# Applying the Multilevel Approach in Estimation of Income Population Differences

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- How do regions and municipalities contribute to the variances of population income and wages?
- (2)
- What factors affect the variances of population income and wages?
- (3)
- What is the relationship between population income, wages and transfers?

## 2. Literature Review

#### 2.1. Income Inequality within and across Territories

#### 2.2. Multilevel Approach in Studies on Population Income

## 3. Research Design

#### 3.1. Multilevel Model with Cross-Classified Random Effects

_{mijt}—dependent variable, characterising observation m related to i-municipality (1…2016) nested in j-region (1…75) in t year (2015…2019).

_{mijt}—random effects for each observation m, where r

_{mijt}~ N(0, σ

^{2});

_{0jt}—random effects for each municipality i, where ε

_{0jt}~ N(0, τ

^{2});

_{000j}—random effects for each region j, where φ

_{00jt}~ N(0, e

_{j}

^{2});

_{000t}—random effects for each year t, where π

_{00jt}~ N(0, e

_{t}

^{2}).

^{2}, τ

^{2}, e

_{j}

^{2}, e

_{t}

^{2}in the total sum of effects. The intra-unit correlation coefficient (IUCC) is represented as follows [46]:

_{mijt}—independent variable attributed to municipality i (1…2017) nested in region j (1...75) in t year (2015…2019), %;

_{jt}—independent variable related to j-region (1...75) in t year (2015…2019), %.

#### 3.2. Data Source

#### 3.3. Factors Determining the Variance of Population Income

- -
- Personal income, including wages and incomes of individual entrepreneurs;
- -
- Pension, social transfers and benefits, including unemployment allowance, social benefits and assistance measures, benefits and compensations to military personnel, maternity benefits, child care and other "child" benefits;
- -
- Insurance indemnities; lottery winnings; interest on deposits, and compensation for deposits, scholarships, and money transfers.

#### 3.3.1. Economy of Territory and Its Structure

#### 3.3.2. Human Capital

#### 3.3.3. Income Inequality

#### 3.3.4. Spatial Correlations

_{mijt}—random effects for each observation m, where r

_{mijt}~ N(0, σ

^{2});

_{0jt}—random effects for each municipality i, where ε

_{0jt}~ N(0, τ

^{2});

_{000j}—random effects for each region j, where φ

_{00jt}~ N(0, e

_{j}

^{2});

_{000t}—random effects for each year t, where π

_{00jt}~ N(0, e

_{t}

^{2}).

#### 3.4. Factors Influencing the Wage Variance

#### 3.4.1. Labour Productivity and Unemployment

#### 3.4.2. Spatial Autocorrelation of the Unemployment Rate and Wages

#### 3.4.3. Open Economy

#### 3.4.4. The Structure of the Employed Population

#### 3.4.5. Income Inequality

#### 3.4.6. Working Conditions and the Trade Union Activity

_{mijt}—random effects for each observation m, where r

_{mijt}~ N(0, σ

^{2});

_{0jt}—random effects for each municipality i, where ε

_{0jt}~ N(0, τ

^{2});

_{000j}—random effects for each region j, where φ

_{00jt}~ N(0, e

_{j}

^{2});

_{000t}—random effects for each year t, where π

_{00jt}~ N(0, e

_{t}

^{2}).

## 4. Results

#### 4.1. Variance of Population Income

#### 4.2. Model of Population Income

_{31jt}) had a positive effect on income per capita (model 4i). Cross-interaction estimates showed that factors attributed to the regional level (Patent) determined the relationship between dependent variables (Income per capita) at the municipal level and predictors (Production per capita).

_{3ijt}= 43.59 + 3.625 * Patent − 2.166 * Transfers.

- -
- A high volume of production was associated with high volume of income per capita (γ
_{30jt}= 43.59); - -
- This relationship would be stronger in regions with a higher share of patent activity (γ
_{31jt}= +3.625); - -
- This relationship would be weaker in regions with a higher share of transfers (γ
_{32jt}= −2.166).

