Predicting convergence of per capita income in Spain: A Markov and cluster approach

Gálvez-Rodríguez, José F.; Manzano-Hidalgo, Miguel; García-Luengo, Amelia V. Article Predicting convergence of per capita income in Spain: A Markov and cluster approach Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Gálvez-Rodríguez, José F.; Manzano-Hidalgo, Miguel; García-Luengo, Amelia V. (2025) : Predicting convergence of per capita income in Spain: A Markov and cluster approach, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 1, pp. 1-16, https://doi.org/10.3390/economies13010017 This Version is available at: https://hdl.handle.net/10419/329297 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Academic Editor: Fabio Clementi Received: 2 December 2024 Revised: 7 January 2025 Accepted: 8 January 2025 Published: 11 January 2025 Citation: Gálvez-Rodríguez, J.F., Manzano-Hidalgo, M., García-Luengo, A.V. (2025). Predicting Convergence of Per Capita Income in Spain: A Markov and Cluster Approach. Economies, 13(1), 17. https://doi.org/10.3390/ economies13010017 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). Article Predicting Convergence of Per Capita Income in Spain: A Markov and Cluster Approach José F. Gálvez-Rodríguez * , Miguel Manzano-Hidalgo and Amelia V. García-Luengo * Department of Mathematics, University of Almería, 04120 Almería, Spain; [email protected] *Correspondence: [email protected] (J.F.G.-R.); amgar[email protected] (A.V.G.-L.) Abstract: In this work we analyze the evolution of productivity, in terms of the convergence of per capita income, of all the Spanish provinces, based on data from the previous decade. On the one hand, a cluster analysis allows us to group the Spanish provinces according to four income levels (low, medium-low, medium-high and high), which can be determined from the quartiles of the distribution, and, on the other hand, Markov chains make it possible to study the long-term evolution of productivity and convergence between the provinces, as well as the speed of convergence towards the equilibrium situation. Moreover, we can obtain the average time to return to an income level in which a province was previously. With the above, predictions of future income levels are made for the provinces, both in the current situation, and if the pandemic caused by COVID-19 had not existed, which leads us to evaluate the impact of the health emergency. Keywords: cluster analysis; Markov chain; per capita income 1. Introduction Markov chains Takács (1960) are an important mathematical tool in the analysis of time series and have been used by several authors when studying the economic growth and other phenomena, even in other disciplines. Markov analysis starts from the present value of a certain variable in order to predict its future. For example, Quah (1993) uses Markov chains to analyze how countries exchange positions over time in terms of their GDP per capita. Each finite and homogeneous Markov chain is characterized by a matrix which contains different probabilities. In the study cited above, the probabilities that a country remains in its current income group or moves to a higher or lower income group are calculated and collected in this matrix. It is concluded that, in general, countries tend to converge in terms of their GDP per capita over time, but the rate of convergence varies depending on the region and the time period studied. This approach has been used by other authors in similar analyses in a wide variety of contexts. Some relevant examples of studies in which the Markov chain methodology has been used to analyze the dynamics of growth and convergence are the following: • Fingleton (1997) uses the evidence given by data from 1975 to 1993 in order to justify that different regions of the European Union seem to be converging towards stable proportions in terms of per capita income levels. • Bode and Nunnenkamp (2011) analyze the impact of foreign direct investment on per capita income and growth, in general, in the United States, since the mid-1970s, demonstrating that the investment in employment has favored the increase in income, although the investment destined for capital has not had the same impact on the states with the greatest poverty, where it has been most intensive. Economies 2025,13, 17 https://doi.org/10.3390/economies13010017 Economies 2025,13, 17 