The World in economic indices that do not depend on a country's size. VII. Statistical characteristic of the world economy on the plane of "rectified" economic indices {s, r} in 2016-2021 Текст научной статьи по специальности «Экономика и бизнес»

Мир в экономических индексах, не зависящих от размера страны. VII. Статистические характеристики мировой экономики на плоскости «выпрямленных» экономических индексов {s, r} в 2016-2021 гг.

Сейдаметова Зарема Сейдалиевна, доктор педагогических наук, профессор

Темненко Валерий Анатольевич, кандидат физико-математических наук, доцент

Крымский инженерно-педагогический университет имени Февзи Якубова, Симферополь, Республика Крым

Цель исследования – изучение временных вариаций в 2016-2021 гг. распределения стран на плоскости «выпрямленных» экономических индексов {s,r}. Индекс r («Продуктивность») связан с величиной ВВП на душу населения. Индекс s связан с известным Индексом Восприятия Коррупции CPI. Выявлено, что параметры уравнения линейный регрессии между «выпрямленными» индексами s и r (включая коэффициент детерминации R^2) мало меняются от года к году. Выявлена также временная устойчивость параметров линейной регрессии между s и r на отдельных участках оси продуктивности r – например, для пятидесяти самых продуктивных стран. Выявлено, что примерно для семидесяти наименее продуктивных стран корреляция между «выпрямленными» индексами s и r незначительна (мал. коэффициенты детерминации R^2). Научная новизна заключается в выявлении этих статистических характеристик распределения стран на плоскости «выпрямленных» экономических индексов и в попытке оценить их временную динамику. Представленные данные позволяют в результате выявить меру некоторой глобальной устойчивости мировой экономики на рассматриваемом временном промежутке; они позволяют также предположить возможность значительных погрешностей статистических данных для стран с высоким уровнем коррупции (малые значения индекса s) и невысокой продуктивностью (малые значения индекса r).

Ключевые слова: мировая экономика; экономические индексы; линейная регрессия; коэффициент детерминации; временная динамика.

Цитировать: Seidametova Z.S., Temnenko V.A. The World in economic indices that do not depend on a country's size. VII. Statistical characteristic of the world economy on the plane of "rectified" economic indices {s, r} in 2016-2021 // KANT. – 2023. – №3(48). – С. 71-82. EDN: FIANDS. DOI: 10.24923/2222-243X.2023-48.12

Seidametova Zarema Seidalievna, DSc of Pedagogical sciences, Professor

Temnenko Valerii Anatolievich, Ph.D. of Physics and Mathematical sciences, Associate Professor

Fevzi Yakubov Crimean Engineering-Pedagogical University, Simferopol, RC

The purpose of the research is to study time variations in 2016-2021 of the distribution of countries on the plane of “rectified” economic indices {s,r}. The index r (“Productivity”) is related to the value of GDP per capita. The index s is related to the well-known Corruption Perceptions Index CPI. It was revealed that the parameters of the linear regression equation between the “rectified” indices s and r (including the coefficient of determination R^2) change little from year to year. The temporal stability of the linear regression parameters between s and r was also found in certain parts of the productivity axis r, for example, for the fifty most productive countries. It was found that for about seventy of the least productive countries, the correlation between the “rectified” indices s and r is insignificant (the coefficients of determination R^2 are small). The scientific novelty lies in identifying these statistical characteristics of the distribution of countries on the plane of “rectified” economic indices and in an attempt to estimate their time dynamics, as a result, the presented data make it possible to identify a measure of some global stability of the world economy in the considered time period, they also allow us to suggest the possibility of significant errors in statistical data for countries with a high level of corruption (small values of the index s) and low productivity (small values of the index r).

Keywords: world economy; economic indices; linear regression; coefficient of determination; temporal dynamics.

УДК 339.97:330.43

ВАК РФ 5.2.5

Seidametova Z.S., Temnenko V.A.

The World in economic indices that do not depend on a country’s size. VII. Statistical characteristic of the world economy on the plane of “rectified” economic indices {s, r} in 2016-2021

Introduction

In the paper [1], a new pair of so-called “rectified” economic indices s and r was introduced:

s=(CPI⁄10)^2, (1)

r=10√EPI. (2)

In formula (1) CPI is the Corruption Perceptions Index. We borrow this index from the database [2]. CPI is a hybrid expert assessment of the level of corruption in a given country on a 100-point scale: the higher the CPI, the lower the level of corruption.

