The World in economic indices that do not depend on a country's size. VIII. The Correlations of the Perfectness Index pf with "rectified" economic indices s and r in 2016-2021 Текст научной статьи по специальности «Экономика и бизнес»

Мир в экономических индексах, не зависящих от размера страны. VIII. Корреляция Индекса Перфектности pf с «выпрямленными» экономическими индексами s и 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 – построение и описание распределения стран на плоскости экономических переменных {Продуктивность r; Индекс Перфектности pf} и {s-индекс; Индекс Перфектности pf} в 2016-2021 гг. для определения параметров уравнения линейной регрессии и оценки их устойчивости во времени, а также для выявления стран, максимально отклоняющихся от прямой линии линейной регрессии. Научная новизна заключается в выявлении некоторых статистических характеристик глобальной экономики, возможно, малоизменяющихся со временем. Полученные данные позволяют, в результате, отождествить страны, максимально отклоняющиеся от уравнения линейной регрессии, со странами, имеющими аномально малые или аномально большие значения Индикатора Равновесия b.

Ключевые слова: мировая экономика; экономические индексы; Индекс Перфектности; «выпрямленные» экономические индексы; линейная регрессия.

Цитировать: Seidametova Z.S., Temnenko V.A. The World in economic indices that do not depend on a country's size. VIII. The Correlations of the Perfectness Index pf with "rectified" economic indices s and r in 2016-2021 // KANT. – 2023. – №3(48). – С. 82-91. EDN: FIKZAD. DOI: 10.24923/2222-243X.2023-48.13

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, Republic of Crimean

The purpose of the study is to build and describe the distribution of countries on the plane of economic variables {Productivity r; Perfectness Index pf} and {s-index; Perfectness Index pf} in 2016-2021 to determine the parameters of the linear regression equation and assess their stability over time, as well as to identify countries with maximum deviation from the straight line of the linear regression. The scientific novelty lies in the identification of some statistical characteristics of the global economy, which may not change much over time. The data obtained make it possible, as a result, to identify countries with maximum deviation from the linear regression equation with countries that have anomalously small or anomalously large values of the Equilibrium Indicator b.

Keywords: world economy; economic indices; Perfectness Index; “rectified” economic indices; linear regression.

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. VIII. The Correlations of the Perfectness Index pf with “rectified” economic indices s and r in 2016-2021

Introduction

The paper [1] proposed two mathematical transformations of a pair of basic economic indices {CPI, EPI} into two new pairs of indices {s, r} and {b, pf}. These transformations make it possible to reveal some fundamental statistical regularities of the world economy, hidden behind the disordered distribution of countries on the base plane of indices {CPI, EPI}.

For the convenience of the reader, we recall here the definition of the economic variables we used. The Corruption Perception Index CPI is a hybrid expert assessment of society using a special methodology on a 100-point scale: the higher the CPI value, the lower the level of corruption in the country. This estimate is updated annually in the database [2] for about 170-180 countries.

The Economic Productivity Index EPI is defined as follows:

EPI=(GDP/PC)/(max⁡{GDP/PC})∙100 (%). (1)

The numerator of formula (1) GDP/PC is the Gross Domestic Product per capita of a given country. The denominator in formula (1) is the maximum value of GDP/PC achieved in some country in the same year. The numerator and denominator must be expressed in current US dollars. In this paper we borrow information about the GDP/PC of the countries of the world from the database of the World Bank [3]. The Index EPI is expressed as a percentage, but we usually omit the percent sign (%) when writing the Index EPI. We only calculate EPI for countries with known CPIs. Among these countries Luxembourg has been the champion in terms of GDP/PC over the past few years.

The Index CPI is a discrete integer. The index EPI is continuous. For our purposes of economic analysis of the global economy, we can neglect the mathematical difference between these two indices and assume that each country in the world with known values CPI and EPI is represented annually by a point on a continuous plane {CPI, EPI} within a square of size 100100.

“Rectified” economic indices {s, r} are defined using the following transformation of variables:

■(s=(CPI/10)^2,@r=10√EPI.) (2)

Transformation (2) is performed independently for each of the base variables CPI and EPI. It is monotonic (does not change the order of countries by CPI or by EPI). It preserves the normalization: the maximum possible values of sand r are equal to 100.

The last pair of economic variables {b, pf} is introduced by transforming the pair {s, r}:

■(pf=100√((r∙s)/(max⁡{r∙s})),@b=√(λ r/s). ) (3)

In formulas (3) max⁡{r∙s} is the maximum value of the product r∙s, achieved in a given year in some country of the world; λ=√(2⁄5) – numerical coefficient. The motivation for such a value λ, as well as the motivation for the economic utility of transformation (3), are given in the paper [1].

