**3. Data**

In order to study the spread of EF around the world, its evolution during this past decade, and subsequently its impact on the richness of the world, it is necessary to obtain the values of the EFW index and of the IEF together with the gross domestic product for the studied countries

The EFW index values, obtained from the portal www.freetheworld.com, accessed on 30 October 2006 [38], are provided for 140 countries in the 2000–2006 period, i.e., over 7 years. The values of the IEF can be found on the site of the "Heritage Foundation" [39]. The indices are given for 157 countries in the (12 years) period 1997–2007. The values of the Gross Domestic Product per capita (GDP) of countries for corresponding periods may be downloaded from the IMF website [40]. All values are annual data.

We point out that it was unfortunately necessary to exclude certain countries for which the data was unavailable for various reasons. This is, for example, the case of Iraq. Iraq's second war has made the measurement of economic indicators quite dubious: the values obtained for the IEF and EFW indices or for GDP could not be considered to be significant. That being said, there are still 908 data points for the EFW index and 1784 for the IEF for the studied periods.

#### *3.1. Statistical Characteristics of Indices Distribution*

The first step in the study of the indices concerns the distribution of their values. The histograms and cumulated probability densities of the EFW and of the IEF are reproduced in Figures 1 and 2, respectively. The main statistical characteristics (mean, standard deviation, variance, coefficient of variation, skewness and kurtosis) of these distributions are included in Table 1.

**Figure 1.** (**a**) Economic Freedom of the World (EFW) histogram for 908 data points, i.e., when available for all (140) countries and for all (7) years; (**b**) cumulative probability density for the EFW and normal distribution fit with mean *μ* = 6.49 and variance *σ*<sup>2</sup> = 0.96.

**Figure 2.** (**a**) Index of Economic Freedom (IEF) histogram for 1784 data points; (**b**) cumulative probability density for the IEF and normal distribution fit with mean *μ* = 58.79 and variance *σ*<sup>2</sup> = 143.34.

Figures 1 and 2 sugges<sup>t</sup> that both indices follow a normal law slightly displaced to the right, i.e., to values greater than the median values, whence the negative skewness. This impression is reinforced by the average values of the indices: 6.49 for the EFW index and 58.79 for IEF, see Table 1. These two averages are greater than the corresponding median values: 5 in the case of EFW and 50 in the IEF. The skewness is negative for both indices: −0.3567 for the EFW index and −0.2373 for the IEF, see Table 1, confirming that the probability densities are no longer important for values above the median. These features show that the economies of the studied countries are generally more free than constrained.

**Table 1.** Summary of (rounded) main statistical characteristics of the economic freedom indicators distributions, i.e., the Economic Freedom of the World (EFW) index and Index of Economic Freedom (IEF), according to the examined time interval Δ*T* for the number *N* of data points.


In order to confirm that the distributions follow a normal law, a Kolmogorov–Smirnov (KS) test is performed. The results of the tests are shown in Table 2. The KS distances, DKS = 0.0399 for EFW and 0.0310 for IEF, are lower than the "critical values" of the normal distribution, 0.0449 for EFW and 0.0321 for the IEF. In addition, *p*−values, 0.1088 for the EFW index and 0.0633 for the IEF, are above the 5% significance level; thus the KS tests are considered to lead to statistically significant features. It is therefore possible to conclude that the EFW index and IEF values follow a normal law with *μ* = 6.49 and 58.79 and variance *σ*<sup>2</sup> = 0.96 and 143.34 respectively, i.e., the standard distribution (SD) is equal to 0.98 and 11.98, respectively.

#### *3.2. EFW Index in Year 2006*

For example, consider a specific year, 2006. Table 3 shows the EFW index values for the 20 freest countries for the year 2006. Hong Kong, Singapore, and New Zealand occupy the first 3 places. The rest of the top 20 is made up of the grea<sup>t</sup> Anglo-Saxon countries (USA, Canada, Australia) and European countries (Switzerland, United Kingdom, Ireland, Estonia, Iceland, Denmark, Finland, Austria, Netherlands, Germany, Slovakia). It should be noted that there is one South American country, Chile (in 6th position) and one country from the Arabian Peninsula, Kuwait (in 19-th position).

**Table 2.** Kolmogorov–Smirnov (KS) test for the adjustment of data from EFW and IEF to a normal distribution. The distances of KS (DKS) and the *p*-values indicate that KS tests are statistically significant. It is therefore allowed to conclude that the EFW and IEF values follow a normal law, with *μ*= 6.49 and 58.79 and variance *σ*<sup>2</sup> = 0.96 and 143.34, respectively.