In formula (2) EPI is the Economic Productivity Index:

EPI=(GDP/PC)/(max⁡{GDP/PC})∙100 (%), (3)

where GDP/PC is the Gross domestic product per capita in a given country in some year, measured in current US dollars; max{GDP/PC} is the maximum value of GDP/PC achieved in some country in the world in the same year. The index EPI is measured as a percentage, but we will usually omit the percent symbol (%) when writing EPI. We borrow GDP/PC values from the World Bank database [3]. The champion country with the highest GDP/PC value is selected among the countries with known CPI value. During the last years Luxembourg is such a champion country. Monaco and Liechtenstein usually outperform Luxembourg in terms of GDP/PC, but no CPI is published for them in [2].

The “rectifying” transformation (1), (2) is performed in the base plane of economic indices {CPI, EPI}, but it is performed separately for each of the two economic indices. This transformation does not mix the two base indices, it is monotonic: it does not change the order of countries in either the EPI (Productivity Ranking) or the CPI. These “rectified” indices are normalized, as are the underlying indices CPI and EPI. The maximum value of the index r is 100. The maximum value of the index s is also 100. It is convenient to call the index r as “Productivity”. We have not entered any special name for the index s. Let’s call it “s-index”.

The number of the country in the annual list of countries compiled in descending order of productivity r will be called the “productivity ranking” and will be denoted as n(r).

It is obvious that the “rectifying” transformation (1), (2) allows the inverse transformation to the base indices CPI and EPI:

CPI=10√s, (4)

EPI=(r⁄10)^2. (5)

The usefulness of the “rectifying” transformation (1), (2) was motivated in the paper [1] by the presence of the statistical dependence between the indices CPI and EPI that we identified earlier in [4], [5]. This dependency can be conditionally written like this:

EPI~CPI^4. (6)

This dependence was revealed in [4] and [5], not at the level of an individual country, but at the level of productivity groups (EPI-groups) see [4].

The presence of dependence (6) means that with some random spread the statistical relation must be satisfied:

√EPI~CPI^2. (7)

Relation (7) means, taking into account the rectifying transformation (1), (2), that between the “rectified” indices s and r we can expect a linear correlation. This assumption was tested and confirmed in the paper [1] on the statistical data of 2021.

The purpose of this paper is to study temporal variations in the distribution of countries on the plane of rectified economic indices {s,r} in 2016-2021 and determine the degree of stability of the linear regression equation coefficients k and d

r=ks+d, (8)

as well as the degree of stability of the coefficient of determination R^2 both for the entire set of countries with known s and r, and for individual subsets of countries in a certain interval of values of the productivity ranking n(r). This goal structures the following tasks: 1) build the distribution of countries on the plane {s,r} for each year from the time period under consideration and determine the parameters of linear regression (8) and the coefficient of determination R^2; 2) construct a distribution on the {s,r} plane for the fifty most productive countries (n(r)50) for each year, determine the linear regression parameters and study their temporal stability (these “top” 50 countries in terms of productivity approximately correspond to the EPI-groups Hot and Warm; for EPI-groups, see, for example, [6], [7] on the material of 2019 and [8]-[12] on the material of 2020); 3) construct a distribution on the plane {s,r} for the “Top Hundred” countries with productivity ranking (n(r)100), determine the parameters of linear regression (8), study their temporal stability (the “Top Hundred” of countries approximately corresponds to the combination of EPI-groups Hot, Warm and UpperCold); 4) plot the distribution on the plane {s,r} for the “tail” of the distribution of countries by productivity (n(r)≥101), determine the parameters of linear regression and give them the interpretation. This “tail” of the distribution contains approximately seventy countries. It approximately corresponds to the union of the EPI-groups MiddleCold, LowCold and the “Least Developed Countries” (LDC).

Main part.

Preliminary remarks. The electronic appendix to this paper (https://t.ly/kPL8L) presents data on rectified economic indices s and r for 2016-2021. In each annual data summary, countries are ordered in descending order of the Productivity r. The number of countries on the lists varies from year to year. The list of each year contains all countries for which the indices s and r are known (therefore, these are countries with known GDP/PC and CPI). Each annual table contains the name of the country, the country code according to the three-letter ISO standard [13], the country’s ranking n(r), the “base” indices CPI and EPI, and the “rectified” indices s and r. All distributions of countries given below on the {s,r} plane are based on the data presented in these tables.