We will denote the index r by the term Productivity. The index pf is named in [1] the “Perfectness Index”. This index pf is proportional to the geometric mean of the two “rectified” economic indices s и r. An index pf with equal weight takes into account the capabilities of the economy (described by Productivity r) and the state of society (described by the s-index).

The economic variable b, introduced using transformation (3), is named in [1] the Socio-Economic Equilibrium Indicator, or in short form: the Equilibrium Indicator. For most countries, the Equilibrium Indicator is close to one. There are only a few countries for which b it noticeably exceeds unity (b1.52.5). We believe that the social and economic equilibrium in these countries has been significantly disturbed.

The purpose of the paper is to study the distribution of countries on the “hybrid” planes of economic indices containing the Perfectness Index pf and one of the “rectified” economic indices s or r. This goal structures the following tasks: 1) build the annual distribution of all countries on the plane {r, pf} according to the statistical data of 2016-2021; 2) determine the annual statistical parameters k_r, d_r of a linear regression equation

pf=k_r r+d_r (4)

and the coefficient of determination R_r^2 corresponding to this equation, and describe the temporal dynamics of these statistical parameters in 2016-2021; 3) identify countries with maximum deviation from the straight line of the linear regression (4), and give an economic interpretation of this “maximum deviation” phenomenon; 4) perform tasks similar to tasks 1-3 for the annual distribution of countries on the plane {s, pf} with the determination of statistical parameters k_s and d_s the linear regression equation on this plane:

pf=k_s s+d_s, (5)

as well as the corresponding coefficient of determination R_s^2. Identify countries with maximum deviation from the straight line of linear regression (5); 5) to study the behavior of the statistical characteristics of linear regression equations (4) and (5) for three parts of the axis of perfectness: a) for a set of fifty countries (the “Golden 50”) with the highest value of the Perfectness Index pf (perfectness ranking n(pf)≤50); b) for the set of one hundred countries (the “Top 100”) with the highest Perfectness Index value (perfectness ranking n(pf)≤100); c) for the “tail” of the perfectness distribution (all countries with a perfectness ranking of n(pf)≥101).

Main part.

Correlation of the Perfectness Index pf with productivity r in 2016-2021. All countries. The electronic application (https://t.ly/teD2a) contains six annual lists of countries with economic indices s, r, pf and b for 2016-2021. In each annual list, countries are ordered in descending order by the Perfectness Index pf. Fig. 1 shows for each year the distribution of countries on the plane {r, pf}, built according to the application data (https://t.ly/teD2a). The dotted line in each figure shows a straight line of linear regression (4). The statistical parameters of this equation k_r and d_r, as well as the coefficient of determination R_r^2 are given in table 1.

Table 1. Statistical parameters of the linear regression equation (4) on the plane k_r, d_r for all countries in 2016-2021

year 2021 2020 2019 2018 2017 2016

kr 1.110 1.130 1.115 1.101 1.084 1.115

dr -0.446 -0.443 -0.851 -0.791 -1.051 -1.302

R2r 0.924 0.930 0.924 0.916 0.918 0.915

Table 1 shows a high and stable over time correlation between Perfectness Index pf and Productivity r. The 6-year average value of the coefficient of determination R_r^2=0.9210.005 with a very small arithmetic mean deviation. The slope of the straight line of linear regression is also very stable over time k_r. Six-year mean k_r=1.1090.011 with a small arithmetic mean spread. Of course, it cannot be argued that this high and stable correlation expresses some eternal law of the world economy. But we empirically revealed here the fact of the existence of such a stable correlation over a fairly long time period of six years.

The names of some countries are inscribed on the fig. 1 coded according to the three-letter ISO standard [4]. These are the countries with the largest deviations locally for a fixed value of Perfectness pf from a straight line of linear regression (4) to the left or to the right along the Productivity axis r. Is there anything special about the economies of these countries?

Fig. 1. Distribution of countries on the plane of economic indices {r, pf} in 2016-2021.

All countries. is a straight line of linear regression.

