Economic Freedom Index Calculation Using FCM

The Index of Economic Freedom is an annual index and ranking created by The Heritage Foundation and The Wall Street Journal in 1995 to measure the degree of economic freedom in the world's nations. There are many kinds of Economic Freedom Indices depending on variables which many institute or company determine for their research. The aim is to predict countries or regions according to economic parameters. In this study, fuzzy clustering algorithm is proposed for economic freedom ındex calculation. By using degree of memberships founded by FCM, Economic Freedom index will be calculated for regions. Results compared with indices calculated by The Heritage Foundation for the year 2013, 2014, 2015 and 2016. It is showed that FCM is an alternative method for index calculating systems.


Introduction
In an economically free society, each person controls the fruits of his or her own labor and initiative.Individuals are empowered-indeed, entitled-to pursue their dreams by means of their own free choice (Miller and Kim, 2015b).Economic freedom and democracy affects economic performance by identifying organizational structure.Than, we have to answer these two questions: What is economic freedom and what is it used for?"Economic freedom" means the degree to which a market economy is in place, where the central components are voluntary exchange, free competition, and protection of persons and property (Gwartney and Lawson, 2002).The goal is to characterize the institutional structure and central parts of economic policy (Berggren, 2003).
Economic freedom is the fundamental right of every human to control his/her own labor and property.In an economically free society, individuals are free to work, produce, consume and invest in any way they please.In an economically free society, governments allow labor, capital and goods to move freely and refrain from coercion or constraint of liberty beyond the extent necessary to protect and maintain liberty itself (www.heritage.org, 2016).The goal of economic freedom is not simply an absence of government coercion or constraint but the creation and maintenance of a mutual sense of liberty for all.
Since Adam Smith's Wealth of Nations, it has been argued that economic freedom is essential to a nation's economic progress.Studies by Dollar (1992) and by Sachs and Warner (1995) concluded that economic growth is faster in countries which are economically more open.It should be noted that economic freedom is not synonymous with political freedom and civil liberty.
Political freedom is concerned with the way in which nations choose their governments and other representatives.On the other hand Civil liberty includes the right of citizens to free assembly (including the right to organize trade unions), freedom of the press, freedom of religion, and due process and equal treatment under the law (Johnson and Lenartowicz, 1998).Some uses of economic freedom can be given below (Miller and Kim, 2015a): i.
Advancing Opportunity: Today's successful economies are not necessarily geographically large or richly blessed with natural resources.Many economies have managed to expand opportunities for their citizens by enhancing their economic dynamism.In general the overarching objective of economic policies must be to create an environment that provides the most opportunity for the widest range of activities that can lead to increased prosperity. ii.
Promoting Prosperity: In many respects, economic freedom is merely shorthand for an openness to entrepreneurial activity that increases opportunity for individuals to succeed in their endeavors.
iii.Antidote to Poverty: By a great many measures, the past two decades during which the Index has been charting the advance of economic freedom have been the most prosperous in the history of humankind.Those countries that have adopted some version of free market capitalism, with economies supported by efficient regulations and open to the free flow of goods, services, and capital, have participated in an era of globalization and economic integration in which solutions to many of the world's development problems have taken hold and generated real improvements in living standards.
iv. Societal development and democratic progress: Growing economic freedom is unequivocally about more than financial success.Achieving greater overall prosperity that goes beyond materialistic and monetary dimensions of wellbeing is equally important.The societal benefits of economic freedom extend far beyond higher incomes or reductions in poverty.Countries with higher levels of economic freedom enjoy higher levels of overall human development as measured by the United Nations Human Development Index, which measures life expectancy, literacy, education, and the standard of living in countries worldwide v.
The Key to Upward Mobility and Greater Social Progress: The massive improvements in global indicators of income and quality of life largely reflect a paradigm shift in the debate over how societies should be structured to achieve the most optimal outcome.Over the past two decades, this debate has largely been won by capitalism.However, fears that the immediate benefits of capitalism are fading has brought to the forefront concerns about economic mobility and economic freedom.
As we summarize benefits of economic freedom, we can say that economic freedom increases in income per capita and most low-income group income, amplify life expectancy and play a key role in the development of society.