In constrast, Table 4 shows the EFW index for the 21 least free countries in 2006. It is remarkable that the least free countries are mainly grouped in Africa: 16 out of the 21 last countries.

**Table 3.** 2006 Economic Freedom of the World (EFW) Index values for the 20 freest countries.


**Table 4.** 2006 Economic Freedom of the World (EFW) Index values for the 21 least free countries. Unlike the 20 freest countries on the planet, the 21 least free countries are almost all in Africa (16 of the 21).


#### *3.3. IEF in Year 2006*

Similarly, Tables 5 and 6 list the IEF values for the 20 freest countries and the 20 least free countries, respectively. The former British colonies still dominate the ranking. Hong Kong and Singapore occupy the top 2 places in the ranking. The big Anglo-Saxon (United States, United Kingdom, Australia, and Canada) countries are also in the top 20. Among all the regions of the world, Europe has the largest number of countries in the top 20 (9 of the 20 countries are European).

As in the case of the EFW index, a large majority of the "less free" countries are in Africa (10 out of 20 countries). The (last) Communist Countries (North Korea and Cuba) are appearing in the 2 last places of the ranking.


**Table 5.** 2006 Index of Economic Freedom (IEF) values for the 20 freest countries.

**Table 6.** 2006 Index of Economic Freedom (IEF) values for the 20 least free countries.


#### *3.4. Regional Evolution of Economic Freedom*

In order to study the geographical distribution of economic freedom, it is possible to calculate an "average freedom value" for the six major continents (Africa, Asia, Europe, North America, Oceania, and South America). The distribution of countries by continent is carried out by following the geographical scheme of the United Nations Statistics Division [41]. This partition has been chosen because it has been developed with the aim of conducting statistical studies relevant to the various regions. However, the calculation of such an average selected is not a simple arithmetic mean. It does not make sense to give a similar weight to the United States and e.g., to Ecuador, to China, or to Vietnam. Instead, we consider that the weight should depend on the country's contribution to the world economy, for example through the GDP. Thereafter, the weight is given by

$$w\_i = \frac{GDP\_i}{\sum\_{j=1}^{N} GDP\_j} \tag{1}$$

where *wi* represents the weight of the country *i* and *GDPj*, the internal product country *j*.

The evolutions of the EF for the 6 continents, obtained by this method are reproduced in Figure 3 for the EFW index, and in Figure 4 for the IEF.

**Figure 3.** Yearly evolution of the Economic Freedom of the World (EFW) Index for the six continents (Africa, Asia, Europe, North America, Oceania, and South America). The index calculation for a region results from a weighted averaging of the indices of the countries belonging to the specific region. The weight of a country is the ratio of the GDP of the country to the GDP of the world economy.

**Figure 4.** Yearly evolution of the Index of Economic Freedom (IEF) for the six continents (Africa, Asia, Europe, North America, Oceania, and South America). The index calculation for a region results from a weighted averaging of the indices of the countries belonging to the specific region. The weight of a country is in the ratio of the GDP of the country to the GDP of the world economy.

For the EFW index, Figure 3 shows that Oceania is the the freest of the six regions, with an index value 8 , relatively stable of the 7 years. Europe, North America, and Asia are *ex aequo* with a value 7.5, which represents the world average value. Africa is the less free region and South America does not fare much better.

For the IEF, Figure 4 also shows that Oceania is the freest region with an ever increasing value. It goes from 73.36 in 1996 to 81 in 2007. Europe and North America follow the same evolution and have almost identical values. Asia regresses in terms of "economic freedom", even though there is a slight improvement in the last two years. It goes from 72.2 to 67.9 with a minimum value equal to 66.4 in 2005. Africa is again the least free region of the

world, but progresses over the 12 years period. Overall, the world average freedom is rising from 68 in 1996 to 71 in 2007.

The "rate changes" appear to be different from one index to the other; this is due to the periods of study. Indeed, if the study period is restricted to 2000–2006 for the IEF, the results so obtained for both indices are almost identical. The slight differences are explained by the fact that the IEF is "more conservative" than the EFW; the IEF leads to values lower than EFW for a country. This topic is discussed further in Section 3.6.

#### *3.5. Exponential Versus Power Law Behaviour*

In this section, countries are ranked according to the value of the indices in a conventional order: a low ranking indicates that the country belongs to the group of the freest countries in the world. Conversely, a "high" rank means that the country has an index value, whence a low EF as compared to others.