#### 4.3. Wage Model

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Variable | Definition | Min | Max | Mean | Standard Deviation |
---|---|---|---|---|---|

Municipal level | |||||

Working population per capita, coef. | the ratio of average number of enterprises’ employees to permanent population | 0.04 | 2.6 | 0.2 | 0.1 |

PopulationSize, thousand of person | the total number of registered population | 0.64 | 12615.3 | 61.8 | 301.0 |

Urban/Rural, % | the share of urban population in total population | 0 | 100 | 43.3 | 36.9 |

Production per capita, mln. RUB | the volume of goods, works, and services produced and shipped divided by the number of permanent population | 0.001 | 50.3 | 0.4 | 1.5 |

Productivity, mln. RUB | the volume of goods, works, and services produced and shipped divided by the number of enterprises’ employees | 0.005 | 118.4 | 1.3 | 2.6 |

WIncome per capita, mln. RUB | volume of social transfers and taxable population income of municipalities in the neighbourhood | 0.07 | 1.6 | 0.2 | 0.1 |

WWage, thousand. RUB | the average monthly wage of municipalities in the neighbourhood | 12.02 | 103.8 | 28.1 | 11.2 |

Regional level | |||||

EmpAgricultura, % | the share of employees in the agriculture and fisheries | 0.2 | 23.7 | 8.4 | 4.2 |

EmpMining, % | the share of employees in the mining industry | 0.03 | 27.5 | 2.6 | 4.9 |

EmpManufacturing, % | the share of employees in the manufacturing industry | 1.3 | 24.8 | 14.0 | 5.5 |

EmpConstruction, % | the share of employees in the construction industry | 3.5 | 15.4 | 8.1 | 2.1 |

EmpCateringHotels, % | the share of employees in the catering and hotels industries | 1.0 | 4.4 | 2.2 | 0.5 |

Entrepreneurship, number per 100 person | the number of entrepreneurs per 100 people | 1.8 | 4.2 | 2.6 | 0.4 |

EmpSmallBusiness, coef. | the average number of employees of small and medium-sized enterprises to the population | 0.03 | 0.2 | 0.1 | 0.03 |

Unemployment, % | the unemployment rate (International Labour Organization methodology) | 1.2 | 18.6 | 5.9 | 2.4 |

WUnemployment, % | the average value of the unemployment rate in neighbouring municipalities | 3.2 | 15.8 | 5.9 | 2.0 |

Export, billion $ | the volume of exports | 0 | 197.1 | 4.7 | 18.6 |

WGRP, mln. RUB/distance | the spatially weighted value of Gross Regional Product | 12.7 | 787.2 | 82.0 | 85.3 |

High school, % | the share of the employed population with higher education | 21.9 | 50.4 | 31.5 | 5.0 |

College education, % | the share of the employed population with secondary education | 31.4 | 58.9 | 46.2 | 5.4 |

Elementary school, % | the share of the employed population with primary education and incomplete school | 0.3 | 13.6 | 4.4 | 2.0 |

Patent, number per 10,000 person | the volume of patents granted for inventions and utility models per 10,000 population | 0 | 8.9 | 1.3 | 1.0 |

Female, % | the share of working women in total number of women in the region | 45.6 | 79.4 | 59.9 | 5.9 |

Younger, % | the share of employees under 20 years of age | 0.1 | 1.9 | 0.5 | 0.2 |

Older, % | the share of employees over 50 years of age | 17.6 | 32.8 | 27.1 | 2.5 |

Harmful, % | the proportion of workers employed in harmful and (or) hazardous working conditions in the total number of employees | 17.6 | 67.7 | 39.1 | 9.9 |

Trade union, units | the number of trade union organisations | 9 | 1866 | 258.9 | 266.2 |

Gini, coef. | the Gini coefficient of population income | 0.34 | 0.44 | 0.38 | 0.02 |

Transfers, % | the share of social transfers in the monetary population income | 10.9 | 35.6 | 22.2 | 4.3 |

## Appendix B

**Figure A1.**Marginal effects of interaction terms from model 4i where income per capita is the dependent variable: (

**a**) production per capita is an independent variable at municipal level, while the Patent is a grouping factor at regional level; (

**b**) production per capita is an independent variable at municipal level, while the Transfers is a grouping factor at regional level; (

**c**) population size is independent variable at municipal scale, and Gini coefficient is a grouping factor at regional level. Minimum and maximum values (lower and upper bounds) of the moderator are used to plot the interaction.

