) to quantify and compare the sustainability of EU economies, and which was the basis for the analysis of the beta and sigma convergence of EU economies for the period 2006–2016.

The study proposes an analysis of the convergence of the sustainability of EU states’ economies grouped in clusters, each containing EU countries with similar/close ISDE-EU levels and dynamics.

The topic proposed by this article aims at combining two concepts: sustainable development, more precisely ways of measuring it, and convergence.

Concerns about the protection of the environment and the conservation of natural resources have existed since ancient times, but never in human history has it been clearer than now that they must become a desideratum of the policies of all states of the world, the most important task being held by developed economies.

In 1987, the UN (United Nations) concludes in the Report of the World Commission on Environment and Development “Our Common Future” [4]:

“[…] the “environment” is where we all live: and “development” is what we all do in attempting to improve our lot within that abode. The two are inseparable. Further, Development issues must be seen as crucial by the political leaders who feel that their countries have reached a plateau towards which other nations must strive. Many of the development paths of the industrialized nations are clearly unsustainable. And the development decisions of these countries. Because of their great economic and political power. Will have a profound effect upon the ability of all peoples to sustain human progress for generations to come.”

This moment is a reference both for the development and popularization of the concept of sustainable development and for the creation of global policies that promote the idea of the need to adhere to a set of principles and rules that lead to achieving a common goal: the sustainable development of the world states.

The discussion on the convergence of EU states would have become useless without an external factor that would lead states to move towards a common denominator on sustainable development. This factor arose with the presentation by the UN of the report “Our Common Future” (1987) and culminated at EU level with the publication in 2001 by the Commission of European Communities (CEC) [5] of the document “A Sustainable Europe for a Better World: A European Union Strategy for Sustainable Development”, which states that:

“Sustainable development is a global objective. The European Union has a key role in bringing about sustainable development, within Europe and also on the wider global stage, where widespread international action is required. To meet this responsibility, the EU and other signatories of the 1992 United Nations’ “Rio declaration” committed themselves, at the 19th Special Session of the United Nations’ General Assembly in 1997, to draw up strategies for sustainable development in time for the 2002 World Summit on Sustainable Development. This strategy forms part of the EU preparations for that summit.”

These statements will create the proper environment for a discussion on convergence of the sustainable development of EU states, and we have in mind the existence of a set of indicators, principles, and rules designed to quantify and compare the intensity with which this new concept of sustainable development is implemented at the level of economic policies of the EU member states.

The current paper tackles the beta and sigma convergence of the sustainability member states of EU 27 (EU states including United Kingdom and excluding Croatia). To reach this objective, a sustainability composite index developed by Turturean et al. [3] is used. The results obtained confirm: the existence of clusters with homogeneous states from the point of view of sustainability and the existence of both beta and sigma convergence at the level of the entire EU 27 as well as for some of the clusters.

This paper is relevant and brings contributions to the existing literature in the field of sustainability. Its first originality is represented by the use of a composite index for the analysis of similar behaviours of sustainability in the EU 27 member states, followed by the identification of five clusters in the EU 27 member states. The second originality element is the beta and sigma analysis of the sustainability convergence in the EU 27 member states. Thirdly, there is the analysis of the convergence at the level of the five clusters based on the composite index of sustainability, and in the fourth place the paper analysed the correspondence between the results obtained for the beta and sigma convergence both at the level of EU 27 and at cluster level.

The paper continues with the following structure: the Section 2 presents the literature and the research hypotheses, the Section 3 describes the Materials and Methods, the Section 4 highlights the Results, ending with the Conclusions.

2. Literature Review and Research Hypotheses

The term sustainable development is relatively new and was first used in 1987 at the UN level in the Report of the World Commission on Environment and Development [1], being defined as follows: “Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs” a definition that tried to outline the idea of responsible development of society taking into account the protection of the environment with all the resources it makes available.

The continuing concern about the degradation of natural resources and their conservation is the basis for the day-to-day growth of indicators that want to quantify each type of resource. Therefore, for researchers, measuring sustainable development is an ongoing challenge. Among the pioneers who worked on the creation of a system of indicators for measuring sustainable development there is Bossel [6], who is one of the first researchers to try to define the components of such a system, broken down by territorial administrative aggregation levels.

Starting with 2002, just one year after the launch of “A Sustainable Europe for a Better World: A European Union Strategy for Sustainable Development” (2001) [3] and the establishment by Eurostat of the EU SDGs [2], the efforts of specialists have been focused on the construction of an aggregate indicator that allows the evaluation of sustainable development—Ronchi et al. [7] for Italy—but who noted that the data available at the time were poor in terms of both recording frequency and number of indicators to quantify different dimensions of sustainable development, which does not allow for statistical quality data processing. In the same year, Barrera-Roldan and Saldivar-Valdes [8] proposed a methodology for building a sustainable development index (SDI) which, in the opinion of the authors, was able to quantify both the degree of greening of urban community activities as well as their socio-economic status. Myriam Nourry [9] conducts a study on the ability to comprehensively characterize sustainable development in France using a set of eight indicators, concluding that none of the indicators provide a comprehensive picture of the phenomenon of sustainable development and highlights the fact that the use of GDP in assessing sustainable development artificially reduces the complexity of the phenomenon.

Bartelmus [10], in “Quantitative Economics: How Sustainable are our Economies?”, presents a set of aggregation methods and tools, needed to build indicators to measure sustainable development. In a study published in 2014, Salvati and Carlucci [11], based on a multidimensional approach, measured the sustainable development, locally, for Italy, using a composite index using exploratory statistics and spatial analysis. In 2015, Mikulic, Kozic, and Kresic [12] make a critical analysis of measuring sustainable development in tourism using weighted indicators. In order to support the authorities in monitoring sustainable development, Tret’yakova and Osipova [13] proposed and developed a set of methodological tools that they applied to several regions in Russia.

In 2019, Turturean et al. [3] proposed to quantify the level of sustainability for EU 27 economies a composite index called the Index of Sustainable Development of EU Countries’ Economies (ISDE-EU), which initially considered all indicators of sustainable development provided by EUROSTAT. It was based on the methodology provided by the OECD, Handbook on Constructing Composite Indicators: Methodology and User Guide [14].

Please note that, during the period analyzed, the UK was an EU state, which is why it was included in the analysis.

Even if in the last six years, between 2017–30 July 2022, according to Clarivate, Web of Science (WoS), there were 15,991 papers in the literature whose title comprises the word Convergence, most of them being in the field of Mathematics, and 30,439 papers which contained in the title the word Sustainability; when we refer to papers which contained both the words Convergence and Sustainability, their number reduces drastically, narrowing down to 28 papers, 27 more precisely since one repeats itself. These 27 papers, even if few in number, have a very heterogeneous structure as regards the distribution by category according to WoS.

The distribution of papers in the above mentioned WoS report by topics related to the theme of our current paper is the following:

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the Category Green sustainable Science and Technology with 9 studies;

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the Category Environmental Studies with 8 papers;

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the Category Economics with 3 studies;

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the Category Development Studies with 2 papers.

They total 22 papers, but we need to mention that their number could be even smaller since the same paper might have been included in two or even three different categories.

A large part of the papers published on the topic of Convergence and Sustainability during the period 2017–30 July 2022 tries to prove the need to introduce the concept of Sustainability and subsequently to reach a convergence in sustainability as standards of corporate management, starting from the social responsibility assumed by most of the corporations, Pazienza et al. [15]. Another category of papers attempts to determine, by means of the bibliometric analysis conducted, the correlation between convergence of innovation and sustainability. De la Vega and De Paula [16], following the bibliometric analysis conducted, coin a new term, “innovability”, which should simultaneously cover the concept of innovation and that of sustainability. Another paper from the same field focuses on the creation of Local Innovative and Productive Systems (LIPS) and on the effect the innovations they produce lead to the increase in the sustainability level of socio-economic processes, giving as an example Brazil [17].

George [18] tries to demonstrate, on a case study performed at CIAL: India’s Green Port, what the framework required to reach a natural convergence of sustainability is, by identifying as main factors the good governance, the social responsibility, and an adequate organisational management.

Harlow et al. [19] propose an efficient management manner of sustainable transition by projecting the “convergent political windows”, which should ensure the fulfilment of the expected outcome by using “a combination between the transition management and the method of multiple flows in order to increase the transforming potential of transitional arenas”.

Simo-Kengne [20] performs an ample study on 148 countries during the period 2006–2016, using panel data, concerning the relationship between the tourism growth and the environmental sustainability and concludes that beside the positive implications the tourism has on the economies of the states comprised in the analysis, there is also the estimation of a negative effect on the environmental sustainability. In this regard, the authors propose the implementation of a tourism management which should set the balance between its economic benefits and the negative impact these have on the environment.

Guidotti [21] suggests that the economic sustainability has a catalytic effect on the health state of the population, by eliminating the treatment disparities. The population’s health state also beneficially influences in its turn economic growth, producing thus a positive chain reaction.