The goal here is to determine, the so called "rank-size" law, once the countries are ranked, in particular whether the indices follow an exponential or a power law (These are the two most simple analytical functions carried over from statistical physics to econophysics; whence their mathematical origin is well known and not further discussed.), i.e.,

$$INDEX \sim \mathfrak{e}^{\lambda r} \tag{2}$$

or

$$INDEX \sim r^{\vee} \tag{3}$$

where *r* is the rank of the country; *λ* and *ν* are characteristic exponents. The latter equation corresponds to the (so called Zipf) rank-size law [42], if *ν* = −1.

Figure 5a,c,e shows that the EFW has an exponential behaviour for countries with a rank higher than 20. The value of the exponent decreases a little bit more each year and ends up to stabilise at −0.0049 in 2005 and 2006 (see Table 7). The low error bars (less than 0.0001) and the high value regression coefficient (the regression coefficient is greater than 93%) confirm that the data perfectly follow the exponential law.

Figure 5b,d,f shows the power-law behaviour of the EFW. Table 8 reports the values of the exponent of the power law for the 6 studied years. It does not vary much between 2001 and 2004; it falls to −0.0743 in 2005 and −0.007 in 2006. Here again, the effectiveness of the regressions is high, between ∼89% and 93%. This indicates that the data follows a power law.

For the IEF, the semi-log graphs, see Figure 6a,c,e, indicates an exponential behavior according to the rank of countries. The exponent decreases every year, going down from −0.006 in 1996 to −0.0036 in 2007 (see Table 9). The regression coefficient shows that the exponential law has been "perfectly" followed since 2003, a year for which the efficiency of the regression exceeds 90%.

Unlike the EFW index, for which the data follow a power law for all ranks, Figure 6b,d,f shows a transition point between 2 different power law for the IEF, near rank 10. The exponent of the law for countries with a rank below 10 "increases" over the years, from −0.0931 in 1996 to −0.0518 in 2007. The exponent for countries with rank higher than 10 remains relatively stable −0.016 over the 12 years here studied (see Table 10).

It should be noted that countries with low EF (those which have a very high rank) follow neither a power law nor an exponential law; this feature holds for both indices. The difficulty of performing economic measures for these countries can explain that the index values are fraught with errors that are not possible to compensate. These countries are often those with a minimally developed economy, weakly connected to their outside world, apparently subject to the will of a dictator.

**Figure 5.** Examples of semi-log [(**<sup>a</sup>**,**c**,**<sup>e</sup>**)] and log–log [(**b**,**d**,**f**)] plots of the rank-size relation between the Economic Freedom of the World (EFW) index and the country rank for the years 2000, 2003, and 2006, respectively: the semi-log plots show that the relationship is exponential for countries of high rank (≥20); the log–log plots point to a behaviour close to a power law.

**Figure 6.** Examples of semi-log [(**<sup>a</sup>**,**c**,**<sup>e</sup>**)] and log–log [(**b**,**d**,**f**)] plots of the rank-size relation between the Index of Economic Freedom (IEF) and the country rank for the years 1997, 2002 and, 2007, respectively; the semi-log plots show that the IEF ranking has an exponential behaviour; the log–log plots point to the existence of a transitional point between two different power laws, i.e., near rank 10.


**Table 7.** Yearly evolution of the *λ* exponent in the assumed empirical exponential law between the EFW index and the rank (*r*), the standard error (Δ*λ*), its relative value (Δ*λ*/*λ*), and the efficiency (*R*2) of the regression. The low error bar values (less than 0.0001) and the effectiveness of the regressions confirmthatthedatathelaw.

#### *3.6. Comparison of Both Indices*

The purpose of this section is to compare the indices, whence it is necessary to restrict the observation "period" at the largest but common year interval. We should also take into account the countries common to both sets. That leaves 138 countries to be examined over a period extending from 2000 to 2006, i.e., 2 sets of 862 data points.

To have a meaningful comparison, it is best to "normalise" the index values in an observation interval; here we choose the interval to be [0, 1]. To do so, it is sufficient to divide the values of the EFW index by 10 and those of the IEF by 100.

The distributions of the 862 data points are reproduced in Figure 7 for both indices. The average of the EFW values is 0.6542, while the average for the IEF is slightly lower at 0.6118. This shows that the EFW gives, on average, an index value slightly greater than that given by the IEF for the same country (see below in Table 11).

**Figure 7.** Histogram of (**a**) Economic Freedom of the World (EFW) and (**b**) Index of Economic Freedom (IEF) values for the 862 data points, common to both indices, normalised over [0, 1].