**Figure A2.**Diagnostics plots of random effects from model 4i: (

**a**) for year, (

**b**) for regions and (

**c**) for municipalities.

**Figure A3.**Marginal effects of interaction terms from model 4i where average wages are the dependent variable, production per capita are the independent variable at the municipal level co cледующими grouping factor at the regional level: (

**a**) the share of employees in the agriculture and fisheries; (

**b**) the share of employees in the manufacturing industry; (

**c**) the share of employees in the mining industryж (

**d**) the volume of patents granted for inventions and utility models per 10,000 population; (

**e**) the Gini coefficient of population income. Minimum and maximum values (lower and upper bounds) of the moderator are used to plot the interaction.

**Figure A4.**Diagnostics plots of random effects from model 4w: (

**a**) for year, (

**b**) for regions and (

**c**) for municipalities.

## References

- Gustafsson, B.; Shi, L. Income inequality within and across counties in rural China 1988 and 1995. J. Dev. Econ.
**2002**, 69, 179–204. [Google Scholar] [CrossRef] - Gravier-Rymaszewska, J.; Tyrowicz, J.; Kochanowicz, J. Intra-provincial inequalities and economic growth in China. Econ. Syst.
**2010**, 34, 237–258. [Google Scholar] [CrossRef] [Green Version] - He, S.; Bayrak, M.M.; Lin, H. A comparative analysis of multi-scalar regional inequality in China. Geoforum
**2017**, 78, 1–11. [Google Scholar] [CrossRef] - Chuliang, L.; Shi, L.; Sicular, T. The long-term evolution of income inequality and poverty in China. In WIDER Working Paper 2018/153; UNU-WIDER: Helsinki, Finland. [CrossRef]
- Wang, Y.; Jiang, Y.; Yin, D.; Liang, C.; Duan, F. Examining multilevel poverty-causing factors in poor villages: A hierarchical spatial regression model. Appl. Spat. Anal. Policy
**2021**, 14, 969–998. [Google Scholar] [CrossRef] - Doran, J.; Jordan, D. Decomposing US regional income inequality from 1969 to 2009. Appl. Econ. Lett.
**2016**, 23, 781–784. [Google Scholar] [CrossRef] - Manduca, R.A. The contribution of national income inequality to regional economic divergence. Soc. Forces
**2019**, 98, 622–648. [Google Scholar] [CrossRef] - Khan, M.S.; Siddique, A.B. Spatial analysis of regional and income inequality in the United States. Economies
**2021**, 9, 159. [Google Scholar] [CrossRef] - Breau, S. Rising inequality in Canada: A regional perspective. Appl. Geogr.
**2014**, 61, 58–69. [Google Scholar] [CrossRef] - Breau, S.; Saillant, R. Regional income disparities in Canada: Exploring the geographical dimensions of an old debate. Reg. Stud. Reg. Sci.
**2016**, 3, 463–481. [Google Scholar] [CrossRef] - Kim, R.; Mohanty, S.K.; Subramanian, S.V. Multilevel geographies of poverty in India. World Dev.
**2016**, 87, 349–359. [Google Scholar] [CrossRef] - Díaz Dapena, A.; Morollón, F.R.; de Moura Pires, M.; da Silva Gomes, A. Convergence in Brazil: New evidence using a multilevel approach. Appl. Econ.
**2017**, 49, 5050–5062. [Google Scholar] [CrossRef] - López-Rodríguez, J.; Faíña, A. Regional wage disparities in Europe: What role for market access? Investig. Reg.
**2007**, 11, 5–23. Available online: https://old.aecr.org/images/ImatgesArticles/2008/01%20Lopez-Rod.pdf (accessed on 8 May 2022). - Pereira, J.; Galego, A. Inter-country wage differences in the European Union. Int. Labour Rev.