The COVID-19 pandemic and its secondary consequence, the drop in the population’s mobility, has resulted in the migration of convergence towards the objectives of sustainable development (ODD) according to the 2030 Agenda of the United Nations published in 2015 [22] by means of:

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the transition to the Digital Economy, Castro et al. [23] and Camodeca and Almici [24];

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the digitalization of the educational process, especially in higher education, Deev et al. [25] and Melles et al. [26].

The Convergence and the Sustainability get new meanings in finances. Thus, Sbarcea [27] considers that the economic sustainability provides sustainability to a country on a long term regarding the main macroeconomic and financial indicators such as: Inflation rate (%), Fluctuation of exchange rate, Long-term interest rate (%), Budget Deficit (%), and Public debt (% of GDP). The adherence to the Euro zone will be ensured by setting the convergence and the stability of these indicators at standard values imposed by the EU. The example provided by the author is for Romania for the period 2006–2018.

Vilas et al. [28] ascertain for the period 1990–2018 the existence of convergence between the conventional stock exchange indices, belonging to the FTSE index family, and sustainable development. This is due to the corporate social performance (CSP), the corporate social responsibility (CSR), and the socially responsible investing (SRI). In the financial field, Lesko and Muchova [29], make an analysis of the sustainability of the convergence of the growth rate of the balance of payment (BOP) of ECE countries towards an equilibrium point (BOP) calculated as the ratio between the elasticities of the incomes of export and import demand and the increase of external demand, according to Thirlwal’s theory [30]. The authors manage to prove that almost all the ECE countries have a growth rate which exceeds the value of the balance point of BOP Thirlwal. As a paradox, the convergence of the ECE region is not sustainable if one takes into account the low ratio of income elasticities and the growth of external debt.

The last category of papers aims to study the economic convergence and its influence on sustainability at international level, Juknys et al. [31]; at European level, Suciu et al. [32]; or at regional level, Di Berardino et al. [33].

The three papers are the closest ones through the methodology and objectives to what the authors of the current paper also aimed at doing. Nevertheless, none has the goal to study directly the convergence of the states/regions observed based on a sustainability composite index.

The analysis proposed by the authors in this paper is original because it directly assesses the convergence in sustainability, by means of the sustainability composite index, ISDE-EU27, built for the 27 EU countries. The above-mentioned papers start from the analysis of the economic growth convergence and end in identifying correlations or associations with economic sustainability.

Many articles use the indicator Gross Domestic Product per capita (GDP/capita) in the convergence analysis. This is a controversial indicator especially for the evaluation (measurement) of development (Stiglitz et al., (2009), pp. 23–58) [34]. In the European Union, this indicator is still used to assess economic cohesion and to allocate resources from European funds.

Studies based on the GDP/capita, which focused on convergence in the European Union, were conducted by Mankiw et al. [35], Ben-David [36], Islam [37], Smolny [38], Barro, Sall-I-Martin [39], De la Fuente [40], and Di Liberto and Symons [41].

In most cases, the presence of the convergence process was confirmed, which was highlighted especially for beta convergence. However, the lack of convergence within the studied regions (countries) in the European Union was also analyzed. On the one hand, we find the lack of beta convergence in the works of Baro, Sala-i-Martin (pp. 479–496) [39], Mankiw et al. (pp. 407–437) [35] and, on the other hand, the lack of sigma convergence in the works of Ben-David (pp. 653–679) [38] or Mello and Perrelli (pp. 643–667) [42] for the countries studied.

As mentioned above, the gross domestic product per capita indicator is used in many studies on convergence. The assessment of the development of some countries and the possible convergence is also studied by authors who used a more complex indicator, which considers not only income but also social aspects. This indicator, namely human development index (HDI) was used to measure the convergence beta of Mazumdar [43], Stutcliffe [44], and Noorbakhsh [45]. Konia and Guisan [46] tested both beta convergence and sigma convergence.

The concept of beta convergence arose from the neoclassical growth models of Solow [47] or Cass [48] and is based on the observation that as capital increases, its profitability decreases. This concept is known as the phenomenon of catching-up. Based on this phenomenon, countries with poor economies could “catch up” with the developed economies. Beta convergence is the inverse relationship between the growth rate, specific to a period of time, and the initial level of economic development. The estimation of a negative coefficient between the two variables reflects the existence of beta convergence.

Beta convergence identifies the extent to which economic growth rates are the cause of convergence identified by sigma convergence. Therefore, the existence of beta convergence for the analyzed countries is a necessary but not sufficient condition for the existence of sigma convergence [49]. Thus, there may be beta convergence, but there should be no reduction of the dispersion between the economic growth of the countries, i.e., there should be no sigma convergence. Therefore, sigma divergence can exist at the same time with both beta divergence and beta convergence [50]. These two methods of identifying convergence are complementary to Baro, Sala-i-Martin [39] but the reduction of disparities between countries is highlighted only by the sigma convergence [51,52].

Beta convergence has emerged as a possibility to assess convergence. The method was initially challenged by researchers such as Friedman [51] and Quah [52], who argued that it was not a relevant possibility to quantify the convergence of economic growth. Gradually, beta convergence has taken an increasingly important place in research, evolving from the initial form of absolute beta convergence to conditioned beta convergence.

In the specialized works [53], there are three ways to quantify the beta convergence: absolute beta convergence, group beta convergence (club), and conditional convergence. Absolute beta convergence considers the basic hypothesis of neoclassical theory, namely the existence of the catching-up phenomenon, without considering the different technological and institutional conditions of the countries included in the analysis. Group (club) beta convergence involves the analysis of the phenomenon of convergence within groups of homogeneous countries in terms of technological, institutional, and economic policy conditions. Conditional beta convergence considers absolute convergence and the determinants of economic growth in estimating the equation used in beta.

If, in 1986, Baumol [54] presented a methodology for the analysis of beta convergence, in 1990, Sala-i-Martin presented the term sigma convergence in his doctoral dissertation for the first time [55]. The concept of sigma convergence can be defined as a long-term decrease in dispersion across a group of countries or regions. Thus, if the value of the indicator by which the sigma convergence is measured decreases over time, σ_t > σ_t+n means that we have a reduction of the dispersion at the level of the regions, which indicates the convergence of the studied phenomenon (sigma convergence was initially used for the study of economic convergence). Otherwise, if σ_t < σ_t+n, there is a divergence in the regions.

In the existing literature, empirical research on the sigma of convergence mainly uses the following indicators: coefficient of variation, standard deviation of the logarithmic variable, Gini index, and Theil Index. The presence of the sigma convergence was confirmed by the coefficient of variation in the works of de la Fuente [56], Giannetti [57], Soukiazis and Castro [58], Sala-i-Martin [59], and Yang et al. [60]. The standard deviation indicator has been used by: Fuente [56], Sa-la-i-Martin [59], Young et al. [49], Yang at al. [60]. The Gini and Theil indices were the basis for the assessment of convergence in the works belonging to Sala-i-Martin [59], Monfort [61] or Yang et al. [60].

As a consequence of these previously presented results, the authors formulate the following hypotheses:

H1. 

At the level of the EU 27 countries there is convergence in sustainability.

H1.1. 

At the level of the EU 27 countries there is beta convergence in sustainability.

H1.2. 

At the level of the EU 27 countries there is sigma convergence in sustainability.

Kowalski and Rybacki [62] study the convergence of innovation performance in the World Economy and reach the conclusion that there is a lack of convergence of innovative potential between countries. They also underline a discrepancy between the innovative potential of developed economies and of the emerging ones. In the same line, there is the research conducted by Kijek and Matras-Bolibok [63] and Park [64].

An important theme in the analysis of economies’ sustainability is represented by the analysis of energy use from renewable resources which is found in the studies conducted by Jankiewicz [65], Butnaru et al. [66], Payne et al. [67], and Kasman et al. [68]. In the research performed by Butnaru et al. [69], the existence of the convergence of renewable energy consumption from non-conventional sources is mentioned but it is also underlined that the results are influenced by the development level of the countries analysed, by the homogeneity or heterogeneity of the states from the group as well as by the measures of economic policy taken in the states under analysis.

The study of the convergence of innovations in the renewable energy is dealt with by Bai et al. [69] in China and Kijek et al. [70] in the European countries. Kijek et al. [70] when studying the European countries prove the existence of three convergence clubs while the identified factors of the occurrence of these clubs are the differences between the human resources in science and technology, the initial expenses per capita for environmental research and development and the environment policy.

Testing the real convergence in order to attain a long-term sustainability was addressed by Suciu et al. [32], based on the data available for the countries that entered the EU between 2004–2013. In this research it is ascertained that the newly entered EU countries which have the single currency (EURO) form a homogeneous cluster of convergence, unlike the newly entered EU states which have not yet adopted the single currency (non-EURO). The adoption of the single currency is an important objective alongside the necessary steps of this approach so that the newly entered EU countries should aim for a sustainable, convergent development towards the developed countries in the EU.