Economic Freedom Index
The Index of Economic Freedom is an annual index and ranking created by The Heritage Foundation and The Wall Street Journal in 1995 to measure the degree of economic freedom in the world's nations.For over twenty years the Index has delivered thoughtful analysis in a clear, friendly, and straight-forward format.With new resources for users and a website tailored for research and education, the Index of Economic Freedom is poised to help readers track over two decades of the advancement in economic freedom, prosperity, and opportunity and promote these ideas in their homes, schools, and communities.With the help economic freedom index, we simply analyses the country's economic freedom levels or categorizes them in to similar groups.
2.1.Selected Literature for Freedom Indices: Bengoa and Robles (2003), explores the interplay between economic freedom, foreign direct investment (FDI) and economic growth using panel data analysis for a sample of 18 Latin American countries for 1970-1999.Haan and Sturm (2000), compared various indicators for economic freedom.The robustness of the relationship between freedom and growth is also examined in the paper.The conclusion is that greater economic freedom fosters economic growth but the level of economic freedom is not related to growth.Gwartney et all. (1999), examines the importance of economic freedom by using an index that measures economic freedom in four basic areas: Money and inflation, economic structure, takings and discriminatory taxation, and international trade.The empirical results show that economic freedom is a significant determinant of economic growth, even when human and physical capital, and demographics are taken into account.
Johnson and Lenartowicz (1998) presented a framework for examining the relationship among cultural values, economic freedom and economic growth.Also they found two important results: Firstly, evidence of strong positive association both between economic freedom and economic growth and weak uncertanity between economic freedom and individualautonomy.Ayal and Karras (1998), examined the relationship between economic growth and economic freedom.Their results are very supportive of the proposition that aggregate "economic freedom" enhances growth both via increasing total factor productivity and via enhancing capital accumulation.Stroup (2006) examines the interaction of economic freedom and democracy on measures of health, education, and disease prevention in society.He has found that greater economic freedom consistently enhances these welfare measures, even among more democratic countries.Democracy has a smaller positive influence that disappears for many welfare measures in countries with more economic freedoms.Heckelman (2000) investigates casuality between economic freedom and economic growth.As for the results; growth may precede one of the component indexes and no relationship is found to exist between growth and two of the indexes.Esposto and Zaleski (1999) attempt to bridge this gap by analyzing the effect of economic freedom on the quality of life.Taking advantage of newly developed measures of economic freedom, we analyze the impact of economic freedom on life expectancy and literacy rates.They also found that greater economic freedom enhances the quality of life both across nations and increases the improvements in the quality of life over time.economic freedom with economic growth and income equality.At the end he gives a short summary of implications for economic policy.

Fuzzy Clustering Analysis
Clustering analysis is a statistical classification technique for discovering whether the individuals of a population fall into different groups by making quantitative comparisons of multiple characteristics.The objective of cluster analysis is the classification of objects according to similarities among them and organizing of data into groups (Balasko et all., 2005).
Fuzzy Clustering Analysis comes into the picture as an appropriate method when the clusters cannot be separated from each other distinctly or when some units are uncertain about membership.Membership grades are assigned to each of the data points.These membership grades indicate the degree to which data points belong to each cluster.Thus, points on the edge of a cluster, with lower membership grades, may be in the cluster to a lesser degree than points in the center of cluster.Fuzzy clusters are functions modifying each unit between 0 and 1 which is defined as the membership of the unit in the cluster.The units which are very similar to each other hold their places in the same cluster according to their membership degree.Similar to other clustering methods, fuzzy clustering is based on distance measurements as well.The structure of the cluster and the algorithm used to specify which of these distance criteria will be used.Some of the convenient characteristics of fuzzy clustering can be given as follows (Naes and Mevik, 1999): i.
It provides membership values which are convenient to comment on. ii.
It is flexible on the usage of distance. iii.
When some of the membership values are known, they can be combined with numeric optimization.
The advantage of fuzzy clustering over classical clustering methods is that it provides more detailed information on the data.Since there will be too much output when there are too many individuals and clusters, it is difficult to summarize and classify the data.Moreover, fuzzy clustering algorithms, which are used when there is uncertainty, are generally complicated (Oliveira and Pedrycz, 2007).