All countries on the plane {s,r}. The fig. 1 shows the annual distribution of all countries on the plane of “rectified” economic indices {s,r} for the time period from 2016 to 2021. This figure shows for each year some conditional “outer perimeter” line for the most productive EPI-groups: the EPI-group Hot and the EPI-group Warm. The EPI-group Hot in this time period has a fixed composition, it contains 24 countries, the list of which does not change. The highest productivity value in this EPI-group belongs to the productivity champion country Luxembourg (r=100). The lowest productivity value in this EPI-group has Japan (2021, r=54; 2019, r=59; 2018, r=61) or United Arab Emirates (2020, r=56; 2016, r=59).

Fig. 1. Distribution of countries on the plane of “rectified” economic indices {r, s}. All countries.

The highest value s-index in the EPI-group Hot has Denmark (2018, s=77 (CPI=88)) or New Zealand (2017, s=79 (CPI=89)), or both (2021 and 2020, s=77 (CPI=88); 2019, s=76 (CPI=87); 2016, s=81 (CPI=90)).

Country with the lowest s-index value in the EPI-group Hot is Qatar (2016, s=37 (CPI=61)) or Israel (2021, s=35 (CPI=59); 2020 and 2019, s=36 (CPI=60); 2018, s=37 (CPI=61); 2017, s=38 (CPI=62)).

The EPI-group Hot is visually quite clearly separated on fig. 1 from the EPI-group Warm. These two EPI-groups partially overlap each other in terms of the range of change s. The highest Productivity value r in this EPI-group Warm in 2016-2021 had Italy: r=52 (2021 and 2020), r=54 (2019, 2018 and 2016), r=55 (2017). Trinidad and Tobago (r=35, 2021) had the lowest value of Productivity of the EPI-group Warm or the lowest Productivity had Croatia (r=35, 2020 and 2016; r=36, 2019, 2018 and 2017).

Uruguay (s=50, 2016) or Estonia (s=50, 2017; s=53, 2018; s=55, 2021 and 2019; s=56, 2020) had the highest s-index value in EPI-group Warm.

Panama (s=14, 2018 and 2019) or Trinidad and Tobago (s=17, 2021; s=16, 2020 and 2019; s=12, 2016) had the lowest s-index EPI-group Warm.

Based on the data presented, it can be concluded that for EPI-groups Hot and Warm the largest and the smallest values of Productivity, as well as the largest and the smallest values of the s-index can be considered almost constants (diffuse constants) in the considered period of time.

Table 1 presents the parameters of the linear regression equation (8) k and d, as well as the coefficient of determination R^2 for the entire set of countries in different years. The straight line of this linear regression is shown as a dotted line in each figure.

Table 1. Linear regression parameters for all countries.

year 2021 2020 2019 2018 2017 2016

k 0.927 0.934 0.963 0.933 0.954 0.909

d 6.61 6.41 7.19 7.57 7.81 8.22

R2 0.747 0.761 0.767 0.741 0.753 0.742

It is obvious that the parameters of linear regression (8) for all countries of the world change insignificantly in these years. We can assume that they are diffuse constants (k0.92, d7.6, R^20.75).

Note that the difference from zero of the statistical parameter d, which is not consistent with the naive formula (7), should not undermine confidence in the use of “rectified” economic indices s and r: only with the help of these “rectified” indices we can write down the linear regression equation (8), which had would be quite ponderous when written in the “base” indexes CPI and EPI.

The distribution of the fifty most productive countries (“Golden 50”) on the {s,r} plane. The fig. 2 shows the distribution on the plane of “rectified” economic indices {s, r} of countries from the “Golden 50”, i.e. the top of the productivity ranking (n(r)50). This “top” of the ranking almost coincides with the unification of the EPI-groups Hot and Warm. The number of countries in these two EPI-groups together (N_HW) varies slightly over the years, but not significantly: N_HW=51 (2021); 49 (2020); 53 (2019, 2018 and 2016); 54 (2017).

Fig. 2. Distribution on the plane of “rectified” economic indices {r, s} of the countries belonging to the “Golden 50”. The “top” part of the productivity ranking (n(r)50).

Table 2 shows the parameters of the linear regression equation (8) k and d, as well as the coefficient of determination R^2 for “Golden 50”.

Table 2. Linear regression parameters (8) for countries belonging to the “Golden 50” (n(r)50).

year 2021 2020 2019 2018 2017 2016

k 0.622 0.631 0.610 0.599 0.614 0.569

d 26.57 26.21 28.98 29.01 29.56 30.71

R2 0.497 0.533 0.531 0.486 0.538 0.551

Table 2 shows that these statistical characteristics of the “Golden 50” are quite stable over the considered time intervals, but obviously differ from similar statistical characteristics of the entire set of countries presented in table 1. The most significant seems decrease in the determination coefficient from a characteristic value of R^20.75 (table 1) to a characteristic value of R^2 0.5 (table 2).