As a consequence of these previously presented results, the authors formulate the following hypotheses:

H2. 

The different evolutions of the sustainability dynamics of the EU 27 countries can be grouped by homogeneous clusters.

H3. 

At the level of the EU 27 countries there is convergence in sustainability.

H3.1. 

At the level of the EU 27 there is beta convergence in sustainability.

H3.2. 

AT the level of the EU 27 clusters there is sigma convergence in sustainability.

Based on the results of the current research, the authors will attempt to establish to which extent the above-mentioned hypotheses are validated or not. Based on the validated/invalidated hypotheses actions to be taken by the European organisations/institutions will be suggested, actions able to monitor the sustainability of the economies of the member countries. Starting from the suggestions made available by this study, measures/policies can be built and implemented in order to lead to and/or to strengthen/accelerate the convergence process in sustainability of the economies of the EU states.

3. Materials and Methods

Figure 1 summarizes, in stages, the methodology used in this article. Subsequently, each stage will be presented in detail.

3.1. Stage 1: Data Collection

The data used in this article is based on the values of an Index of Sustainable Development of EU Countries’ Economies (ISDE-EU), which was originally based on 101 independent variables structured in 17 goals and represented by the indicators in chapter Sustainable Development Indicators on the Eurostat website [71]. The ISDE-EU calculation methodology and its values are explained in detail in the article Composite Index of Sustainable Development of EU Countries’ Economies (ISDE-EU) published by Turturean et al. [3] and are not covered by this article. The values of the ISDE-EU indicator for the 27 countries corresponding to the analyzed 11-year period, 2006–2016, are presented in

Table 1. ISDE-EU for 2006–2016 for 27 Countries, original data.

		ISDE-EU
		Years
		2006	2007	2008	2009	2010	2011	2012	2013	2014	2015	2016
Country	AT	0.080	0.100	0.220	0.150	0.200	0.150	0.210	0.240	0.250	0.250	0.280
	BE	0.160	0.150	0.120	0.140	0.180	0.180	0.200	0.190	0.210	0.210	0.250
	BG	0.320	0.430	0.200	0.210	0.260	0.180	0.210	0.240	0.140	0.190	0.210
	CY	−0.190	−0.170	−0.240	−0.270	−0.230	−0.240	−0.200	−0.190	−0.180	−0.130	−0.150
	CZ	−0.370	−0.360	−0.370	−0.380	−0.330	−0.320	−0.240	−0.290	−0.250	−0.220	−0.210
	DK	0.320	0.370	0.430	0.490	0.520	0.460	0.460	0.520	0.490	0.550	0.510
	EE	−0.220	−0.190	−0.220	−0.240	−0.220	−0.140	−0.160	−0.180	−0.130	−0.140	−0.120
	FI	0.290	0.300	0.340	0.350	0.310	0.330	0.320	0.280	0.300	0.340	0.270
	FR	−0.010	−0.010	−0.020	0.050	0.080	0.050	0.060	0.130	0.120	0.120	0.150
	DE	0.080	0.130	0.170	0.170	0.170	0.210	0.220	0.220	0.280	0.290	0.330
	GR	−0.060	−0.060	−0.060	−0.080	−0.110	−0.020	0.080	0.140	0.130	0.140	0.160
	HU	−0.170	−0.300	−0.330	−0.310	−0.290	−0.230	−0.120	−0.080	−0.100	−0.120	−0.150
	IE	−0.030	−0.010	0.030	0.020	0.050	0.110	0.200	0.180	0.210	0.200	0.180
	IT	−0.060	−0.030	−0.010	−0.030	0.030	0.130	0.200	0.250	0.270	0.290	0.400
	LV	0.080	0.000	0.050	0.060	−0.010	0.040	−0.020	−0.010	−0.070	−0.070	−0.140
	LT	0.040	−0.060	−0.120	−0.150	0.000	−0.090	−0.120	−0.070	−0.100	0.010	0.030
	LU	0.260	0.250	0.290	0.350	0.330	0.280	0.290	0.340	0.360	0.360	0.420
	MT	−0.490	−0.460	−0.440	−0.490	−0.400	−0.360	−0.330	−0.230	−0.250	−0.270	−0.290
	NL	0.100	0.120	0.120	0.150	0.160	0.180	0.180	0.200	0.220	0.240	0.290
	PO	−0.080	−0.150	−0.230	−0.240	−0.220	−0.240	−0.220	−0.220	−0.190	−0.190	−0.150
	PT	−0.140	−0.090	−0.060	−0.080	−0.050	−0.050	−0.070	−0.010	0.030	0.000	−0.010
	RO	0.050	0.110	0.010	−0.050	−0.060	−0.080	−0.030	−0.020	−0.010	0.050	0.030
	SK	−0.310	−0.380	−0.390	−0.380	−0.320	−0.330	−0.320	−0.290	−0.230	−0.200	−0.230
	SL	−0.240	−0.220	−0.180	−0.150	−0.120	−0.080	−0.010	0.000	−0.020	−0.010	0.010
	ES	0.040	0.040	0.100	0.120	0.150	0.100	0.150	0.150	0.200	0.210	0.230
	SE	0.360	0.250	0.320	0.380	0.380	0.410	0.440	0.450	0.460	0.470	0.520
	UK	0.170	0.170	0.200	0.190	0.220	0.180	0.210	0.240	0.230	0.250	0.260

Source: Adapted from Turturean et al. [3] (calculations using EUROSTAT data).

.

The EU countries not included in this table did not have sufficient data at the date when calculations were performed, to allow the calculation of the ISDE-EU for the entire analyzed period.

3.2. Stage 2: Rescaling Original Values of ISDE-EU to a Fixed 0.01–0.99 Range

In order to increase the degree of comparability and interpretation of the original ISDE-EU values presented in Table 1, we opted for their transformation so that the range of variation is reduced to the range [0.01; 0.99], where 0.01, represents countries with very weak sustainable economy while 0.99 represents countries with very strong sustainable economy.

The relationship used to rescale the original data, is called min-max normalization between specific range [a; b], ISDE-EU and is of the form [72]:

ISDE-EU’_c,y = a + [(ISDE-EU_c,y − ISDE-EU_min)(b − a)/(ISDE-EU_max − ISDE-EU_min)]

(1)

Normalization based on an arbitrary set of values [a, b] will have the following effects on the initial data:

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the normalized values will take values from the interval [a, b] where a = 0.01 and b = 0.99, which is finite and allows us to calculate the indicators for beta and sigma convergence (e.g., logarithms);

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the influence of the outliers on the series is diminished;

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the level of homogeneity of the series increases the data becoming more compact.

The data resulting from the normalization [0.01; 0.99], based on the relation (1), are presented in

Table 2. ISDE-EU for 2006–2016 for 27 Countries Rescaling data to a fixed 0.01–0.99 range.