If we compare the list of countries with the largest deviation to the left of the 2021 linear regression line with the list of countries in paper [5] with lower values of the Equilibrium Indicator b (“countries with underutilized economies”) this year, we can see that these two lists almost coincide. The difference between these two lists is that the list of countries with low values b includes the additional countries Chile and Barbados – on fig. 1 for 2021, these two countries are near the point depicting Seychelles (SYC), but closer to the linear regression line – and St. Lucia – the dot depicting this country is located next to the dot depicting St. Vincent and the Grenadines (VCT), but closer to a linear regression line. In addition, the list of countries with a lowered value in 2021 b does not include Denmark (DNK), marked in fig. 1. We can identify these lists and accept that not only in 2021, but also in other years, the countries with the maximum left local deviation from the linear regression line (with a fixed pf), are “countries with an underutilized economy”. In other words, we are inclined to accept that the underutilization of the economy is a real economic phenomenon, and an abnormally small value of the Equilibrium Indicators b and/or a large deviation to the left from the linear regression line on the plane {r, pf} are two manifestations of this phenomenon.

Some countries consistently appear in the list of countries with the largest deviations to the left of the straight line of linear regression (4) for six years. These are Denmark (DNK), New Zealand (NZL), Estonia (EST), Uruguay (URY). These countries have excessively large CPI values in their EPI-groups that are not “supported” by sufficiently large EPI values. This discrepancy between the capabilities of society (low level of corruption, reflected in the excessively large CPI index) and insufficiently high economic productivity, we designated the special term “underutilization” of the economy [6], [7].

All countries with the largest local deviations to the right of the straight line of the linear regression in 2021 are included in the lists of countries with disturbed socio-economic equilibrium given in the paper [5]. These are Luxembourg (LUX), Ireland (IRL), USA, Qatar (QAT), Israel (ISR). They are included in the list of countries with a weak violation of the socio-economic equilibrium. In paper [5] these five countries are described as the “Model II of the socioeconomic equilibrium." In addition to these five “Model II” countries, Kuwait (KWT), Bahrain (BHR) and Panama (PAN) demonstrate locally maximum deviations to the right from the straight line of linear regression (4) in 2021. These three countries are included in another list of countries with a weak violation of the socio-economic equilibrium of the paper [5]: the “Model III of the socio-economic equilibrium”. This list contains 15 countries in 2021, but in fig. 1, only these three countries from this list have locally the largest deviations from the regression line to the right (with a fixed value of the Perfectness Index pf).

On the fig. 1 (2021) five more countries are marked as countries with the locally largest deviation to the right of the regression line with small values of the perfectness index (pf≤20). These are Russian Federation (RUS), Turkmenistan (TKM), Equatorial Guinea (GNQ), Libya (LBY) and Yemen (YEM). These countries belong to the group of countries with significant socio-economic disequilibrium (the “disequilibrium countries” [5]).

We can say that a significant deviation to the right from the regression line (4) indicates a violation in the socio-economic equilibrium in the country. However, this feature does not identify all countries with disequilibrium and does not allow distinguishing countries with weak disequilibrium (Model II and Model III) from countries with significant disequilibrium.

From fig. 1 we can see that some countries have the maximum local deviation to the right from the regression axis (4) during the entire period under consideration. Apparently, it can be argued that for these countries the disequilibrium is a stable phenomenon. The essence of this violation is the discrepancy between the level of productivity and the level of corruption (too low value of the index CPI for the value of the index EPI corresponding to the given country).

Correlation of the Perfectness Index pf with s-index in 2016-2021 All countries. Fig. 2 shows the distribution of countries on the plane {s, pf} in 2016-2021. The distribution is based on the Application data (https://t.ly/teD2a). The dotted line in each figure shows a straight line of linear regression (5). Table 2 shows the parameters of the linear regression equation k_sand d_s, as well as the coefficient of determination R_s^2.

Table 2. Statistical parameters of the linear regression equation (5) on the plane {s, pf} for all countries in 2016-2021

year 2021 2020 2019 2018 2017 2016

ks 1.198 1.217 1.242 1.209 1.207 1.193

ds 3.041 3.042 3.254 3.332 3.388 3.715

R2s 0.936 0.940 0.947 0.939 0.944 0.941

Table 2 shows a high and stable over time correlation between the Perfectness Index pf and s-index. The six-year average value of the coefficient of determination R_s^2=0.9410.003 with a small arithmetic mean deviation. The slope of the direct linear regression k_s=1.211 with an arithmetic mean deviation 0.012 is also very stable.

In fig. 2, as in fig. 1, the abbreviated three-letter names of those countries are indicated that have (on some small interval pf) the maximum deviations to the left and to the right from the straight line of linear regression (5). The lists of left and right “deviators” in fig. 1 and in fig. 2 almost mirror each other: the left deviation in fig. 1 turns into the right deviation in fig. 2 and, conversely, the right one turns into the left. The qualifying term " almost " is generated by the fact that these lists do not exactly match.