Fuzzy C-Means (FCM) Algorithm
Fuzzy C-Means algorithm forms the basis of all clustering techniques that depend on objective function.It was developed by Bezdek (1974a and1974b).When the FCM algorithm comes to a conclusion, the dots in the p dimension space become a sphereshaped figure.It is assumed that these clusters are approximately the same size.
Cluster centers represent each cluster and they are called prototypes.Euclidean distance ik d between the data and the cluster center is used as the distance measurement and can be calculated by formula given in Equation.1.
where k x represents the position observation value in the coordinated system, and i v represents the cluster center.It is necessary to know the number of clusters and the membership degrees of the individuals beforehand to be able to put this technique into practice.Since it is difficult to know these parameters before the application, it is possible to find these values through the method of trial and error or through some techniques developed.
The objective function used for this clustering method is as follows: This function is the weighted least square function.n parameter represents the number of observations, and c represents the number of clusters.
, function is minimized for each value of c, in other words if it is derived from the 1st degree according to j v 's and made equal to 0, the prototype of FCM algorithm can be given in Equation .3: In equation.3, it symbolizes; the number of cluster with c, fuzziness index with m, process ending criteria with  and membership degrees matrix with U of FCM algorithm generate cluster prototypes at random.By taking means of these values, membership degrees matrix is calculated as given in Equation.4:(Sintas et all., 1999).
U cluster prototypes are updated in all iteration and the processes are repeated until value reach to previously determined error term.After FCM algorithms is implemented membership degrees are used in other to decide which individual will participate in which cluster.For each individual; the highest cluster membership is observed and this individual is added to that cluster.However each individual can participate in other clusters with a certain membership degree (Sintas et all., 1999).

Fuzzy Clustering Validity Index
A good clustering method will produce high quality clusters with high intra-class similarity and low inter-class similarity.The quality of a clustering result depends on both the similarity measure used by the method and its implementation.The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.
Aim of clustering analysis is to put similar objects into same groups.In many clustering algorithms, it is hard to know the actual num¬ber of cluster before the Alphanumeric Journal Volume 6, Issue 1, 2018 application.In studies based on real data, if the researchers do not have preliminary information about the number of cluster, it cannot be known whether the number of cluster which calculated is more or less than the real num¬ber of cluster.Determination processes of the opti¬mal number of clusters are generally called as Cluster Validity.So, after clustering processes are carried out the validity of the number of cluster which calculated can be determined (Halkidi et all, 2001, Erilli et all, 2011).
Many fuzzy clustering analysis validity indexes are used in literature (Bezdek, 1974a and1981;Rezaee et all., 1998;Kwon, 1998;Xie and Beni, 1991).Conveni¬ent clustering validity analyses are used depending on data structure and the number of variables.In this study, Artificial Neural Networks Based Cluster Valid¬ity Index was used for the optimum number of cluster detection.

Artificial Neural Networks Based Cluster Validity Index
This method was proposed by Erilli et all. (2011).Optimum number of cluster is decided by artificial neural network.In this method at first the lowest and the highest number of cluster which are convenient to data are decided.The most convenient determined number of cluster will be in this interval.Let the optimal number of cluster is opt c , maximum number of the cluster is maks c and minimum number of the cluster is min c , are defined.The relation between them will be like that; Then, feed-forward artificial neural networks are implemented for each possible numbers of clusters in the manner that its output will be data matrix and its target value will be the number of cluster to which each data is appointed as a result of fuzzy clustering.The median of RMSE (root-mean-square error) value which is obtained through artificial neural networks according to several hidden layer unit number are calculated for each number of clusters.The graph or obtained median values of each number of clusters or classification error is drawn and the first jumping (where median value of RMSE overgrows for the first time) is observed.Then prejumping value is determined as the most convenient number of cluster (Erilli et all., 2011).