		ISDEEU
		Years
		2006	2007	2008	2009	2010	2011	2012	2013	2014	2015	2016
Country	AT	0.5471	0.5660	0.6790	0.6131	0.6602	0.6131	0.6696	0.6979	0.7073	0.7073	0.7356
	BE	0.6225	0.6131	0.5848	0.6037	0.6413	0.6413	0.6602	0.6508	0.6696	0.6696	0.7073
	BG	0.7733	0.8769	0.6602	0.6696	0.7167	0.6413	0.6696	0.6979	0.6037	0.6508	0.6696
	CY	0.2927	0.3115	0.2456	0.2173	0.2550	0.2456	0.2833	0.2927	0.3021	0.3492	0.3304
	CZ	0.1231	0.1325	0.1231	0.1137	0.1608	0.1702	0.2456	0.1985	0.2362	0.2644	0.2738
	DK	0.7733	0.8204	0.8769	0.9335	0.9617	0.9052	0.9052	0.9617	0.9335	0.9900	0.9523
	EE	0.2644	0.2927	0.2644	0.2456	0.2644	0.3398	0.3210	0.3021	0.3492	0.3398	0.3587
	FI	0.7450	0.7544	0.7921	0.8015	0.7638	0.7827	0.7733	0.7356	0.7544	0.7921	0.7262
	FR	0.4623	0.4623	0.4529	0.5188	0.5471	0.5188	0.5283	0.5942	0.5848	0.5848	0.6131
	DE	0.5471	0.5942	0.6319	0.6319	0.6319	0.6696	0.6790	0.6790	0.7356	0.7450	0.7827
	GR	0.4152	0.4152	0.4152	0.3963	0.3681	0.4529	0.5471	0.6037	0.5942	0.6037	0.6225
	HU	0.3115	0.1890	0.1608	0.1796	0.1985	0.2550	0.3587	0.3963	0.3775	0.3587	0.3304
	IE	0.4435	0.4623	0.5000	0.4906	0.5188	0.5754	0.6602	0.6413	0.6696	0.6602	0.6413
	IT	0.4152	0.4435	0.4623	0.4435	0.5000	0.5942	0.6602	0.7073	0.7262	0.7450	0.8487
	LV	0.5471	0.4717	0.5188	0.5283	0.4623	0.5094	0.4529	0.4623	0.4058	0.4058	0.3398
	LT	0.5094	0.4152	0.3587	0.3304	0.4717	0.3869	0.3587	0.4058	0.3775	0.4812	0.5000
	LU	0.7167	0.7073	0.7450	0.8015	0.7827	0.7356	0.7450	0.7921	0.8110	0.8110	0.8675
	MT	0.0100	0.0383	0.0571	0.0100	0.0948	0.1325	0.1608	0.2550	0.2362	0.2173	0.1985
	NL	0.5660	0.5848	0.5848	0.6131	0.6225	0.6413	0.6413	0.6602	0.6790	0.6979	0.7450
	PO	0.3963	0.3304	0.2550	0.2456	0.2644	0.2456	0.2644	0.2644	0.2927	0.2927	0.3304
	PT	0.3398	0.3869	0.4152	0.3963	0.4246	0.4246	0.4058	0.4623	0.5000	0.4717	0.4623
	RO	0.5188	0.5754	0.4812	0.4246	0.4152	0.3963	0.4435	0.4529	0.4623	0.5188	0.5000
	SK	0.1796	0.1137	0.1042	0.1137	0.1702	0.1608	0.1702	0.1985	0.2550	0.2833	0.2550
	SL	0.2456	0.2644	0.3021	0.3304	0.3587	0.3963	0.4623	0.4717	0.4529	0.4623	0.4812
	ES	0.5094	0.5094	0.5660	0.5848	0.6131	0.5660	0.6131	0.6131	0.6602	0.6696	0.6885
	SE	0.8110	0.7073	0.7733	0.8298	0.8298	0.8581	0.8863	0.8958	0.9052	0.9146	0.9617
	UK	0.6319	0.6319	0.6602	0.6508	0.6790	0.6413	0.6696	0.6979	0.6885	0.7073	0.7167

Source: Data calculated by authors based on data using the relation (1).

.

3.3. Stage 3: Grouping Countries by the Dynamics of ISDE-EU

The 27 EU countries have different levels of sustainability of their national economies, which is why, in order to obtain groups of countries with similar values and behaviors of ISDE-EU for the period 2006–2016, we used Hierarchical Cluster Method, implemented in IBM-SPSS 20 Software (IBM, Armonk, NY, USA).

Cluster grouping of the sustainability of EU states will enable the adoption by the EU of differentiated measures or policies adjusted to the specifics of the countries forming each cluster, which should ensure the convergence to the objectives set up in the 2030 Agenda for Sustainable Development [22].

When choosing the measure of the distance between the clusters, we considered the fact that the original data were min-max normalized, resulting in a more compact data set with a diminished outlier’s effect [73]. The advantages resulting from the min-max normalization will recommend, as a measure of the distances between the clusters, one of the components of the Minkovski family of measures, among which are: Manhattan Distances, Euclidean Distance, and so on [74].

The Euclidean distance is frequently used and easy to calculate, being adapted to work with data sets for which there are compact or isolated clusters [73,75].

In our analysis, we chose the Squared Euclidean Distance (SED) measure with the intention of amplifying, by squaring, the advantages of the Euclidean Distance and creating clusters with a different behavior. Additionally, the Squared Euclidean Distance (SED) is frequently used for continuous variables, to measure the distances between two indents, x and y, belonging to two different clusters, C_i and C_j, and is calculated as the sum of the squared differences between the values for the instances [76].

$$SED\left(x_i,y_j\right)={\displaystyle \sum }_{\begin{array}{c}i=1, card \left(C_i\right)\\ j=1, card \left(C_j\right)\end{array}}{\left(x_i-y_j\right)}^2$$

(2)

Hierarchical Clusters Agglomeration Algorithm involves the iterative traversal of a set of steps, involving the merging of instances into sub-clusters and the sub-clusters into larger and larger clusters, respecting a set of principles related to the similarity of the instances included in them [77].

The algorithm will stop when it reaches a finite number of clusters that are different enough from each other so that they can no longer be attached to larger clusters.

Before explicitly presenting the steps, we must go through, we will have to specify that the method of calculating the distances between two clusters is the average between linkages and it was based on the metric of the square of the Euclidean distances, as it is described in the relation (2). This method assumes that the distance between two clusters, C_i and C_j, is measured by averaging the distances between all pairs of instants of the form (x, y), where instant x belongs to cluster C_i and y belongs to cluster C_j [78], as you can see in Figure 2.

In relation (3) we have presented the way of calculating the distance between the pair of clusters C_i and C_j based on the average between linkage method [79]:

$$d\left(C_i,C_j\right)=\overline{d^2}=\frac{{{\displaystyle \sum }}_{k=1}^{card\left(C_i\right)\times card\left(C_j\right)}(d_k^2)}{card\left(C_i\right)\times card\left(C_j\right)}=\frac{{{\displaystyle \sum }}_{\begin{array}{c}i=1, card \left(C_i\right)\\ j=1, card \left(C_j\right)\end{array}}{\left(x_i-y_j\right)}^2}{card\left(C_i\right)\times card\left(C_j\right)}$$

(3)

The application of the Hierarchical Clusters agglomeration algorithm [80] using the average between linkage method involves the iterative completion of the following steps:

• Each instance will initially be considered a separate cluster;
• All cluster pair distances will be evaluated based on the calculation ratio specific to the average between linkage method, relation (3), using the square metric of Euclidean distances, relation (2);
• Construction of the matrix containing the distances between the pairs of clusters calculated in the previous step;
• Choosing the pairs of clusters for which, in the distance matrix, we record the smallest distances;
• Based on a similarity criterion, we attach the pairs of clusters at a distance less than a reference value;
• Resuming the previous steps until the clusters can no longer be attached;
• Once the goal from the previous step has been reached, the algorithm will stop and provide the last clustered structure obtained.

3.4. Stage 4: Convergence Analysis

3.4.1. Beta Convergence Methodology

Baumol [51] was the first to develop a methodology for studying the real convergence of certain countries’ economies. He relied on the estimation of a regression model that he identified graphically. The regression model he used is:

$$\frac{1}{T}\mathit{ln}\left(y_{i,t_0+T}-y_{i,t_0}\right)={\beta }_0+{\beta }_1\mathit{ln}\left(y_{i,t_0}\right)+{\epsilon }_{it}$$

(4)

where:

$T$—last recorded time period

$y_{t_0}$—the value of GDP per capita at the beginning of the period

$y_t$—the value of GDP at the end of the period

${\beta }_0$—intercept

${\beta }_1$—regression coefficient, an indicator that measures beta convergence.

Mankiw et al. [35] and Barro and Sala-i-Martin [81] created the methodology for estimating the beta convergence based on economic growth modeling. The study performed by Mankiw et al. [35] is based on the theoretical model of Solow [47] and Swan [82] and Barro and Sala-i-Martin [81] deduce the model of regression based on the theoretical model of Ramsey [83], Cass [48], and Koopmans [84].

$$\frac{1}{T}\mathit{log}\left(\frac{y_{i,t_0+T}}{y_{i,t_0}}\right)={\beta }_0+{\beta }_1\mathit{log}\left(y_{i,t_0}\right)+{\epsilon }_{it}$$

(5)

where:

$y_{i,t_0}$—represents the per capita income of the region or country i in period t

$T$—represents the number of years for which beta convergence is estimated

${\epsilon }_{it}$—the random factor.

If ${\beta }_1$ is negative and statistically significant, then there is the phenomenon of beta convergence. If this parameter is positive, it will indicate a divergence phenomenon.

Based on the parameters estimated by the previous relation, the convergence rate can be calculated, which represents the time period necessary to halve the difference between the incomes at the level of the individual states to the equilibrium state. The beta convergence rate, according to Barro and Sala-i-Martin [39], is calculated as follows:

$$\beta =-\frac{1}{T}\mathit{ln}\left(1-{\beta }_{1}T\right)$$

(6)

This indicator expresses the rate of convergence, i.e., the annual rate at which poorer economies catch-up with the developed economies.

The main disadvantage of this method is that it considers the average annual growth rate of economic development for the period studied. During this period, the conditional average rates may not be constant or, in other words, economic growth for a country may both increase and decrease. Therefore, the slope of the regression line that estimates the absolute beta convergence indicator should not be constant. For this reason, in the literature, the estimate of the regression equation was used, that allows the determination of beta convergence, but on a panel data [85,86]. Moreover, another advantage of using panel data is that the equilibrium may not be kept constant, as in the original methodology, using fixed effects.