2 of 16 • Lipták (2011) examines the evolution of unemployment in the Hungarian labor market during the period 1992–2009. Despite being a methodology exposed at the end of the previous century, Markov chains keep on appearing in the nowadays literature to study the evolution of economic variables. We refer the reader to the works by Wenxuan (2023), Rey (2023), Papanikolaou (2020), Arreola and Montiel (2024), Kerkouch et al. (2024), Chen et al. (2022), Karahasan (2020) and Haller et al. (2020). However, knowing the probability of transition between states along a period of time is not enough, since we still do not know which region continues in the same situation or change it after several periods of time. That is the reason why considering a cluster analysis can be of some interest in order to know, also, the elements of each group with respect to the past data. Cluster analysis, for what a good reference is Everitt et al. (2011), is a statistical technique whose primary objective is to group data according to their similarities and differences based on certain characteristics. It is a widely used method in Economics and Finance, to analyze the structure of markets, identify patterns and trends, and classify countries or companies into different groups in accordance with their economic and financial properties. A particular example of a study with cluster analysis is the one carried out by Yang and Hu (2008), who analyze data from the China Human Development Index in 1982, 1995, 1999 and 2003, in order to classify its provinces into four levels of income based on the three basic aspects contemplated in that index. We can also refer to other recent studies which show the interest of clustering in analyzing some economic variables. For example, Gostkowski et al. (2021) study the relationship between the national level of economic development and energy consumption in the main sectors of industry of the countries that belong to the Visegrad Group; He et al. (2021) analyze the socio-economic spatial structure of urban agglomeration in China by using clustering, and Zarikas et al. (2020) use clustering to group countries with respect to active cases, active cases per population and active cases per population and per area, so that the impact of the COVID-19 pandemic can be explored. In this sense, different economic variables such as GDP per capita, economic growth rate, foreign investment, international trade, among others, can be used to group countries into different categories based on their level of economic development and its growth dynamics. For example, cluster analysis can be used to identify groups of countries that converge with each other, that is, which have similar levels of economic development and are growing at a similar rate. In this way, as we will develop later, patterns of convergence or divergence between countries can be identified and the underlying causes of these trends can be analyzed. We can conclude from this that cluster analysis is a useful tool for analyzing the structure and dynamics of countries, and can be used in combination with other statistical techniques, such as time series analysis and Markov chains, to obtain a more complete and accurate view of economic convergence between countries. The main goal of this work is the application of the previously mentioned methodologies to the study of the convergence of per capita income of the Spanish provinces (in fact, we consider the fifty provinces together with the two autonomous cities in Spain) based on the GDP per capita of each of them in the years of the last decade. Moreover, we can evaluate the impact of COVID-19 pandemic on this convergence and study its speed of this convergence, both with and without COVID-19. Indeed, there are some recent works in which the authors face COVID-19 to income inequality, such as the one carried out by Deaton (2021). Additionally, a cluster analysis can let us have an idea of the possible groups of income in the future. Hence, the structure of the work is as follows: Section 2 provides some concepts and results that the reader should recall, which have to do with these two statistical methodologies as well as some reasons to chose both of them in the study. Moreover, Section 3includes the application of them to the study of the evolution of Economies 2025,13, 17 3 of 16 per capita income