The Data
There are many institutions which measures and calculates economic freedom.Some of the organizations are; Heritage, Fraser Institute, Free the world, Cato Institute, Buck Eye Institute, Ratio Institute etc.In this article, it has been used heritage data for calculation indices with FCM method.Heritage organization uses 10 measured aspects of economic freedom which can be grouped into four broad categories (Miller and Kim, 2015b) Rule of Law (Property Rights, Freedom from corruption) ii.
Government Size (Fiscal Freedom, Government Spending) iii.
Regulatory Efficiency (Business Freedom, Labor Freedom, Monetary Freedom) iv.
Market Openness (Trade Freedom, Investment Freedom, Financial Freedom) Alphanumeric Journal Volume 6, Issue 1, 2018 Each of 10 economic freedoms within these categories is graded on a scale of 0 to 100.A country's overall score is derived by averaging these 10 economic freedoms with equal weights being given to each.
There are also 11 variables given in Heritage reports.These are; 5 important subject about tax (Tariff Rate %, Income Tax Rate %, Corporate Tax Rate %, Tax Burden % of GDP and Gov't Expenditure % of GDP) and 6 important subject of economy (Population (Millions), GDP Growth Rate (%), 5 Year GDP Growth Rate (%), Unemployment (%), Inflation (%) and Public Debt (% of GDP) ).
The data includes 185 countries.But 8 countries variables are mostly missing (Afghanistan, Iraq, Kosovo, Libya, Liechtenstein, Sudan, Syria, Somalia), so we analyze application for 177 countries for the year 2013, and 178 countries for the years 2014, 2015 and 2016 (Brunei Darussalam is added).
Fuzzy clustering analyses are used to categorize the countries with these 21 variables.
After FCM administration to the data, degree of membership for each country can be calculated.With the help of membership degrees, ranking for countries is calculated and compared with the list of Heritage Foundation for the year 2013, 2014, 2015 and 2016.Correlation coefficient and significant level summarize the power of proposed method for calculating economic freedom index.Analysis is performed with Matlab.2009b and SPSS.21 package programs.

Classification Results
For the data 2013, it has been calculated 6 clusters as well.Separation of cluster centers can be seen easily in Figure .1.Countries have to belong to the clusters with membership degrees with a coefficient between 0 and 1. Whichever is greater from a country's coefficient, the country will be assigned to that cluster.Every observation is ranked from big to small within the cluster they belong to according to their membership degrees.In addition, the focal point of each cluster is calculated and these are also ranked according to their sizes.Thus, the whole series is separately ranked from big to small and the ranking calculation is completed.
In  As we look to the Table .1,15 of 20 countries are take part in both list.Harmony of two lists compared with Spearman Rank Correlation coefficient.Spearman's rho between FCM and Heritage for the whole data is given in Table .2.The calculated coefficient is calculated as 0,748 and it is significant at level 0,01 (p=0,000).

Table.2. Spearman Rank Correlation Results for Data 2013
For the data 2014, it is performed 5 clusters.Separation of cluster centers can be seen easily in Figure.In Table .3,it is given first 20 countries arranged in order for the data 2014.Spearman's rho coefficient between FCM and Heritage is given in Table .4.Correlation coefficient calculated as 0,829 and it is significant at level 0,01 (p=0,000).For the data 2015, it is performed 5 clusters.Separation of cluster centers can be seen in Figure .3.As for the results given in Table .5 it can be seen for the first 20 countries, 13 of 20 countries are take part in both list.For the data 2016, it is also performed 5 clusters.As we look for the first 20 countries in Table .7,13 of 20 countries are take part in both list.