Barro and Sala-i-Martin [39] note that the convergence rates of economic development of countries determined using fixed effects panel data are much higher than the value of 2% around which the results of most research are located, identifying values between 12 and 20%. The possible explanation identified by them is represented by a few values recorded in time. Therefore, in estimating convergence, short-term adjustments are captured around the trend instead of determining long-term convergence.

Considering the methodology for assessing the convergence of economies, we will test the existence of the beta convergence of the sustainability of the member states of the European Union based on the sustainable development index created, ISDE.

The equation that will be estimated, to determine the beta convergence, is of the form:

$$\frac{1}{T}\mathit{ln}\left(\frac{ISD{E}_{i,t_0+T}}{ISD{E}_{i,t_0}}\right)={\beta }_0+{\beta }_1\mathit{ln}\left(ISD{E}_{i,t_0}\right)+{\epsilon }_{it}$$

(7)

where:

$ISD{E}_{i,t_0}$—the sustainable development index at the initial moment of analysis

$ISD{E}_{i,t_0+T}$—sustainable development index at the end of the period considered in the analysis

${\beta }_0$—constant (intercept) of the regression model

${\beta }_1$—the slope of the regression that quantifies the existence of beta convergence.

If ${\beta }_1$ is statistically negative and significant, then the countries considered in the analysis show beta convergence in sustainability; the growth rate of the sustainability of the less developed countries is higher compared to the more sustainably developed countries.

Based on the above equation, we will also estimate the convergence rate of sustainability in order to identify how less sustainably developed economies are closer to the equilibrium of sustainability. The study is performed on cross-section data. We will not address the beta estimation of absolute convergence on panel data, because we only have records for eleven years for the sustainability quantification indicator.

The beta analysis of sustainability convergence will be performed, both on all countries included in the study and on the homogeneous groups of countries in terms of sustainability identified above.

3.4.2. Sigma Convergence Methodology

The sigma convergence analysis methodology provides for modelling the dynamics of ISDE-EU dispersion/concentration indicators. The most common indicators used in the sigma convergence analysis, found in the literature, are: the coefficient of variation, standard deviation of the logarithmic variable, Gini INDEX, and Theil INDEX.

Coefficient of Variation

The first indicator used in the sigma convergence analysis is the coefficient of variation. It has been used by authors such as de la Fuente [56], Giannetti [57], Soukiazis, Castro [58], Sala-i-Martin [59], and Yang et al. [60].

Thus, the coefficient of variation ${\sigma }_t$in year t can be calculated according to the formula:

$${\sigma }_t=\frac{\sqrt{\frac{{\displaystyle {{\displaystyle \sum }}_{i=1}^n}{(y_i-\overline{y})}^2}{n}}}{\overline{y}},$$

(8)

where:

$y_i$—represents the value of the variable in region (country) i

$\overline{y}$—is the average of the variable

$n$—number of regions (countries).

Considering the methodology for assessing the convergence of sustainability, we will verify the existence of the sigma of the convergence of sustainability of the member states of the European Union based on the sustainable development index created, ISDE.

The equation used to determine the sigma of sustainability convergence in this article will take the form of:

$$Sigma{(CV)}_{ISDE}=\frac{\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n{(ISD{E}_i-\overline{ISDE})}^2}{n}}}{\overline{ISDE}}$$

(9)

where:

ISDE_i represents the index of sustainable development in the country $i$,

$\overline{ISDE}$ average of the sustainable development index of the EU countries under study,

$n$ the number of countries, and Sigma(CV)_ISDE represent the coefficient of variation.

SD of Logarithm

The second method, often used in empirical research, considers the standard deviation of the logarithmic variable, calculated according to the formula [81]:

$${\sigma }_t=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n\left[\mathit{ln}(y_i\right)-\overline{\mathit{ln}(y)}{]}^2}{n-1}}$$

(10)

To determine the sigma of the convergence of the Sustainable Development Index (ISDE), we use the equation:

$$Sigma=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^n\left[\mathit{ln}(iSD{E}_i\right)-\overline{\mathit{ln}(iSDE)}{]}^2}{n-1}}$$

(11)

where $iSD{E}_i=\frac{ISD{E}_i}{\overline{ISDE}}$ is the proportion of the sustainable development of country i, compared to the average sustainable development index of all countries (cluster or total).

This standardized index may reflect changes in the relative position of different countries [60].

Gini INDEX

The Gini concentration index was first used by Corado Gini to study the inequality (degree of concentration) of income in a population. By definition, it ranges from 0 to 1. A low value corresponds to a more equal distribution, while a high value corresponds to a more unequal distribution.

This Gini INDEX can also be used to measure convergence across countries or regions [59]. One of the Gini INDEX calculation methods is shown below.

$$Gini=\frac{1}{2n^2\overline{y}}{\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^N\left|y_i-y_j\right|$$

(12)

Another Gini INDEX calculation method is the trapezoid method.

$$Gini=1-{\displaystyle \sum }_{i=1}^n\left(Y_i+Y_{i-1}\right)\left(F_i-F_{i-1}\right)$$

(13)

where:

$Y_i$ represents the value of the cumulative proportion of the study variable (e.g., Income, GDP/loc),

$F_i$ represents the value of the cumulative proportion of the number of individuals (regions, countries).

Knowing that $F_i=F_{i-1}+f_i\Rightarrow f_i=F_i-F_{i-1}$, then we can rewrite the above formula as follows:

$$Gini=1-{\displaystyle \sum }_{i=1}^n\left(Y_i+Y_{i-1}\right)\cdot f_i$$

(14)

To simplify the calculation of Gini INDEX for the sustainable development index, in our article, we condition that ISDE_i−1 < ISDE_i (the values of sustainable development indices are ordered in ascending order), and we obtain:

$$Gini=1-{\displaystyle \sum }_{i=1}^n\left(ISD{E}_i+ISD{E}_{i-1}\right)\cdot f_i$$

(15)

Theil Index

The fourth indicator used to measure the convergence of the sustainable development index is Theil INDEX [59].

By definition, it varies between zero and infinity. Theil INDEX is, together with Gini INDEX, part of the family of concentration/diversification indicators that can measure the degree of inequality between various entities. The general formula is:

$$Theil=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left[\frac{y_i}{\overline{y}}\cdot \mathit{ln}\left(\frac{y_i}{\overline{y}}\right)\right]$$

(16)

In our analysis, in order to measure the convergence of the sustainable development index between states, we used the formula:

$$Theil=\frac{1}{n}{\displaystyle \sum }_{i=1}^n\left[\frac{ISD{E}_i}{\overline{ISDE}}\cdot \mathit{ln}\left(\frac{ISD{E}_i}{\overline{ISDE}}\right)\right]$$

(17)

The existence of the convergence sigma implies a decreasing evolution/a negative trend of the dynamics of the indicators used, hence the need to verify the significance of the slope of the existing trend and the significance of the regression model:

$$IN{D}_t={\beta }_0+{\beta }_{1}t+{\epsilon }_t$$

(18)

where: $IN{D}_t$—represents any of the indicators used in the sigma convergence analysis: coefficient of variation, standard deviation of the logarithmic variable, Gini INDEX and Theil INDEX corresponding to the time period t.

4. Results

4.1. Results of Grouping Countries by the Dynamics of ISDE-EU

Following the clustering of EU countries, except Croatia (EU 27), based on the values recorded for ISDE-EU interval analyzed, we obtained the following groups/clusters of countries with a higher level of homogeneity in terms of ISDE-EU values and its dynamics over time.

The five clusters are shown in both Figure 3 and

Table 3. Grouping of EU 27 countries 2017 (excluding Croatia) * based on ISDE-EU using the Hierarchical Clusters method.

Cluster 1	Cluster 2	Cluster 3	Cluster 4	Cluster 5
Austria	Cyprus	Czechia	Denmark	Latvia
Belgium	Estonia	Malta	Sweden	Lithuania
Bulgaria	Hungary	Slovakia		Romania
Finland	Poland
France	Portugal
Germany	Slovenia
Greece
Ireland
Italy
Luxembourg
Netherlands
Spain
United Kingdom

* According to the situation in 2017, from which Croatia was excluded due to lack of data. Grouping obtained by authors based on ISDE-EU and the clustering method according to the presented methodology.

below. It is observed that there are two clusters with more than five countries and three clusters with a maximum of three countries. The different evolutions of the sustainability dynamics of EU 27 countries were grouped by homogeneous clusters, following the application of the cluster methodology; therefore, the H2 hypothesis is valid.

4.2. Beta Convergence Analysis Results

The period covered by the study is determined by the availability of indicators for quantifying sustainable development based on which the sustainable development index of the 27 member countries of the European Union 2006–2016 was calculated.

Table 4. Beta Convergence indicator values across the 5 clusters and across the EU 27.