Fig. 2. Distribution of countries on the plane of economic indices {s, pf} in 2016-2021 All countries. is a straight line of linear regression.

On the plane {s, pf} it is impossible to consider essentially non-equilibrium countries in the “tail” of the distribution for small s and pf. Not all countries with underutilized economies can be considered as “evasive” from linear regression. For example, Denmark (DNK), which was noticeable as an underutilized economy on the {r, pf} plane, does not show itself on the {s, pf} plane in 2016-2020. In 2016 and 2017, the plane {s, pf} does not show a large deviation from the straight line of the linear regression for Estonia (EST), Uruguay (URY), Chile (CHL), Botswana (BWA). On the {r, pf} plane, these countries visibly report underutilization of their economies.

Linear regression (4) on the plane {r, pf} for countries belonging to separate sections of the perfectness axis. We have studied the “internal” stability of linear regression (4) by constructing the distribution on the plane of economic indices {r, pf} not for all countries, but for countries belonging to individual sections of the Perfectness axis. We looked at three such sites: the “Golden 50” (fifty countries with a perfectness ranking n(pf)50); the “Top 100” (one hundred countries with a perfectness ranking n(pf)100); “tail” of the distribution by perfectness (perfectness ranking n(pf)≥101). For the first two sections (the Golden 50 and the Top 100) in the table 3 and table 4 shows the statistical parameters k_rand d_r linear regression equations (4), as well as the coefficient of determination R_r^2. These statistics for the Golden 50 and the Top 100 are very stable over time and are close to the corresponding statistical characteristics of all countries (see table 1).

Table 3. Statistical parameters of the linear regression equation on the plane {r, pf} for the Golden 50 in 2016-2021 (n(pf)50)

year 2021 2020 2019 2018 2017 2016

kr 1.022 1.036 1.050 1.021 1.022 1.018

dr 6.318 6.797 5.276 6.047 7.332 7.332

R2r 0.808 0.839 0.838 0.804 0.837 0.837

Table 4. Statistical parameters of the linear regression equation on the plane {r, pf} for the Top 100 in 2016-2021 (n(pf)100)

year 2021 2020 2019 2018 2017 2016

kr 1.090 1.107 1.125 1.085 1.086 1.110

dr 1.089 1.189 -0.928 0.756 -0.672 -0.531

R2r 0.894 0.906 0.895 0.889 0.887 0.878

The six-year average value of the coefficient of determination, equal to 0.8290.005 for the “Golden 50” and equal to 0.8910.006 for the “Top 100” is somewhat smaller than the value R_r^2 for all countries (0.921 0.005).

The statistical characteristics of the distribution of countries on the plane of economic indices {r, pf} change dramatically when moving to the “tail” of the distribution of countries by perfectness (n(pf)≥101). This distribution is shown in fig. 3. Statistical parameters of the linear regression equation (4) for this distribution are given in table 5.

Fig. 3. Distribution on the plane of economic indices {r, pf} of countries belonging to the “tail” of the distribution according to the Perfectness Index (n(pf)≥101).

is a straight line of linear regression.

Table 5. Statistical parameters of the linear regression equation (4) on the plane {r, pf} for the “tail” of the distribution of countries according to the Perfectness ranking (n(pf)≥101)

year 2021 2020 2019 2018 2017 2016

kr 0.463 0.484 0.396 0.372 0.370 0.378

dr 6.503 6.572 7.767 7.488 7.455 6.913

R2r 0.314 0.352 0.293 0.329 0.288 0.287

The general view of the distribution shown in fig. 3 and the statistical parameters of the linear regression equation (table 5) show that the “tail” of the perfectness distribution is very poorly approximated by linear regression (4). The six-year average value of the coefficient of determination in the “tail” of the distribution by Perfectness is significantly lower than for all countries, or for the “Golden 50” or the “Top 100”: R_r^2=0.310 0.021. The economic patterns operating in the “tail” of the distribution are different from those patterns that are subject to the countries included in the “Top 100”.

Linear regression (5) on the plane {s, pf} for countries belonging to separate sections of the perfectness axis. In tables 6, 7, 8 and in fig. 4 we present the results of the same study carried out on the plane of economic indices {s, pf}. At the same time, we investigate the “internal” stability of the correlation of the Perfectness Index with the s-index in certain sections of the perfectness axis. As above, we consider separately for each year the three segments of this axis: the “Golden 50” (countries with a perfectness ranking n(pf)50); the “Top 100” (countries with a perfectness ranking of n(pf)100); “tail” of distribution by perfectness (n(pf)≥101).