in Spain, by provinces, based on historical data, analyzing the behavior of the Spanish economy in the long term, both in the current situation and in case the pandemic generated by COVID-19 had not existed. Furthermore, in this section, a discussion is given together with the results. Finally, Section 4collects the main conclusions of the work. Provincial Income Disparities in Spain: A Literature Overview As stated at the beginning of this section, Markov chains have become an interesting approach to deal with the evolution of economic variables and, in particular, of per capita income over the years, thanks to many works in the economic literature. In this subsection, we focus on Spanish studies which are related to this topic. On the one hand, it is worth mentioning the work by Le Gallo and Chasco (2008), who study the evolution of the population growth among the group of 722 municipalities included in the Spanish urban areas over the period 1900–2001. Furthermore, Ayuda et al. (2010) analyze the disparities in long-run regional population growth in continental Europe, concluding that there is a common pattern of divergence in economic growth of Europe. Markov chains are used in this paper and they also consider Spain as a particular case in the study, for what a cluster analysis is useful. On the other hand, Gardeazábal (1996) studies the Spanish provinces dynamics, in terms of their income, in the time period between 1967 and 1991, concluding that they tend to the equilibrium distribution, being more concentrated in medium levels of income. However, Tirado et al. (2016) explore per-capita GDP disparities across Spanish provinces from 1860 to 2010. The previous cited works suggest us considering Markov chains and cluster analysis in order to study the evolution of per capita income in Spain for the last decade, which has not been analyzed in the literature yet. 2. Methodology In this section, we collect some concepts and results related to Markov chains and cluster analysis, which will be used in the following section to get the results of the main study. Moreover, we introduce the data we will work with and discuss the hypothesis on the Markov chains approach and the suitability of the type of clustering used in the study. 2.1. Markov Chains We refer the reader to Appendix Ain order to recall some basic preliminaries on Markov chains. In this paper, we will work with finite and homogeneous discrete-time Markov chains. The Markov chain must be finite because we want to group the provinces into a finite number of income states, as usual in the related literature. Finally, homogeneity assumption has to do with the idea of getting a unique matrix which gathers all the information according to the collected data. 2.1.1. Long-Term Behavior Dynamic models which are based on Markov chains have, as a point of interest, the analysis of the convergence of the transition probabilities when the time tends to infinity. This leads us to study the long-term behavior, also known as stationary or limit, and which is fundamental in this work to carry out a dynamic analysis of economic aspects. Particularly, if {Xn:n∈N} is a homogeneous and finite Markov chain, with k possible states, the limiting distribution is the probability distribution given by lim n→∞P1(1)P2(1). . . Pk(1)      p11 p12 . . . p1k p21 p22 . . . p2k . . .. . ..... . . pk1pk2. . . pkk       n . Economies 2025,13, 17 4 of 16 Getting the limiting distribution in the study of this paper means that we can know what is likely to happen in the long term according to the distribution of the per capita income. Roughly speaking, the Spanish provinces tend to group into some states according to the proportions given by the limiting distribution. However, the limiting distribution does not have to exist and, if it exists, it may not be unique, since it will depend on the initial distribution. In case the previous limit does not change for any initial distribution, we say that it is the equilibrium distribution of the chain. Furthermore, the stationary distribution is the one that does not change after each transition of the chain according to the transition probability matrix, that is, it is a row matrix p such that pP =p . It should be taken into account that when the equilibrium distribution exists and is unique, then it meets the stationary one. 2.1.2. States