	${\widehat{\mathit{\beta }}}_0{}^{(*)}$	${\widehat{\mathit{\beta }}}_1{}^{(*)}$	${\mathit{\beta }}^{ (**)}$
Cluster 1	−0.2293 (0.017)	−0.0783 (0.000)	−0.0565
Cluster 2	−0.1288 (0.068)	−0.1247 * (0.045)	−0.0785
Cluster 3	−0.1037 (0.060)	−0.0817 (0.025)	−0.0583
Cluster 4	N.A.	N.A.	0
Cluster 5	−0.4163 (−0.132)	−0.6208 (0.137)	−0.1871
All	−0.0239 * (0.004)	−0.0556 * (0.000)	−0.0434

⁽*⁾ ${\widehat{\beta }}_0$ and ${\widehat{\beta }}_1$ represent the estimated parameters of the regression model according to Equation (7); ⁽**⁾ $\beta$—convergence rate. In parentheses are presented the p-value corresponding to the Student’s t test for testing the significance of the regression parameter for 5% significance level.

shows both the estimated beta convergence indicators for each country cluster, but also for all countries, as well as the beta convergence rate.

The analysis of all countries from the point of view of the beta convergence of the sustainability index in the period 2006–2016 highlights the existence of a negative link between the average annual growth rate of the sustainable development index and the sustainable development index. This element is highlighted in Figure 4.

The graphical representation of the average annual growth rate of the sustainability index in the period 2006–2016 and the sustainable development index for all member states of the EU 27, highlights the existence of an inverse linear link.

The estimation of the beta convergence indicator for all countries considered in the study and presented in Table 4, confirms the existence of the beta convergence of sustainability. The convergence rate of countries’ sustainability is about 4.34%. This value is mainly found in determining the convergence beta of economic growth [87].

A more detailed analysis of the sustainability beta convergence of the member states of the EU 27 is further carried out on the clusters of homogeneous countries in terms of sustainability, previously determined.

Cluster one includes Austria, Belgium, Bulgaria, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, the Netherlands, Spain, and the United Kingdom. The graphical representation of the average growth rate of the sustainable development index according to the sustainable development index, namely Figure 5, highlights the existence of an inverse linear link specific to the existence of the convergence beta. The estimated ${\widehat{\beta }}_1$ parameter is negative and statistically significant, which confirms the existence of beta convergence. Therefore, the countries that are included in cluster 1 are characterized by the beta sustainability convergence, i.e., the less sustainable developed countries tend to recover, on average, 5.65% annually the difference up to the steady state of the sustainability of cluster 1, according to the beta convergence rate shown in Table 4.

The second cluster includes the countries of Cyprus, Estonia, Hungary, Poland, Portugal, and Slovenia. The negative slope of the scatterplot determined based on the average growth rate of the sustainable development index, highlights the existence of an inverse linear link corresponding to the existence of beta convergence (Figure 6). The existence of the beta convergence of the sustainability of the countries in cluster 2 is also supported by the estimated parameter ${\beta }_1$ (−0.1247), which is statistically negative and significant. The convergence rate of sustainability is 7.85%, the highest value for the five clusters.

We also note that in the analysis period 2006–2016, Poland has an average negative annual growth rate, which suggests the divergence of the sustainability index, the value of their sustainability index registering, on average, a decrease.

Cluster three, shown in Figure 7, comprises three countries, the Czech Republic, Malta, and Slovakia, a small sample to estimate a regression model to identify the beta sustainability convergence. The existence of the beta convergence of the sustainability of the countries in cluster 3 is, however, supported by the parameter ${\widehat{\beta }}_1$ estimated (−0.1247) which is negative and statistically significant. The convergence rate of sustainability is 5.83%, close to that of the first cluster.

Denmark and Sweden are the countries that are part of cluster 4. We cannot perform a regression analysis in this cluster.

The last cluster, represented in Figure 8, includes Latvia, Lithuania, and Romania. The estimated regression coefficient, ${\widehat{\beta }}_1$ is negative (−0.6208), but not statistically significant. We also note that in the analysis period 2006–2016, all three countries have an average negative annual growth rate, which suggests the divergence of the sustainability index, the value of their sustainability index registering, on average, a decrease.

4.3. The Results of the Sigma Convergence Analysis

The analysis of ISDE convergence in EU 27 countries with the help of beta convergence has the disadvantage that beta convergence focuses only on the analysis of average values for the period studied. Sigma convergence, on the other hand, can measure the degree of convergence or divergence at each moment in time, in the study period.

4.3.1. Sigma Convergence in the EU 27

Table A1. Sigma convergence at the level of EU 27 countries.

Anul	Sigma CV	Sigma SD Log ISDE	Gini Index	Theil Index
2006	0.4393	0.8464	0.2249	0.1110
2007	0.4611	0.6807	0.2351	0.1174
2008	0.4809	0.6670	0.2344	0.1269
2009	0.5113	0.9026	0.2582	0.1460
2010	0.4592	0.5656	0.2289	0.1103
2011	0.4270	0.5137	0.2164	0.0952
2012	0.3968	0.4656	0.1977	0.0817
2013	0.3834	0.4398	0.1896	0.0756
2014	0.3722	0.4098	0.1849	0.0700
2015	0.3666	0.4033	0.1835	0.0677
2016	0.3845	0.4307	0.1919	0.0752

in the Appendix A shows the convergence sigma indicators, the coefficient of variation (Sigma CV), the standard deviation of ln (ISDE), Sigma (ISDE), the Gini concentration index (Gini INDEX), and the Theil concentration index (Theil INDEX), for the studied EU 27 countries.

We further aim to highlight the downward evolution of these sigma convergence indicators, an evolution that would confirm the existence of convergence at the ISDE level in the studied EU countries.

Figure 9 shows that both the sigma convergence coefficients Sigma CV and Sigma (ISDE) and the concentration indices Gini INDEX and Theil INDEX have a downward trend in the period 2006–2016 for all EU 27 countries studied.

All these indicators show that ISDE, for the different EU 27 countries, are converging at a certain speed. However, this rate is relatively low, which confirms previous results of beta-convergence. At the same time, there is an increase in sigma convergence indicators (indicating a divergence) starting in 2006 and culminating in 2009, when EU 27 countries were already affected by the financial crisis and recession. Since 2010, there has been a sustainable convergence of ISDEs in EU 27 countries.

The presence of the decreasing trend for the evolution of the four indicators measuring sigma convergence is not sufficient, considering the relatively short study period (2006–2016) of 11 years and the rather low speed of convergence.

For this reason, in addition to estimating the trend equation, trend tests of these convergent sigma indicators in Figure 9 were also performed. Linear trend patterns and significance levels were summarized in

Table 5. Testing the significance of the sigma convergence trend in the period 2006–2016 in the EU 27.

	Constant	b1	b1 Stdz	R Sq.
Sigma CV	24,124 (0.003)	−0.012 (0.003)	−0.804 (0.003)	0.647 (0.003)
Sigma	91,715 (0.001)	−0.045 (0.001)	−0.851 (0.001)	0.723 (0.001)
Gini	12,797 (0.001)	−0.006 (0.002)	−0.828 (0.002)	0.686 (0.002)
Theil	13,222 (0.001)	−0.007 (0.002)	−0.828 (0.002)	0.686 (0.002)

.

We can see that the time series for Sigma CV and Sigma have negative and significant trends, which indicates a real convergence. The trend slope for Sigma CV is $b_{1(SigmaCV)}=-0.012$ and for Sigma is $b_{1(Sigma)}=-0.045$, both parameters being statistically significant. This convergence is also supported by the so-called indicators of inequality or concentration, namely Gini INDEX and Theil INDEX. Additionally, these two indicators have a downward trend with negative and statistically significant trends, respectively $b_{1(Gini)}=-0.006$, $b_{1(Theil)}=-0.007$.

Figure 9 shows that the trends of the four sigma convergence indicators are similar (the slope of the trend models is approximately the same). The same can be seen based on the third column in Table 5, which estimates the regression coefficients of the standardized beta coefficients of the convergence sigma indicators. These estimates have values between (−0.804 and −0.851), which are relatively close.

4.3.2. Sigma Convergence in Each Cluster

For a more detailed presentation of the sigma convergence of sustainability of the studied EU member states, we further perform an analysis on clusters of homogeneous countries in terms of sustainability, clusters that have been previously determined.

Thus, the first cluster, the graphical representation of the sigma convergence indicators in Figure 10 highlights the existence of a negative trend for all these indicators. The four graphs in Figure 10 were obtained based on the data recorded in the Appendix, respectively

Table A2. Sigma convergence in Cluster 1 countries.

Anul	Sigma CV	Sigma SD Log ISDE	Gini Index	Theil Index
2006	0.2155	0.2061	0.1174	0.02299
2007	0.2294	0.2135	0.1221	0.02550
2008	0.1932	0.1905	0.1055	0.01885
2009	0.2006	0.1985	0.1070	0.02032
2010	0.1840	0.1948	0.0977	0.01785
2011	0.1378	0.1357	0.0724	0.00959
2012	0.1015	0.1001	0.0512	0.00522
2013	0.0827	0.0788	0.0443	0.00340
2014	0.0958	0.0922	0.0516	0.00458
2015	0.0956	0.0922	0.0518	0.00457
2016	0.1090	0.1031	0.0585	0.00586

.