The first two groups of countries (the “Golden 50” and the “Top 100”) have approximately the same statistical characteristics as all countries together. Table 6 presents the statistical parameters k_sand d_s and the coefficient of determination R_s^2 of the linear regression equation (5) on the plane {s, pf} for the “Golden 50” in 2016-2021.

Table 6. Statistical parameters of the linear regression equation (5) on the plane {s, pf} for the “Golden 50” in 2016-2021

year 2021 2020 2019 2018 2017 2016

ks 1.049 1.103 1.105 1.076 1.105 1.099

ds 12.615 10.575 11.516 11.999 10.089 9.771

R2s 0.858 0.868 0.879 0.873 0.884 0.871

The value of the coefficient of determination R_s^2 in table 6 indicates a very close statistical relationship between Perfectness and s-index. This relationship is stable over time. The changes R_s^2 are small. The average value for six years of the coefficient of determination R_s^2= 0.872 0.006. It is lower than for all countries (R_s^2=0.941 0.003). The six-year average slope of the linear regression for the “Golden 50” k_s=1.0900.020. This slope is less than for all countries and its temporal variations are greater than for all countries (k_s=1.211 0.012).

Fig. 4. Distribution on the plane of economic indices {s, pf} of countries belonging to the “tail” of the distribution according to the Perfectness Index (n(pf)≥101).

is a straight line of linear regression.

Table 7 presents the same statistical parameters of the linear regression equation (5) for the “Top 100”.

The high value of the coefficient of determination in table 7, close to one, shows that for this group, one hundred countries with n(pf)100 the statistical relationship of the Perfectness Index with the s-index is very strong and stable over time. The six-year average value of the coefficient of determination R_s^2=0.916 0.006. This value is higher than for the “Golden 50” and lower than for all countries.

Table 7. Statistical parameters of the linear regression equation (5) on the plane {s, pf} for the “Top 100” in 2016-2021

year 2021 2020 2019 2018 2017 2016

ks 1.148 1.174 1.184 1.161 1.150 1.133

ds 5.839 5.435 6.356 5.852 6.405 6.870

R2s 0.908 0.913 0.933 0.910 0.919 0.915

The slope of the straight line of linear regression (5) is also quite stable over time. The six-year average of the slope for the “Top 100” is k_s=1.158 0.015. This value is higher than for the “Golden 50” and lower than for all countries.

The strong and stable statistical relationship between the Perfectness Index and s-index breaks down in the “tail” of the distribution of countries by perfectness (n(pf)≥101). Figure 4 shows the distribution on the plane {s, pf} of the countries belonging to this “tail” of the distribution. Table 8 shows the statistical parameters of the linear regression equation (5) for this “tail” of distributions.

Table 8. Statistical parameters of the linear regression equation (5) on the plane {s, pf} for the “tail” of the distribution of countries by the perfectness index in 2016-2021

year 2021 2020 2019 2018 2017 2016

ks 0.661 0.644 0.621 0.707 0.717 0.712

ds 6.347 6.819 7.100 6.320 6.021 5.924

R2s 0.596 0.578 0.554 0.538 0.590 0.589

Fig.4 and table 8 show that the linear regression equation (5) does not adequately describe the statistical relationship s-index with perfectness index. Such a relationship is most likely non-linear and weighed down by a large scatter.

The six-year average value of the coefficient of determination for the “tail” of the distribution by perfectness R_s^2=0.574 0.018. This value is much less than unity, but more than the value of the similar coefficient of determination on the plane {r, pf}: R_r^2=0.310 0.021. Consequently, in the “tail” of the distribution, the perfectness index correlates badly with both productivity r and s-index, but the correlation with s-index is still somewhat better expressed.

Conclusions

According to the statistical data describing the state of the world economy in 2016-2021, the existence of a high and stable over time correlation between the Perfectness Index pf and Productivity r, as well as between the Perfectness Index pf and s-index, which is calculated by the Corruption Perception Index CPI, has been established. The coefficient of determination for the linear regression equations in the {r, pf} and {s, pf} planes R^2 exceeds 0.9 and changes very little over time. It has been established that countries with a local maximum deviation to the left or to the right from the straight line of linear regression (on a certain small interval of the index pf) are either countries with an “underutilized” economy or countries with a disturbed socio-economic equilibrium.

It has been established that the revealed correlations between Productivity r and the Perfectness Index pf, as well as between s-index and pf, are well expressed for the upper part of the Perfectness ranking (n(pf)100) and do not provide an adequate description of the state of that part of the economy that corresponds to the “tail” of the distribution by perfectness (perfectness ranking n(pf)≥101).

References:

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