This means that for the “Golden 50” countries, the individual spread of indices s and r is just as important as the general statistical trend described by the regression equation (8). For the entire set of countries, the general trend (8) dominates over the random spread of the indices. Note that the characteristic slope of the linear regression k from table 2 (k0.61) is less than the above average annual slope from table 2 (k0.93). This means that for the “Golden 50” countries, the statistical increase in productivity r with the growth of the s-index is slower than for the whole set of countries. The difference between these two values of the slope of the linear regression k means that the linear regression (8) with constant coefficients k and d is not an exact expression of some law of the world economy, but, apparently, reflects some main feature of this law, which we previously called the “law of remuneration for good behavior” [5].

Distribution on the plane of “rectified” economic indices {s,r} of countries with a ranking n(r) from n(r)=51 to n(r)=100 (“Silver 50”). Fig. 3 shows the distribution on the plane {s, r} for fifty countries with a productivity ranking n(r) from n(r)=51 to n(r)=100. We have named this group of countries “Silver 50”. This group of countries is close in composition to the EPI-group UpperCold. Statistical parameters k, d and R^2 of the linear regression equations (8) are shown in table 3. The straight line of linear regression itself is shown as a dotted line in each figure.

According to table 3, the slope k and coefficient of determination R^2 for the “Silver 50” countries differ significantly from the corresponding parameters for the entire set of countries (table 1) and the “Golden 50” countries (table 1). The statistical relationship between the s-index and productivity r for these countries is very weakly expressed (small values of the parameters k and R^2). In addition to weakness, this relationship is also characterized by volatility: the statistical parameters k and R^2 are unstable over time.

Table 3. Linear regression parameters (8) for countries belonging to the “Silver 50” (51n(r)100).

year 2021 2020 2019 2018 2017 2016

k 0.123 0.149 0.188 0.125 0.156 0.207

d 22.64 21.70 22.83 23.88 23.92 22.06

R2 0.069 0.097 0.138 0.059 0.076 0.208

Fig. 3. Distribution on the plane of “rectified” economic indices {r, s} of countries belonging to “Silver 50” (51≤n(r)100).

The group of “Silver 50” countries is heterogeneous along the axis s (i.e., in terms of the level of corruption). It has a small subset of nine countries with a sustained overshoot of s=25 (CPI=50) during the time period under consideration: Costa Rica, St. Lucia, Mauritius, Grenada, St. Vincent and Grenadines, Dominica, Botswana, Georgia, Namibia. The last country on this list is Namibia. This country did not reach the minimum level s=25 only in 2021. These countries are in the right half of each annual chart in fig. 3. These countries, considered in isolation, would provide higher values for the slope parameter k in the linear regression equation (8). But this subgroup of countries is, by itself, too small to provide reliable statistics.

“Top Hundred” ranking on the plane {s,r}. Fig. 4 shows the distribution on the plane of “rectified” economic indices {s, r} for the “Top Hundred” countries (n(r)100). This “Top Hundred” in terms of composition of countries is close to the union of the EPI-groups Hot, Warm and UpperCold. The total number of countries in these three EPI-groups N_HWUC varies slightly from year to year, but not significantly: N_HWUC=90 (2021); 89 (2020); 96 (2019), 95 (2018), 94 (2017), 95 (2016).

Fig. 4. Distribution of countries on the plane of “rectified” economic indices {r, s}. “Top Hundred” countries (n(r)100).

Table 4 presents the parameters of the linear regression equation (8) for the “Top Hundred” countries. Table 4 shows that the parameters of the linear regression for the “Top Hundred” countries changed little over the years in this time period. These parameters can be considered diffuse constants (k0.8, d15.8, R^20.71). The linear regression parameters for the “Top Hundred” countries are between those for all countries, table 1, and the “Golden 50” parameter (n(r)50).

Table 4. Linear regression parameters for the “Top 100” countries (n(r)100).

year 2021 2020 2019 2018 2017 2016

k 0.796 0.822 0.805 0.787 0.802 0.763

d 14.82 13.43 15.99 16.48 16.88 17.05

R2 0.710 0.719 0.719 0.708 0.721 0.704

Obviously, the distribution on the plane of “rectified” economic indices {s, r} for the “Top Hundred” countries, as well as the distribution of all countries on this plane, are well described by linear regression equations (8) with almost constant statistical parameters over this time period 2016-2021 k, d and R^2.