**2018**, 157, 101–128. [Google Scholar] [CrossRef] [Green Version] - Bosco, B.; Poggi, A. Middle class, government effectiveness and poverty in the EU: A Dynamic Multilevel Analysis. Rev. Income Wealth
**2019**, 66, 94–125. [Google Scholar] [CrossRef] - Ngarambe, O.; Goetz, S.J.; Debertin, D.L. Regional economic growth and income distribution: County-level evidence from the US. J. Agric. Appl. Econ.
**1998**, 30, 325–337. [Google Scholar] [CrossRef] [Green Version] - Stansel, D. Local decentralization and local economic growth: A cross-sectional examination of US metropolitan areas. J. Urban Econ.
**2005**, 57, 55–72. [Google Scholar] [CrossRef] - Higgins, M.J.; Levy, D.; Young, A.T. Growth and convergence across the United States: Evidence from county-level data. Rev. Econ. Stat.
**2006**, 88, 671–681. [Google Scholar] [CrossRef] - Baddeley, M.; Martin, R.; Tyler, P. Regional wage rigidity: The European Union and United States compared. J. Reg. Sci.
**2000**, 40, 115–141. [Google Scholar] [CrossRef] - Pannenberg, M.; Schwarze, J. ‘Phillips Curve’ or ‘Wage Curve’: Is there really a puzzle? Evidence for West Germany. In DIW Discussion Papers 139; Deutsches Institut für Wirtschaftsforschung (DIW): Berlin, Germany, 1998; Available online: http://hdl.handle.net/10419/95840 (accessed on 10 May 2022).
- Bockerman, P.; Ilmakunnas, P. Do job disamenities raise wages or ruin job satisfaction? Int. J. Manpow.
**2006**, 27, 290–302. [Google Scholar] [CrossRef] [Green Version] - Malkina, M.Y. Evaluation of the factors of intra-regional income differentiation of the Russian population. Spat. Econ.
**2015**, 3, 97–119. (In Russian) [Google Scholar] [CrossRef] - Kosfeld, R.; Dreger, C. Towards an East German wage curve: NUTS Boundaries, Labour Market Regions and Unemployment Spillovers. In IZA Discussion Papers 10892; Institute of Labor Economics (IZA): Bonn, Germany, 2017; Available online: http://hdl.handle.net/10419/170876 (accessed on 6 April 2022).
- Demidova, O.; Timofeeva, E. Spatial aspects of wage curve estimation in Russia. J. New Econ. Assoc.
**2021**, 3, 69–101. (In Russian) [Google Scholar] [CrossRef] - Nolan, B.; Richiardi, M.G.; Valenzuela, L. The drivers of income inequality in rich countries. J. Econ. Surv.
**2019**, 33, 1285–1324. [Google Scholar] [CrossRef] - Meschi, E.; Vivarelli, M. Trade Openness and Income Inequality in Developing Countries. CSGR Working Paper Series 232/07. 2007. Available online: http://wrap.warwick.ac.uk/1876/1/WRAP_Meschi_wp23207.pdf (accessed on 12 June 2022).
- Messina, J.; Silva, J. Twenty years of wage inequality in Latin America. In IDB Working Paper Series IDB-WP-1041; Inter-American Development Bank (IDB): Washington, DC, USA, 2019. [Google Scholar] [CrossRef]
- Wimer, C.H.; Parolin, Z.; Fenton, A.; Fox, L.; Jencks, C.H. The direct effect of taxes and transfers on changes in the U.S. income distribution, 1967–2015. Demography
**2020**, 57, 1833–1851. [Google Scholar] [CrossRef] [PubMed] - Brady, D.; Bostic, A. Paradoxes of social policy: Welfare transfers, relative poverty, and redistribution preferences. Am. Sociol. Rev.
**2015**, 80, 268–298. [Google Scholar] [CrossRef] [Green Version] - Blundell, R.; Joyce, R.; Keiller, A.N.; Ziliak, J.P. Income inequality and the labour market in Britain and the US. J. Public Econ.
**2018**, 162, 48–62. [Google Scholar] [CrossRef] - Durand-Lasserve, O.; Blöchliger, H. The drivers of regional growth in Russia: A baseline model with applications. In OECD Economics Department Working Papers 1523; OECD Publishing: Paris, France, 2018. [Google Scholar] [CrossRef]
- Zubarevich, N.V.; Safronov, S.G. Spatial inequality of money incomes in Russia and large post-soviet countries. Reg. Issled.