It is also observed here, in cluster 1 (Figure 10), that there is a sigma convergence, as in the case of the analysis of all EU 27 countries. Thus, for the countries: Austria, Belgium, Bulgaria, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Netherlands, Spain, and United Kingdom, which were grouped in the first cluster according to ISDE, there are decreases of the four calculated indicators for sigma convergence, in the period 2006–2016, and linear trend models approximate these downward evolutions with greater accuracy.

If we compare the values in

Table 6. Testing the significance of the sigma convergence trend in the period 2006–2016 in Cluster 1 of the EU 27.

	Constant	b1	b1 Stdz	R Sq.
Sigma CV	30,830 (0.000)	−0.015 (0.000)	−0.912 (0.000)	0.832 (0.000)
Sigma	29,931 (0.000)	−0.015 (0.000)	−0.902 (0.000)	0.814 (0.000)
Gini	16,704 (0.000)	−0.008 (0.000)	−0.909 (0.000)	0.827 (0.000)
Theil	4741 (0.000)	−0.002 (0.000)	−0.917 (0.000)	0.841 (0.000)

with those in Table 5, we find that for the four sigma convergence indicators the trend models are better for cluster 1 (R Square are significantly higher) and the regression coefficients of standardized variables (standardized beta coefficients) of the sigma convergence indicators also have higher negative values for cluster 1.

We see in this cluster only developed countries in Western Europe (except Bulgaria) that joined the EU before 2000. This may be one of the reasons why the level of the sigma convergence is more pronounced.

In the second cluster, the four sigma convergence indicators have a negative trend, with the highest slope, compared to the five clusters and the group of all EU 27 countries. In this cluster we find the countries: Cyprus, Estonia, Hungary, Poland, Portugal, and Slovenia. Figure 11 contains the second cluster convergence sigma indicators, calculated based on

Table A3. Sigma convergence in Cluster 2 countries.

Anul	Sigma CV	Sigma SD Log ISDE	Gini Index	Theil Index
2006	0.1768	0.1580	0.0900	0.0153
2007	0.2249	0.2225	0.1133	0.0264
2008	0.3047	0.2815	0.1472	0.0458
2009	0.2961	0.2616	0.1469	0.0424
2010	0.2787	0.2471	0.1361	0.0377
2011	0.2534	0.2289	0.1252	0.0317
2012	0.2159	0.1950	0.1109	0.0230
2013	0.2486	0.2283	0.1248	0.0309
2014	0.2187	0.1963	0.1112	0.0235
2015	0.1896	0.1706	0.0932	0.0177
2016	0.1843	0.1612	0.0856	0.0165

from the Appendix A.

Figure 11, which shows the sigma convergence indicators for the countries in cluster 2, indicates that the values for the years 2006 to 2016 have a rather low downward trend and the variability is high, so there are many increases and decreases during the period studied, respectively 2006–2016.

The values in

Table 7. Testing the significance of the sigma convergence trend in the period 2006–2016 in Cluster 2 of the EU 27.

	Constant	b1	b1 Stdz	R Sq.
Sigma CV	9733 (0.277)	−0.005 (0.288)	−0.352 (0.288)	0.124 (0.288)
Sigma	10,559 (0.196)	−0.005 (0.204)	−0.415 (0.204)	0.172 (0.204)
Gini	5232 (0.225)	−0.003 (0.235)	−0.391 (0.235)	0.153 (0.235)
Theil	2467 (0.236)	−0.001 (0.241)	−0.386 (0.241)	0.149 (0.241)

in columns 4 and 5, for the four sigma convergence indicators, mark the existence of a decreasing trend, but this is not statistically significant.

For the third cluster we find, in Figure 12, the evolution of the sigma convergence indicators. The evolution is downward for this third cluster, which has only three countries, namely the Czech Republic, Malta, and Slovakia.

The linear trend models for the four sigma convergence indicators, related to cluster 3, were estimated and summarized in

Table 8. Testing the significance of the sigma convergence trend in the period 2006–2016 in Cluster 3 of the EU 27.

	Constant	b1	b1 Stdz	R Sq.
Sigma CV	130,229 (0.003)	−0.065 (0.003)	−0.809 (0.003)	0.655 (0.003)
Sigma	192,212 (0.010)	−0.095 (0.011)	−0.731 (0.011)	0.534 (0.011)
Gini	55,251 (0.002)	−0.027 (0.002)	−0.824 (0.002)	0.678 (0.002)
Theil	69,311 (0.016)	−0.034 (0.016)	−0.703 (0.016)	0.494 (0.016)

, based on the data in

Table A4. Sigma convergence in Cluster 3 countries.

Anul	Sigma CV	Sigma SD Log ISDE	Gini Index	Theil Index
2006	0.8286	1.2818	0.3616	0.45459
2007	0.5259	0.5529	0.2209	0.15948
2008	0.3584	0.3298	0.1546	0.06882
2009	0.7565	1.1458	0.2912	0.38998
2010	0.2894	0.2634	0.1180	0.04478
2011	0.1270	0.1071	0.0542	0.00823
2012	0.2419	0.1877	0.0981	0.02821
2013	0.1502	0.1182	0.0578	0.01100
2014	0.0449	0.0362	0.0173	0.00100
2015	0.1332	0.1123	0.0575	0.00906
2016	0.1618	0.1381	0.0691	0.01346

, in the Appendix A. It is observed that the parameters of the trend models are significant. Therefore, we can say that there is significant sigma convergence at the level of the third cluster.

The convergence sigma indicators, related to the fourth cluster, can be found in Figure 13. The evolution of these indicators, during the period 2006–2016, shows a slightly upward trend. Thus, for the countries in cluster 4, namely Denmark and Sweden, there is sigma convergence, from the ISDE point of view.

It should be noted, however, that the downward trend for sigma convergence indicators is not a significant trend. If we look at the results in

Table 9. Testing the significance of the sigma convergence trend in the period 2006–2016 in Cluster 4 of the EU 27.

	Constant	b1	b1 Stdz	R Sq.
Sigma CV	12,561 (0.060)	−0.006 (0.061)	−0.581 (0.061)	0.337 (0.061)
Sigma	8915 (0.060)	−0.004 (0.061)	−0.581 (0.061)	0.338 (0.061)
Gini	4436 (0.060)	−0.002 (0.061)	−0.581 (0.061)	0.337 (0.061)
Theil	0.719 (0.073)	−3.57 × 10−4 (0.074)	−0.559 (0.074)	0.313 (0.074)

(results obtained based on the data in

Table A5. Sigma convergence in Cluster 4 countries.

Anul	Sigma CV	Sigma SD Log ISDE	Gini Index	Theil Index
2006	0.0336	0.0238	0.0119	0.00057
2007	0.1047	0.0742	0.0370	0.00548
2008	0.0888	0.0629	0.0314	0.00395
2009	0.0831	0.0589	0.0294	0.00346
2010	0.1041	0.0738	0.0368	0.00543
2011	0.0378	0.0267	0.0134	0.00071
2012	0.0149	0.0105	0.0053	0.00011
2013	0.0502	0.0355	0.0178	0.00126
2014	0.0217	0.0154	0.0077	0.00024
2015	0.0560	0.0396	0.0198	0.00157
2016	0.0070	0.0049	0.0025	0.00002

, in the Appendix A) we find that the parameters of the trend model, for each of the four sigma convergence indicators, are not statistically significant. Therefore, in cluster 4 we cannot specify whether there is convergence or divergence.

Finally, for cluster 5, the evolutions of the convergence sigma indicators in Figure 14 are presented. This cluster consists of Latvia, Lithuania, and Romania, Eastern European countries that joined the EU after 2004.

Figure 14 shows that the values for the years 2006 to 2016 have a slightly upward trend.

The values in

Table 10. Testing the significance of the sigma convergence trend in the period 2006–2016 in Cluster 5 of the EU 27.

	Constant	b1	b1 Stdz	R Sq.
Sigma CV	−2830 (0.825)	0.001 (0.817)	0.079 (0.817)	0.006 (0.817)
Sigma	−3311 (0.762)	0.002 (0.754)	0.107 (0.754)	0.011 (0.754)
Gini	−0.216 (0.968)	0.000 (0.960)	0.017 (0.960)	0.000 (0.960)
Theil	−0.066 (0.971)	0.000 (0.966)	0.015 (0.966)	0.000 (0.966)

(results obtained from the data in

Table A6. Sigma convergence in Cluster 5 countries.