The “tail” of the productivity ranking (n(r)≥101) on the plane {s, r}. Fig. 5 shows the distribution on the plane of “rectified” economic indices {s, r} for the “tail” of the distribution of countries by productivity r(n(r)≥101). The number of countries N_t included in this “tail” of the distribution varies from year to year, but insignificantly: N_t=76 (2021), N_t=70 (2020 and 2018), N_t=67 (2019), N_t=69 (2017), N_t=68 (2016). Table 5 presents the parameters of the linear regression equation (8) for these countries.

Fig. 5. Distribution of countries on the plane of “rectified” economic indices {r, s}. “Tail” of the distribution of countries by productivity (n(r)≥101).

With the exception of some “splash”, the value of the parameters k and R^2 in post-covid 2020, we can assume that the statistical characteristics of the “tail” of the distribution do not change very much over the years. But noteworthy is the sharp discrepancy between the data in table 5 with the data of table 1 and table 4. In the “tail” of the productivity distribution, the correlation of economic indices s and r is weakly expressed: the average value R^20.13. The slope of the linear regression k for the “tail” of the distribution (k0.2) is significantly smaller than for all countries (k0.93) or the “Top Hundred” (k0.8).

Table 5. Linear regression parameters for the “tail” of the productivity distribution (n(r)≥101).

year 2021 2020 2019 2018 2017 2016

k 0.212 0.158 0.230 0.192 0.203 0.190

d 9.19 10.02 9.43 9.69 9.73 9.54

R2 0.170 0.097 0.117 0.129 0.130 0.114

Linear regression equation for a pair of indices {r, CPI} in the “tail” of the productivity distribution. The weak correlation of a pair of indices s and r in the “tail” of the productivity distribution r allows us to assume that at low values of the Economic Productivity Index EPI, not the trend relation (7) is realized, but another relation with a slower statistical growth √EPI and an increase in CPI, for example, the growth equation of the form:

√EPI~CPI. (9)

Fig. 6 shows the distribution on the plane of indices {CPI, r} of countries belonging to the “tail” of the distribution in terms of productivity ranking (n(r)≥101). The figure for each year from 2016 to 2021 shows a straight line of linear regression of the variables {CPI, r}:

r=k_c∙CPI+d_c. (10)

Table 6 shows the statistical parameters of equation (10) k_c, d_c and R_c^2.

Table 6. Linear regression parameters (10) in the variables {CPI, r} for the “tail” of the distribution by productivity (n(r)≥101).

year 2021 2020 2019 2018 2017 2016

kc 0.170 0.129 0.174 0.156 0.162 0.150

dc 6.18 7.68 6.40 6.86 6.84 6.91

R2c 0.204 0.118 0.141 0.157 0.155 0.137

The data in table 6 show that the characteristic value of the determination parameter R_c^2 for the linear regression equation (10) does not differ significantly from the value R^2 from table 5, which describes the linear regression equation (8) for a pair of indices {s, r}. Table 5 and table 6 show that in the “tail” of the distribution by productivity ranking (n(r)≥101) there is probably a statistical correlation between CPI and EPI, but it is so weakly expressed that it is difficult to establish an explicit functional form of this weak dependence: this dependence “sinks” in a sea of random factors.

Fig. 6. Distribution on the plane {CPI, r} of countries belonging to the “tail” of the productivity distribution (n(r)≥101).

Linear regression of the form (10) describes quite well the statistics of the distribution of all countries on the plane of indices {CPI, r}. Table 7 shows the statistical parameters of this linear regression for all countries in 2016-2021.

Table 7. Linear regression parameters (10) in variables {CPI, r} for all countries

year 2021 2020 2019 2018 2017 2016

kc 0.917 0.937 0.975 0.929 0.960 0.916

dc -12.58 -13.64 -13.92 -12.01 -12.66 -11.11

R2c 0.714 0.729 0.739 0.705 0.720 0.714

The data in table 7 show that the statistical parameters of the linear regression equation (10) for all countries are very stable over time.