Classification A first classification of the states of a Markov chain has to do with the access between them: • We will say that a state xj is accessible from another state xi when pij > 0 at some instant of time. Furthermore, when it is probable to go from one state to another in both directions, we will say that both states communicate. If all the states of a Markov chain communicate, we say that the Markov chain is irreducible. • However, if there is a state that cannot be reached from any other in the Markov chain, we will say that it is ephemeral. • On the other hand, if there is a state from which we cannot reach any other one, we say that it is absorbing. Mathematically, the state xiwill be absorbing if pii =1. We can consider another criterion to classify the states of a Markov chain, which has to do with the probability of coming back to a certain state at some point of time. If this probability is 1, we say that the state is recurrent, while if it is less than 1, we will say that it is transitory. A well-known mathematical result establishes that two states that communicate are both recurrent or both transitory. Additionally, if we consider a recurrent state and define the random variable that describes the number of transitions needed to come back to that state, we can find its expected value, which will provide us the mean recurrence time, that is, the mean time since the state is left until the Markov chain returns to it. In particular, we talk about a positive recurrent state if its mean recurrence time is finite. On the other hand, a state is said to be periodic if, starting from it, it is only possible to return to it in a number of stages multiple of an integer greater than 1. It will be aperiodic if we can return to it after each transition. In this case, the period is 1. Table 1summarizes the classification of the states of a Markov chain according to both criteria. Table 1. States classification. Type of State Conditions xjaccesible from xipij >0 xjephemeral pij =0 for each i xiabsorbing pii =1 Recurrent Probability of coming back to it =1 Positive recurrent Finite mean recurrence time Transitory Probability of coming back to it <1 Aperiodic Period =1 Economies 2025,13, 17 5 of 16 The next result is one of the keys of this work: Theorem 1. If {Xn:n∈N} is a homogeneous, finite, irreducible and aperiodic Markov chain, then the stationary distribution exists, is unique and meets the equilibrium distribution. Moreover, the mean recurrence time of each state is given by the inverse of the respective probability in the equilibrium distribution. This theorem gives special emphasis to homogeneous, finite and discrete-time Markov chains. It is worth noting that the software R has a package which is especially focused on the study of these stochastic processes. It is called “markovchain” and can simplify some calculations, so that we can get conclusions from a research work based on the application of dynamic probabilistic models. In order to show this methodology, we refer the reader to Appendix B.1 so that the basic codes in this software can be seen. 2.1.3. Estimating the Transition Probability Matrix Suppose that {Xn:n∈N} is a homogeneous and finite Markov chain, with k possible states. Then the transition probability matrix is constant. If the transition probabilities are not known in advance, we can estimate them if we have data for individual transitions between two consecutive instants of time. In other words, if nij is the number of individuals that were in state xi at time t , and reach state xj at time t+ 1, then the maximum likelihood estimator of the transition probability pij is given, according to Anderson and Goodman (1957), by b pij =nij ∑k j=1nij . Therefore, the probability of moving from state xi to state xj can be calculated as the proportion of individuals who, being in the state xi in a certain instant of time, reach the state xj in the following period. In fact, this estimator is justified to be consistent, that is, the larger the sample size, the better the estimation made. Moreover, it is known that, although this estimator is biased, its bias decresases as the number of individuals under study increases. 