Anul	Sigma CV	Sigma SD Log ISDE	Gini Index	Theil Index
2006	0.0374	0.0303	0.0160	0.00069
2007	0.1667	0.1343	0.0730	0.01374
2008	0.1849	0.1593	0.0786	0.01768
2009	0.2314	0.1918	0.1028	0.02691
2010	0.0674	0.0560	0.0279	0.00230
2011	0.1582	0.1244	0.0632	0.01219
2012	0.1241	0.1054	0.0501	0.00789
2013	0.0688	0.0573	0.0285	0.00240
2014	0.1040	0.0839	0.0454	0.00536
2015	0.1229	0.1028	0.0536	0.00766
2016	0.2071	0.1821	0.0797	0.02247

, Appendix A), in columns 4 and 5 for the four sigma convergence indicators, are among the lowest, in absolute terms, compared to the other clusters. There is a divergence, but this is not statistically significant.

5. Conclusions

Table 11. Summary of the presence and statistical significance of beta and sigma convergence in the EU 27 and the five clusters.

Clusters/Total EU 27	Total EU 27	Cluster 1	Cluster 2	Cluster 3	Cluster 4	Cluster 5
Beta Convergence
Regression Slope (β1) (Positive/Negative)	N	N	N	N	N/A	N
Statistical significance (Yes/No)	Y	Y	Y	Y	N/A	N
Convergence/Divergence (C/D)	C	C	C	C	N/A	C
Sigma Convergence
Regression Slope (β1) (Positive/Negative)	N	N	N	N	N	P
Statistical significance (Yes/No)	Y	Y	N	Y	N	N
Convergence/Divergence (C/D) *	C	C	C/n *	C	C/n *	D
Beta and Sigma Convergence match
Concordance between β and σ Convergence (YES/NO)	Y	Y	Y	Y	N/A	N
Convergence/Divergence (C/D)	C	C	C	C	N/A	C/D

Note: *—“C/n” means convergence, but not statistically significant.

summarizes the research results. Their cumulative analysis shows that, at the level of all analysed states (total EU 27), the simultaneous existence of both beta and sigma convergence in sustainability is validated.

In conclusion, summarizing the research results, we can state the following:

At the level of the EU 27 countries one can notice that the hypotheses H1.1 (there is beta convergence at sustainability level) and H1.2 (there is sigma convergence at sustainability level) are validated because the regression slopes (β₁) are negative and statistically significant. Since there is consistency between beta and sigma convergence, we can therefore state that the hypothesis H1 is validated.

-

Cluster 1, consisting of 16 EU countries, mostly belonging to W and SW Europe, is characterized by an average level of ISDE-EU, which is on an upward trend, above the European average, and a degree of dispersion of ISDE-EU values, located on a decreasing trend, below the degree of dispersion characteristic of the analysed EU countries, Figure 15a,b.

For this cluster, the existence of beta and sigma convergence regarding the sustainability of national economies is confirmed, Table 11.

At the level of this cluster, hypotheses H3.1 (there is beta sustainability convergence at cluster level) and H3.2 (there is sigma sustainability convergence at cluster level), while the concordance between beta and sigma convergence determines the validation of H3 hypothesis.

-

Cluster 2, made up of six EU countries, most of them in N-E Europe, is characterized by an average level of ISDE-EU, located on an upward trend, slightly below the European average and a degree of dispersion of ISDE-EU values, located on a slightly upward trend, below the degree of dispersion characteristic of the analyzed EU countries, Figure 15a,b. For this cluster, the existence of beta convergence in sustainability of the member states is confirmed while sigma convergence in sustainability is uncertain, according to Table 11, because the estimated value of the right slope, although negative, is not significant, as Table 7 and Table 11 show. At the level of this cluster, the H3.1 hypothesis is validated. The H3.2 hypothesis cannot be totally validated because, even if there is a negative regression slope, it is not statistically significant. In regard of the concordance between the beta and sigma convergence, this exists and would determine the validation of the H3 hypothesis, provided that the sigma convergence is not statistically significant.

-

Cluster 3 consisting of three EU countries, mostly located in the central area of Europe, is characterized by the lowest average level of ISDE-EU, which is on an upward trend, well below the European average but also by a degree of dispersion of values ISDE-EU, which is on a downward trend, the lowest, below the degree of dispersion characteristic of the analyzed EU countries (Figure 15a,b). For this cluster, the existence of beta and sigma convergence in the sustainability of national economies is validated. An additional remark is that two of the three states—the Czech Republic and Slovakia—together formed a single state before 1993, and this is probably why this cluster, although it has the lowest level of ISDE-EU average, is still the most homogeneous cluster in the five studied. For this cluster, the H3.1 and H3.2 hypotheses are validated and the concordance between beta and sigma convergence determines the validation of the H3 hypothesis.

-

Cluster 4 consists of only two countries—Denmark and Sweden—and from a statistical point of view it is almost impossible to estimate the convergence in sustainability of national economies. Regarding sigma convergence in sustainability, although the estimates of the straight slope are negative, they are not significant, making its assessment uncertain. Cluster 4 is characterized by the highest ISDE-EU average, which is on an upward trend, far exceeding the European average and a low degree of dispersion for the values recorded for ISDE-EU, which is on a strong downward trend compared to the values recorded for all EU countries. Despite the small volume of cluster 4 which makes it impossible to assess convergence in sustainability, it should be noted that it is a leading cluster in terms of sustainability at EU level. At the level of this cluster, neither H3.1, H3.2 or consequently H3 hypotheses are validated.

-

Cluster 5, consisting of three countries, located in E and N-E Europe, is characterized by an average level of ISDE-EU, located on an upward trend, slightly below the European average and a degree of dispersion of ISDE-EU values, located on a strong upward trend, far above the degree of dispersion characteristic of the analyzed EU countries, as it is shown in Figure 15a,b. According to Table 10, for this cluster the existence of beta and sigma convergence cannot be confirmed or refuted because the estimated slopes for the models are statistically insignificant, both for beta and for sigma convergence (Table 10 and Table 11). Even though cluster 5 is characterized by an average of ISDE-EU level close to that of the EU, it is the most heterogeneous, which does not allow a meaningful assessment of convergence in its sustainability. As a negative aspect of this cluster, we notice that the slope of the standard deviations is increasing strongly, which suggests that, within this cluster, there are sustainability behaviors that tend to differ more and more one from another.

In conclusion, we can say that, at EU level, it was found that there is both beta and sigma convergence in sustainability of the economies of its member states, which is also validated at the level of the analyzed clusters. For clusters containing the largest share of EU states, which is cluster 1 (16 countries) and cluster 3, including three countries, both beta and sigma convergence sustainability is validated. For cluster 2, consisting of six EU countries, only beta convergence in sustainability is validated, the existence of sigma convergence in sustainability being uncertain. Due to its small size, cluster 4, although it cannot be assessed in terms of beta and sigma convergence in sustainability, is undeniably a leading cluster in terms of the sustainability of national economies at EU level and beyond.

The “problem” cluster is cluster 5 and not necessarily in terms of the average level of ISDE-EU and its corresponding trend, but in terms of heterogeneity. This heterogeneity can be attributed to the fact that, imposing in the grouping process a maximum number of 5 clusters, it is possible that in this cluster were allocated countries for which the ISDE-EU component in the associated period was not found in any of the four clusters formed, this cluster being the “victim” of a “residual effect” of clustering on clusters. This requires a closer analysis of the ISDE-EU evolutions for the observed period 2006–2016 corresponding to the 3 states that make up cluster 5 as presented in Figure 16.

Figure 16 shows that there is a state, Lithuania, which has a discordant evolution compared to the other two member states, Romania and Latvia, which, although slightly different, are on a trend forward convergence. Lithuania has a negative evolution towards the two states, moving away from the evolutions of the two states. It is possible that the inclusion of this state in cluster 5 represents the residual effect of the clustering algorithm.

For this cluster, none of the hypotheses H3.1, H3.2, and consequently H3 are validated.

The analyses performed in this study aimed at validating the hypotheses presented in chapter 2, literature review and research hypotheses.

In order to sum up the validation or invalidation of the hypotheses we must mention that the hypotheses H1 (existence of convergence of the entire EU 27 region), H1.1. (existence of beta convergence at the entire EU 27 level), H1.2 (existence of sigma convergence at the level of EU 27), and H2 (the different evolutions of the sustainability dynamics for the EU 27 countries can be grouped by homogeneous clusters) have been entirely validated. The hypotheses H3 (existence of convergence at cluster level), H3.1 (existence of beta convergence at cluster level) and H3.2 (existence of sigma convergence at cluster level) have been fully validated for clusters 1 and 3, partially for cluster 2, and have been invalidated for clusters 4 and 5.

Thus, according to the specificity of each cluster, measures/policies to support/encourage the transition process to sustainably convergent economies can be customised in compliance with the objectives set up in the 2030 Agenda for Sustainable Development [22].

Future research trends should include extensive methods of convergence analysis as well as spatial models of beta convergence or conditional beta convergence. New variables to determine a more performant sustainability index could also be introduced, which would highlight the long-term effects of sustainability.