Comparison of table 7 and table 1 leads to a discouraging conclusion: linear regression equations (8) for a pair of rectified indices {s, r} and (10) for a combined pair of indices {CPI, r} describe the distribution of countries equally well, the determination parameters R^2(0.75) and R_c^2(0.72) practically coincide. See also slope parameters k(0.93) and k_c(0.94). The proximity of these parameters means that everything that can be asserted with confidence for all countries of the world in 2016-2021 can be formulated as follows: there is a strong time-stable statistical relationship between the economic indices CPI and EPI. However, it is not possible to distinguish the statistics of the basic trend (7) (EPI~CPI^4) from the statistics of the basic trend (9) (EPI~CPI^2). This impossibility is generated by the large dispersion of data and, probably, by the presence of at least one other factor that affects the economic productivity of the EPI almost as much as the index CPI, which describes the state of society.

However, the data in table 7 show that the linear regression equation (10) for the pair {CPI, r} is less reliable than the linear regression equation (8) for the pair {s, r}. According to table 7 statistical parameter d_c<0. This means that there is a value CPI_* such that at CPI<CPI_* the linear regression equation (10) becomes unusable: it predicts negative values of productivity r. Equation (10) implies that

CPI_*=(-d_c)/k_c . (11)

Calculating by formula (11) the value CPI_* can be found that in 2016-2021 CPI_*=12-15. Therefore, for countries with CPI less than this critical value, the linear regression equation is not suitable. And, of course, it gives a large prediction error r even at large values CPI. In 2016-2021, only a few countries had CPI15, from one country to four countries in different years. But if we expand the zone of expected unreliability of the forecast according to formula (10) to CPI=25, then about two dozen countries are already in this zone of unreliability, from seventeen countries in 2019 to 27 countries in 2021. Therefore, we can with great confidence refer to the linear regression equation (8) for the pair {s, r} than to the linear regression equation (10) for the pair {CPI, r}.

Are CPI data reliable at low CPI and EPI? The countries belonging to the “tail” of the productivity distribution (n(r)≥101) have not only low productivity values r (r≲20) and, accordingly, small EPI values (EPI≲4%), but also small values s of the s-index and, accordingly, the CPI Corruption Perceptions Index (i.e. these are countries with a high level of corruption). In this “tail” of the productivity distribution, according to the data of the electronic supplement to this paper (https://t.ly/kPL8L) only three countries consistently have CPI>45 values: Bhutan (CPI=65-68), Cabo Verde (CPI=57-59), Rwanda (CPI=53-56). Vanuatu (CPI=45-46) and Sao Tome and Principe (CPI=45-47). In 2021, n(r)≥101 Fiji (CPI=55) and Jordan (CPI=45) were added to this limited set of “low-corruption” countries. All other countries in the tail of the productivity distribution (n(r)≥101) have a CPI<45.

We tend to assume that for countries with low productivity (EPI≲4) and fairly high levels of corruption (CPI<45), the Corruption Perceptions Index (CPI) itself is probably losing relevance and cannot describe the state of society so adequately that one can confidently predict statistically expected productivity r from it. The statistical relationship between economic indices s and r (Eq. (8)), works satisfactorily for the “Top Hundred” productivity distribution (n(r)100). We are sure that in this “tail” of the distribution there is a connection between the state of society and the productivity of the economy, but we are not sure that the index CPI is an adequate assessment of the state of society when CPI<45. Perhaps, with a decrease in CPI, the subjective error of the expert assessment of CPI increases and/or some questions used to determine CPI lose relevance for those who answer these questions (the method for obtaining data for calculating the CPI index is described in [14]). Even the Economic Productivity Index EPI at low EPI is likely to be determined with a significant error (in particular, due to the inevitable growth of the contribution of the shadow economy).

Conclusions

The paper presents the distribution of countries on the plane of “rectified” economic indices {s, r} in 2016-2021. Statistical parameters k, d and R^2 linear regression equations (8) are also given. This equation is the simplest linear version of the statistical relationship between the indices s and r. These statistical parameters are given both for all countries with economic indices known over the years s and r, and for individual parts of the distribution of countries by productivity r: 1) “Golden 50” (n(r)50); 2) “Silver 50” (countries with a productivity ranking from n(r)=51 to n(r)=100 in these years); 3) “Top Hundred” (countries with n(r)100); 4) “Tail” of the productivity ranking (countries with a ranking of n(r)≥101). It has been established that the parameters of the linear regression equation (8) change little over the years in this time period, but differ for different productivity ranking zones. It has been established that the statistical characteristics of the “tail” of the distribution differ sharply from the statistical characteristics of the “Top Hundred” of the distribution. It has been suggested that the Corruption Perceptions Index CPI for countries with CPI<45 and low economic productivity (EPI≲4) does not provide an adequate assessment of the state of society.

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