2.2. Data and Treatment Next step is choosing the data we are going to work with. We collect the GDP per capita, in euros, for the fifty Spanish provinces together with the two autonomous cities, in the time period between 2010 and 2020. We have not considered more years, since the idea is to compare the future estimation with and without COVID-19 pandemic. The source for these data is the National Statistics Institute (2023) (Spain). The data are collected in Table 2, in which the numbers have been rounded to three decimal places. Table 2. GDP per capita relative to the Spanish average for each Spanish province/ autonomous city and year (2010–2020). Province/Auton. City 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Almería 0.802 0.754 0.767 0.760 0.783 0.801 0.838 0.857 0.839 0.841 0.862 Cádiz 0.735 0.736 0.729 0.717 0.699 0.691 0.691 0.695 0.693 0.700 0.679 Córdoba 0.717 0.715 0.696 0.711 0.706 0.717 0.709 0.711 0.703 0.681 0.707 Granada 0.708 0.713 0.719 0.719 0.732 0.737 0.715 0.707 0.705 0.713 0.725 Huelva 0.747 0.779 0.781 0.732 0.719 0.729 0.735 0.763 0.781 0.757 0.767 Jaén 0.706 0.713 0.661 0.715 0.675 0.731 0.699 0.691 0.709 0.670 0.718 Economies 2025,13, 17 6 of 16 Table 2. Cont. Province/Auton. City 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Málaga 0.758 0.748 0.731 0.724 0.730 0.725 0.715 0.720 0.726 0.729 0.710 Sevilla 0.814 0.815 0.818 0.799 0.803 0.790 0.780 0.782 0.783 0.785 0.795 Huesca 1.124 1.132 1.118 1.164 1.139 1.105 1.168 1.136 1.117 1.119 1.216 Teruel 1.045 1.045 1.062 1.085 1.084 1.023 0.994 0.955 0.977 0.965 0.994 Zaragoza 1.093 1.087 1.076 1.085 1.086 1.071 1.076 1.091 1.096 1.096 1.127 Asturias 0.917 0.914 0.905 0.892 0.883 0.882 0.872 0.878 0.880 0.879 0.887 Baleares 1.059 1.059 1.067 1.065 1.077 1.077 1.087 1.085 1.081 1.071 0.913 Las Palmas 0.838 0.836 0.824 0.833 0.820 0.801 0.811 0.810 0.810 0.800 0.716 Sta. Cruz de Tenerife 0.890 0.886 0.878 0.860 0.851 0.843 0.824 0.827 0.816 0.807 0.742 Cantabria 0.945 0.937 0.934 0.921 0.927 0.910 0.913 0.912 0.918 0.922 0.934 Ávila 0.788 0.801 0.818 0.808 0.802 0.788 0.775 0.776 0.784 0.791 0.828 Burgos 1.111 1.132 1.156 1.129 1.112 1.100 1.112 1.128 1.145 1.128 1.144 León 0.870 0.867 0.874 0.856 0.848 0.838 0.818 0.818 0.823 0.829 0.867 Palencia 1.022 1.043 1.024 1.028 1.010 1.027 1.062 1.004 1.054 1.041 1.069 Salamanca 0.804 0.810 0.810 0.798 0.796 0.797 0.810 0.808 0.811 0.822 0.855 Segovia 0.939 0.930 0.921 0.917 0.926 0.931 0.907 0.850 0.858 0.860 0.887 Soria 1.004 1.015 0.993 1.016 1.022 1.018 0.999 0.986 1.082 1.066 1.074 Valladolid 1.026 1.020 1.018 1.020 1.025 1.022 1.045 1.057 1.074 1.064 1.092 Zamora 0.798 0.823 0.851 0.832 0.819 0.819 0.810 0.743 0.756 0.770 0.807 Albacete 0.801 0.793 0.798 0.801 0.783 0.798 0.793 0.806 0.815 0.823 0.851 Ciudad Real 0.835 0.837 0.841 0.826 0.798 0.830 0.834 0.836 0.839 0.824 0.857 Cuenca 0.839 0.859 0.870 0.875 0.852 0.868 0.866 0.867 0.882 0.856 0.891 Guadalajara 0.832 0.836 0.830 0.814 0.770 0.742 0.757 0.776 0.789 0.794 0.816 Toledo 0.760 0.743 0.730 0.731 0.718 0.716 0.721 0.716 0.724 0.717 0.746 Barcelona 1.172 1.167 1.172 1.181 1.193 1.193 1.201 1.208 1.205 1.208 1.201 Gerona 1.156 1.143 1.151 1.144 1.149 1.143 1.154 1.093 1.080 1.084 1.083 Lérida 1.191 1.190 1.219 1.247 1.241 1.240 1.171 1.091 1.091 1.105 1.119 Tarragona 1.168 1.156 1.153 1.157 1.170 1.190 1.206 1.211 1.175 1.154 1.113 Alicante 0.769 0.749 0.737 0.738 0.749 0.747 0.758 0.762 0.756 0.754 0.762 Castellón 0.970 0.998 0.972 0.989 0.992 1.021 1.037 1.090 1.067 1.062 1.048 Valencia 0.940 0.939 0.930 0.935 0.944 0.934 0.920 0.909 0.922 0.921 0.929 Badajoz 0.716 0.709 0.694 0.702 0.690 0.702 0.704 0.715 0.711 0.704 0.738 Cáceres 0.715 0.701 0.716 0.723 0.720 0.721 0.730 0.752 0.765 0.771 0.786 La Coruña 0.949 0.938 0.931 0.940 0.924 0.935 0.941 0.930 0.940 0.935 0.952 Lugo 0.865 0.880 0.899 0.918 0.936 0.950 0.924 0.892 0.910 0.893 0.889 Orense 0.803 0.824 0.842 0.839 0.829 0.825 0.838 0.839 0.852 0.875 0.890 Pontevedra 0.856 0.842 0.840 0.852 0.856 0.854 0.850 0.871 0.859 0.869 0.904 Madrid 1.340 1.360 1.377 1.372 1.372 1.373 1.373 1.370 1.364 1.369 1.370 Murcia 0.832 0.819 0.823 0.831 0.822 0.838 0.833 0.830 0.816 0.818 0.834 Economies 2025,13, 17 7 of 16 Table 2. Cont. Province/Auton. City 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Navarra 1.229 1.234 1.226 1.236 1.239 1.228 1.224 1.220 1.205 1.210 1.221 Álava 1.474 1.496 1.507 1.532 1.550 1.515 1.546 1.524 1.515 1.475 1.506 Vizcaya 1.235 1.227 1.239 1.231 1.245 1.246 1.237 1.212 1.217 1.221 1.220 Guipúzcoa 