In order to confirm that the IEF is more conservative than the EFW index, it is interesting to represent the EFW values according to the IEF values. This is done in Figure 8. By calculating the linear regression coefficient, the slope is found to be 0.7294. This value is markedly less than 1, whence confirming that the EFW gives index values greater than the IEF for a given country.

**Figure 8.** Scatter plot of the relationship between the Economic Freedom of the World (EFW) index and the Index of Economic Freedom (IEF) normalised values. The regression slope points to a linear relationship of ∼0.7294. This value, statistically significant, lower than 1, confirms that the IEF is "more conservative" than the EFW index. The worst EFW country (Zimbabwe) position is emphasised for framing of the data.

**Table 8.** Yearly evolution of the *ν* exponent in the empirical power law between the EFW and the rank (*r*), the standard error (Δ*ν*), its relative value (Δ*ν*/*ν*), and the efficiency (*R*2) of the regression. The low error bar values (Δ*ν*/*ν* 3%) and the effectiveness of the regressions confirm that the data is well following a power law.


#### **4. Relationship between Economic Freedom and Wealth of Countries**

As recalled here above, many studies show a strong relationship between economic freedom and the wealth of a country, i.e., between EF and the country gross domestic product (GDP). In this section, the goal is to evidence this relationship.

A graphic representation of EF according to the GDP, on Figures 9 and 10, shows that the relationship translates into a power law, i.e., thereby defining the exponent *γ*,

$$INDEX \simeq GDP^{\gamma} \,. \tag{4}$$

A positive exponent ( *γ* > 0) indicates a "positive relationship" between EF and the GDP. This would mean that the freest countries are the richest ones. A negative exponent indicates a negative correlation: the freest countries would be the less rich ones.

The existence of this law is very important from an economic point of view. Indeed, it allows us to know the wealth which a country should have as a function of its level of economic freedom. By estimating the influence that a governmen<sup>t</sup> decision will have on the economic freedom index of that country, it is possible to directly measure the impact of a governmen<sup>t</sup> policy on the economy of the country. Moreover, the existence of this (simple) law will enable countries to be classified according to their position on the power law. Countries that are located above the law are countries that have a lower gross domestic product than they should for their level of economic freedom. These countries can be said to be 'underperforming'.

On the other hand, the countries that are located below the law are countries that have a gross domestic product greater than that which it should have. These countries are 'over-performing'.

On Table 9, we report the exponential law parameter ( *λ*) between the IEF and the rank (*r*) of the IEF, the Standard Error ( Δ *λ*) and its Relative Error ( Δ *λ*/*λ*), together with the efficiency of the regression ( *R*2). The *λ* value decreases each year (in absolute value); it increases from −0.006 in 1996 to −0.0036 in 2007. The efficiency of the regression shows that the data follow an exponential law, rather perfectly since 2003, when the efficiency of the regression exceeds 90%.


**Table 9.** Yearly evolution of the *λ* exponent in the empirical exponential law between the IEF and the rank (*r*), the standard error (Δ*λ*), its relative error (Δ*λ*/*λ*), and the efficiency ( *R*2) of the regression. The low error bar values (Δ*λ*/*λ* 2 to 4%) and the effectiveness of the regressions confirm that the data are closely following a power law.

On Table 10, we report the (Zipf) rank-size law exponent (*ν*) between the IEF and the rank (*r*) of IEF, the Standard Error ( Δ*ν*), the Relative Standard Error ( Δ*ν*/*ν*), and the yearly regression coefficients ( *R*2), for the observed different regimes. While the exponent for countries of rank below 10 decreases over the years the exponent for countries of rank higher than 10 remains relatively stable, near the value −0.016 over the 12 years of the study.

**Table 10.** Yearly evolution of the Zipf law exponent (*ν*) between the IEF and the rank (*r*) of IEF, the Standard Error (Δ*ν*), the Relative Standard Error (<sup>Δ</sup>*ν*/*ν*), and the Regression Coefficient (*R*2). While the exponent for countries of rank below 10 decreases over the years, the exponent for countries of rank higher than 10 remains relatively stable, near the value −0.016 over the 12 years of the study.


On Table 11, we report the main characteristics (average and standard deviation) of the normalised EFW and IEF data for the 138 countries out of the 7 years (i.e., 862 data points). The EFW mean is slightly higher than that for the EFW data. The coefficient of variation (*σ*/*μ*) shows a weak dispersion in both cases.

**Table 11.** Summary of (rounded) main statistical characteristics for the so called "normalized" EFW and IEF distributions of the economic freedom indicators, according to the number of countries *Nc*, the examined time interval Δ*T*, whence the number *N* of data points.