Heritage
Countries FCM Spearman's rho between FCM and Heritage is 0,934 (highest against all years) and is significant at level 0,01 given in Table.8 (p=0,000).Finally, in Table.9, the first three grades resulting from the last 4 analyzes are summarized.In all Heritage result, first 2 countries are all same: Hong Kong and Singapore.Third palce is repeated by Australia and New Zealand.There are only 4 different countries in 12 steps.But in FCM results, there 9 different countries in 12 steps.Also there are 3 different leaders in whole FCM results.The correlation coefficient used in the comparison of FCM-Heritage sequences shows that all the results are in the same direction and highly correlated.Moreover, all the coefficients were found to be statistically significant.These results also show that althought the analysis methods are different but results are similar to those used by large organizations such as Heritage or Fraser.It has been proved that successful results can be obtained when using alternative methods in these types of comparisons.

Conclusion
Economic freedom is the key to greater opportunity and an improved quality of life.Economic freedom index is one of the way to calculate economic freedoms and levels.While a simple concept, it is an engine that drives prosperity in the world and is the difference between why some societies thrive while others do not.The goal of economic freedom is to characterize the institutional structure and central parts of economic policy (Berggren, 2003).Also it is not simply an absence of government constraint but the creation and maintenance of a mutual sense of liberty for all.
Alphanumeric Journal Volume 6, Issue 1, 2018 The Index of Economic Freedom is an annual index and ranking created by The Heritage Foundation and The Wall Street Journal in 1995 to measure the degree of economic freedom in the world's nations.For over twenty years the Index has delivered thoughtful analysis in a clear, friendly, and straight-forward format.There are many institutions which measures and calculates economic freedom.All they are using different types of variables and different methods.Most methods based on mathematical calculations.In this study, it is used Fuzzy Clustering Analysis to determine Economic Freedom Index.Analysis is applied for 4 different Data sets.Data sets for the years 2013, 2014, 2015 and 2016 is taken from Heritage web site.
Correlation coefficients between FCM and Heritage takes place 0,748 and 0,934 and all they are significant at level 0,01.Overall correlation coefficient is 0,881 and it is clearly high level.The high correlation coefficients between the suggested index rankings and the Heritage rankings also indicate the strength of the study results.The fact that the coefficients are statistically significant also indicates that there is not much difference between the calculations.Fuzzy clustering and FCM algorithm increased its popularity recently.It can give better results when the number of data or the number of variables increases.Clustering analysis has been shown to give effective results when we have difficulty in deciding individuals.While classifying, it can produce more clear results with complicated data structures when compared with other clustering or classifying methods.With this study, it has been presented that fuzzy clustering analysis can be successfully used for index calculation or ranking measures.
We can simply notice that, results of Fuzzy Clustering Analysis is clearly satisfactory for ranking the countries via economic freedom index calculation.With different analysis methods, organizations can better analyze their current situation.With the help of this study, it has been presented that fuzzy clustering analysis (classification methods) can be successfully used for index calculation or ranking measures Shen and Williamson (2005) searched structural equation-based analysis of data for 91 nations includes several important determinants of cross-national variation in perceived levels of corruption.The analyses yield four major findings: 1) democracy, as measured by indicators of political rights, civil liberties, and press freedom, has a positive effect on perceived level of corruption control; 2) state strength has a positive direct effect; 3) openness of the economy, as measured by economic freedom, has a positive effect; and 4) ethnolinguistic fractionalization has both direct and indirect negative effects.Berggren (2003) analyses benefits of economic freedom as a survey.He explains the concept and importance of economic freedom by giving examples.He utilize Alphanumeric Journal Volume 6, Issue 1, 2018

Table . 1
. 2013 Ranking Results of FCM and Heritage for the first 20 Countries

Table . 3
Table.3 shows that 15 of 20 countries are take part in both list. .2014 Ranking Results of FCM and Heritage for the first 20 Countries

Table . 5
. 2015 Ranking Results of FCM and Heritage for the first 20 Countries Spearman's rho between FCM and Heritage is found 0,772 for Data 2015 given in Table.6 and it is significant at level 0,01 (p=0,000).

Table . 6
. Spearman Rank Correlation Results for Data 2015

Table . 7
. 2016 Ranking Results of FCM and Heritage for the first 20 Countries

Table . 8
. Spearman Rank Correlation Results for Data 2016