1.289 1.299 1.311 1.298 1.286 1.271 1.264 1.297 1.289 1.297 1.288 La Rioja 1.082 1.082 1.082 1.087 1.102 1.096 1.068 1.063 1.068 1.061 1.087 Ceuta 0.850 0.835 0.822 0.839 0.821 0.814 0.805 0.782 0.786 0.795 0.839 Melilla 0.794 0.777 0.752 0.759 0.751 0.743 0.741 0.717 0.724 0.728 0.765 The main idea is to follow the methodology proposed by Quah (1993), to analyze of the evolution of GDP per capita based on data from the last decade. This analysis, with a Markovian approach, will have a double goal: evaluate the impact of COVID-19 pandemic on economic convergence by provinces in Spain; and study the speed of this convergence, both with and without COVID-19. Specifically, we start from a homogeneous and finite Markov chain, with which to study the transitions between productivity states. Productivity will be grouped into four states, “Low income”, “Medium-low income”, “Medium-high income” and “High income”, which will be given by the quartiles of the overall GDP per capita distribution of the provinces relative to the national average of each year, that is, taking as data the result of dividing the GDP per capita of each province by the Spanish GDP per capita of the corresponding year. Next step is constructing the Markov chain transition probability matrix. With that purpose, we obtain a transition probability matrix between each pair of years of the considered period, which will give us a total of ten. To obtain each of them, the maximum likelihood estimator is used (see Section 2.1.3), according to which each transition probability can be obtained as the proportion of provinces that, being in a certain state in year t , change to a certain state in year t+ 1. The general matrix is the result of averaging the numbers in the same position of each of the ten matrix we have constructed before. Then we check if the transition probability matrix satisfies conditions of Theorem 1, so that the stationary and equilibrium distribution can be found and meet, as well as the mean recurrence time of each state. Finally, once we know the stationary distribution of the Markov chain, we try to get the convergence speed of the provinces towards this situation. For this purpose, Shorrocks (1978) proposes an index to analyze the mobility between states given by the transition probability matrix in which the elements of the main diagonal are each greater than or equal to the entries of the matrix that are located in the remaining positions. Specifically, this index is I1=n−tr(P) n−1, where tr(P) denotes the trace of the matrix P and n is the number of states of the Markov chain. This index gives us values between 0 and 1: mobility is null when I1= 1, because in this case tr(P) = n and, consequently, all states are absorbent; 0 means perfect mobility. Additionally, Sommers and Conlisk (1979) give another index with which to know the speed at which the Markov chain reaches the steady state, and it is I2=1− |λ2|, Economies 2025,13, 17 8 of 16 where λ2 is the eigenvalue of the transition probability matrix with the second highest modulus (in fact, in each Markov chain, λ1= 1 is always an eigenvalue, and it is the one having the greatest modulus). For further reference about measures of mobility, see, for example, Formby et al. (2004). 2.3. Cluster Analysis Cluster analysis is a data analysis technique whose main goal is to group the data in a homogeneous way, which means that the elements of the same group are similar to each other in terms of the characteristic which has been analyzed, as long as the discrepancies between individuals from different groups are significant. In other words, this technique tries to minimize the intra-group variability while trying to maximize the inter-group one. In order to determine which individuals in the sample have a certain similarity, distances are generally used and, in this work, we will operate with the most classic distance: the Euclidean one. This distance, in essence, measures the longitudinal magnitude in a straight line from one point to another. There are other distances used in the construction of a cluster, such as the Manhattan distance, the Mahalanobis one or the maximum one. We will have to minimize this distance between individuals to conclude which of them are the most similar. Individual grouping methods are classified into two groups: hierarchical and nonhierarchical