On Table 12, we list countries (The ISO 3166-1 code is used to facilitate the presentation of data) for which the EFW Index does not comply with the power law, i.e., the data points are located outside the area limited by twice the standard deviation from the power law.

**Table 12.** List of countries for which the EFW Index does not comply with the power law, i.e., are located outside the area limited by twice the standard deviation from the power law.


Figures 9 and 10 clearly show that all countries, with a few exceptions obey the power law. The variation coefficient (*σ*/*μ*) shows a weak dispersion on both cases, because the

countries are almost all in an interval corresponding to twice the standard deviation. For the EFW, the countries that pose a problem are Algeria, the Republic of Congo, Burma, and Zimbabwe, but also Venezuela since 2002. As regards the IEF, the problematic countries are more numerous: among these are Angola, Bosnia, Iran, Laos, Libya, and Zimbabwe. Venezuela is only an IEF problem since 2004. The lists of such countries are included in Tables 12 and 13 for each year of interest. In Table 12, we report the list of countries *i* for which the EFW Index values do not comply with the power law. In Table 13, we report the list of countries *i* for which the IEF Index does not comply with the power law.

**Figure 9.** Examples of log–log plot of the Economic Freedom of the World (EFW) Index with respect to country's gross domestic product (GDP) for the years (**a**) 2000, (**b**) 2003, and (**c**) 2006. This relationship is characterised by a power law, with an exponent *γ* 0.674. The dotted lines encompass the region for which the data is within twice the standard deviation away from the trend.

The exponent *γ* values for the period 2000 to 2006 relationship between EFW and GDP are reported in Table 14, while the *γ* values for the IEF for the 1996 to 2007 period are shown in Table 15. In the case of the EFW (see Table 14), the exponent of the law in question remains stable on the 7 years with an average value 0.0674. Notice that the regressions coefficients for the EFW–GDP relation are not as high as in the case of the exponential and power (rank-size) laws. For the IEF, there are 3 periods on the 12 years during which the exponent holds different behaviours. For the 1996 to 2000 years, the exponent has an average value equal to 0.0948, which remains stable around this value over these 5 years. The second phase, which extends over the years 2001 to 2005, is a transition period during which the value of the exponent falls down. It ends up to some stabilisation around 0.0666 during the third period (2006–2007). The efficiency of the regressions is not very good, except for the third period during which *R*<sup>2</sup> is approaching 50%. Therefore, it may be conjectured that the IEF corrections, added in 2006, are bearing fruit.

**Figure 10.** Examples of log–log plots of the Index of Economic Freedom (IEF) relationship to the country's gross domestic product (GDP) for the years (**a**) 1997, (**b**) 2002, and (**c**) 2007. This relationship is characterised by an evolutive power law. The dotted lines limit the region for which the data are located within a maximum distance equal to twice the standard deviation; the few outliers have been removed for calculating the power law exponent *γ*.

**Table 13.** List of countries fo which the IEF does not comply with the power law, i.e., are located outside the area limited by twice the standard deviation from the power law.



**Table 14.** Yearly evolution of the power law exponent (*γ*) between the EFW and GDP, the standard error (Δ*γ*), the relative error bar (Δ*γ*/*γ*) and the efficiency (*R*2) of the regression. The power law exponent remains rather stable over the 7 years with an average value 0.0674 (±0.004).

**Table 15.** Yearly evolution of the power law exponent (*γ*) between the IEF and GDP, the standard error (Δ*γ*), the relative error (Δ*γ*/*γ*), and the efficiency (*R*2) of the regression. There are 3 periods to be noticed in which the exponent adopts different behaviours. For the years 1996 to 2000, the exponent has an average value 0.0948 and remains stable (0.09) for about 5 years. The second phase spreads over the years 2001 to 2005, is a transitional period during which the value of the exponent falls down. It ends up stabilising around 0.0666 on the third and latest period (2006–2007). Notice that the regression coefficient (*R*2) is not very high.


In Table 15, we report the power law exponent (*γ*) between the IEF and GDP, the standard error (Δ*γ*), the relative error (Δ*γ*/*γ*), and the (*R*2) regression coefficient. There are 3 periods to be noticed in which the exponent adopts different behaviours. For the years 1996 to 2000, the exponent has an average value 0.0948 and remains stable for about 5 years. The second phase, which is spread over the years 2001 to 2005, is a transitional period during which the value of the exponent falls down. It ends up stabilising around 0.0666 on the third and latest period (2006–2007). Notice that the regression coefficient is not very high: *R*<sup>2</sup> ∼ 0.376.