clustering. In the first case, we can give a tree-based representation (called dendrogram) so that in each iteration, an order is followed and the structure to create the groups is kept. Moreover, they can be classified into two groups: • Agglomerative: they start from simple groups which become more sophisticated as more iterations are taken. It is, therefore, an ascending approach between individuals. • Divisive: we start from the sample as a group and, at each step, smaller groups are built until the desired number of clusters is achieved. It is, therefore, a descending approach. In non-hierarchical clustering, the number of groups is chosen and, subsequently, individuals are included in each group, being able to move from one group to another at each step, until a certain optimality criterion is got. In this work we will focus on the agglomerative hierarchical clustering based on Ward’s method, using the Euclidean distance to find the distance between elements. The method used has to do with the way of calculating the distance between groups. It does make sense to consider the hierarchical clustering in this work, since its representation (dendrogram) is quite useful for the reader in order to have a quick idea of the relationship between provinces in terms of their per capita income. What is more, thanks to this representation, one can group provinces into a different desired number of clusters. Particularly, we have chosen the agglomerative one, which is the most common type of hierarchical clustering, indeed Kassambara (2017). Since we want to group provinces in terms of their per capita income, we start by treating each province as a singleton cluster and, next, pairs of clusters are successively merged until all of them belong to a single cluster, containing all provinces. Finally, it is worth noting that we have considered four clusters as recommended by the Elbow method and in order to meet the number of states in the Markov chains analysis. For that purpose, software R is used in order to get the clustering. In order to get the final groups, we have added Appendix B.2, in which the used code is explained. 3. Results and Discussion 3.1. Evolution of per Capita Income in Presence of COVID-19 Let us consider the distribution of the per capita income of the fifty provinces and two autonomous cities in Spain for all years between 2010 and 2020 relative to the annual Economies 2025,13, 17 15 of 16 steadyStates(mc) Moreover, we can check if it is aperiodic by using the command period(mc) If the result is 1, the Markov chain is aperiodic. We can also get the mean recurrence time of each state: meanRecurrenceTime(mc) Finally, finding out if it is irreducible is possible by writing is.irreducible(mc) and checking if the result is “TRUE” or “FALSE”. As a complement, the command plot(mc) gives us the transition diagram, which is a plot where we can see the connection between states through the transition probabilities. Appendix B.2. Cluster Analysis Next, we expose the steps that have been followed in order to get the final groups when clustering (agglomerative hierarchical clustering) provinces according to their income in the period 2010–2020: 1. Standarize the variables, that is, subtract the mean from the value of each one and divide the result by the standard deviation of the values of the variable. If the data contain the information to be processed, we must implement df=as.data.frame(scale(data)) 2. Calculate the proximity matrix by using the Euclidean distance: d_eu <-dist(df, method =’euclidean’ ) 3. Find the agglomerative hierarchical cluster with Ward’s method: cluster <- hclust(d_eu, method = ’ward.D’) 4. Draw the dendrogram: plot(as.dendrogram(cluster)) 5. Draw rectangles that group a certain number, k, of the individuals in the sample: rect.hclust(cluster, k = 4) References Anderson, T. W., & Goodman, L. A. (1957). Statistical inference about Markov chains. The Annals of Mathematical Statistics,28(1), 89–110. [CrossRef] Arreola, D., & Montiel, L. V. (2024). Approximating income inequality dynamics given incomplete information: An upturned Markov chain model. Computational Statistics,39(2), 629–651. [CrossRef] Ayuda, M. I., Collantes, F., & Pinilla, V. (2010). Long-run regional population disparities in Europe during modern economic growth: A case study of Spain. The Annals of Regional Science,44, 273–295. [CrossRef] Bode, E., & Nunnenkamp, P. (2011). Does foreign direct investment promote regional development in developed countries? A Markov chain approach for US states. Review of World Economics,147, 351–383. [CrossRef] Chen, Y., Mamon, R., Spagnolo, F., & Spagnolo, N. (2022). Renewable energy and economic growth: A Markov-switching approach. Energy,244, 123089. [CrossRef] Economies 2025,13, 17 16 of 16 Deaton, A. (2021). COVID-19 and global income inequality. LSE Public Policy Review,1(4), 1. [CrossRef] Everitt, B., Landau, S., Leese, M., & Stahl, D. (2011). Cluster Analysis (5th ed.). John Wiley & Sons. Fingleton, B. (1997). Specification and testing of Markov chain models: An application to convergence in the European Union. Oxford Bulletin of Economics and Statistics,59(3), 385–403. [CrossRef] Formby, J. P., Smith, W. J., & Zheng, B. (2004). Mobility measurement, transition matrices and statistical inference. Journal of Econometrics, 120(1), 181–205. [CrossRef] Gardeazábal, J. (1996). Provincial income distribution dynamics: Spain 1967–1991. Investigaciones Económicas,20(2), 263–269. Gostkowski, M., Rokicki, T., Ochnio, L., Koszela, G., Wojtczuk, K., Ratajczak, M., Szczepaniuk, H., Bórawski, P., & Bełdycka-Bórawska, A. (2021). Clustering analysis of energy consumption in the countries of the visegrad group. Energies,14(18), 5612. [CrossRef] Haller, A., Gherasim, O., & B˘alan, M. (2020). Medium-term forecast of European economic sustainable growth using Markov chains. Zb. rad. Ekon. fak. Rij.,38(2), 585–618. He, L., Tao, J. G., Meng, P., Chen, D., Yan, M., & Vasa, L. (2021). Analysis of socio-economic spatial structure of urban agglomeration in China based on spatial gradient and clustering. Oeconomia Copernicana,12(3), 789–819. [CrossRef] Karahasan, B. C. (2020). Can neighbor regions shape club convergence? Spatial Markov chain analysis for Turkey. Letters in Spatial and Resource Sciences,13(2), 117–131. [CrossRef] Kassambara, A. (2017). Practical guide to cluster analysis in R: Unsupervised machine learning. Sthda. Kerkouch, A., Bensbahou, A., Seyagh, I., & Agouram, J. (2024). Dynamic analysis of income disparities in Africa: Spatial markov chains approach. Scientific African,24, e02236. [CrossRef] Le Gallo, J., & Chasco, C. (2008). Spatial analysis of urban growth in Spain, 1900–2001. Empirical Economics,34, 59–80. [CrossRef] Lipták, K. (2011). The application of Markov chain model to the description of hungarian market processes. Zarz ˛adzanie Publiczne, 16(4), 133–149. National Statistics Institute. (2023). Regional Accounting of Spain. Results. GDP and GDP per Capita. 2000–2022 Series. Available online: https://www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica _ C&cid=1254736167628&menu=resultados&idp= 1254735576581 (accessed on 2 October 2023). Papanikolaou, N. (2020). Markov-switching model of family income quintile shares. Atlantic Economic Journal,48(2), 207–222. [CrossRef] Quah, D. (1993). Empirical cross-section dynamics in economic growth. European Economic Review,37, 426–434. [CrossRef] Rey, S. (2023). Intersectional urban dynamics: A joint Markov chains approach. Letters in Spatial and Resource Sciences,16(1), 36. [CrossRef] Shorrocks, A. F. (1978). The measurement of mobility. Econometrica: Journal of the Econometric Society,46(5), 1013–1024. [CrossRef] Sommers, P. S., & Conlisk, J. (1979). Eigenvalue immobility measures for Markov chains. Journal of Mathematical Sociology,6, 253–276. [CrossRef] Takács, L. (1960). Stochastic processes problems and solutions. Chapman and Hall. Tirado, D. A., Díez-Minguela, A., & Martinez-Galarraga, J. (2016). Regional inequality and economic development in Spain, 1860–2010. Journal of Historical Geography,54, 87–98. [CrossRef] Wenxuan, Y. (2023). Human capital dynamics across provinces in china: A spatial markov chain approach. Forum of International Development Studies,53(8), 1–17. Yang, Y., & Hu, A. (2008). Investigating regional disparities of China’s human development with cluster analysis: A historical perspective. Social Indicators Research,86, 417–432. [CrossRef] Zarikas, V., Poulopoulos, S. G., Gareiou, Z., & Zervas, E. (2020). Clustering analysis of countries using the COVID-19 cases dataset. Data in Brief,31, 105787. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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