**2014**, 4, 100–110. Available online: http://media.geogr.msu.ru/RI/RI_2014_04(46).pdf (accessed on 6 April 2022). - Zubarevich, N.V.; Safronov, S.G. Regional inequality in large post-soviet countries. Reg. Res. Russ. (RRR)
**2011**, 1, 15–26. [Google Scholar] [CrossRef] - Manaeva, I.V. Specifics of socio-economic inequality in Russian cities. Econ. Anal. Theory Pract.
**2017**, 16, 960–970. (In Russian) [Google Scholar] [CrossRef] - Kozlova, O.A.; Shelomentsev, A.G.; Goncharova, K.S. Assessment of the impact of age and gender composition of the population on the income differentiation. Fundam. Res.
**2017**, 11, 403–407. (In Russian). Available online: https://fundamental-research.ru/en/article/view?id=41957 (accessed on 26 April 2022). - Ovcharova, L.N.; Popova, D.O.; Rudberg, A.M. Decomposition of Income Inequality in Contemporary Russia. J. New Econ. Assoc.
**2016**, 3, 170–186. (In Russian) [Google Scholar] [CrossRef] - Ivanova, V. Spatial convergence of real wages in Russian cities. Ann. Reg. Sci.
**2018**, 61, 1–30. [Google Scholar] [CrossRef] - Malkina, M.Y. Spatial wage inequality and its sectoral determinants: The case of modern Russia. Oeconomia Copernic.
**2019**, 10, 69–87. [Google Scholar] [CrossRef] - Goldstein, H. Multilevel Statistical Models; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2010. [Google Scholar] [CrossRef] [Green Version]
- Raudenbush, S.W.; Bryk, A.S.; Cheong, Y.F.; Congdon, R.T., Jr.; du Toit, M. HLM 7: Hierarchical Linear and Nonlinear Modeling; Scientific Software International: Linconwood, IL, USA, 2011. [Google Scholar]
- Moellering, H.; Tobler, W. Geographical variances. Geogr. Anal.
**1972**, 4, 34–50. [Google Scholar] [CrossRef] - Wang, Y.; Liang, C.; Li, J. Detecting village-level regional development differences: A GIS and HLM method. Growth Change
**2019**, 50, 222–246. [Google Scholar] [CrossRef] [Green Version] - Masache, K.C.; López, R.A. Concentración espacial de capital humano calificado y desigualdad regional de ingresos en Ecuador. Paradig. Económico
**2017**, 9, 5–26. (In Spanish). Available online: https://paradigmaeconomico.uaemex.mx/article/view/4845 (accessed on 12 April 2022). - Ketels, C.; Protsiv, S. Cluster presence and economic performance: A new look based on European data. Reg. Stud.
**2021**, 55, 208–220. [Google Scholar] [CrossRef] - Kopecny, S.; Hillmert, S. Place of study, field of study and labour-market region: What matters for wage differences among higher-education graduates? J. Labour Mark. Res.
**2021**, 55, 19. [Google Scholar] [CrossRef] - Shi, Y.; Leite, W.; Algina, J. The impact of omitting the interaction between crossed factors in cross-classified random effects modelling. Br. J. Math. Stat. Psychol.
**2010**, 63 Pt 1, 1–15. [Google Scholar] [CrossRef] - Meyer, R. Deviance information criterion (DIC). Wiley StatsRef: Statistics Reference Online; Balakrishnan, N., Colton, T., Everitt, B., Piegorsch, W., Ruggeri, F., Teugels, J.L., Eds.; John Wiley & Sons Ltd: Hoboken, NY, USA, 2016; pp. 1–6. [Google Scholar] [CrossRef]
- Fallah, B.; Partridge, M. The elusive inequality-economic growth relationship: Are there differences between cities and the countryside? Ann. Reg. Sci.
**2007**, 41, 375–400. [Google Scholar] [CrossRef] - Goodman, C.B. Political fragmentation and economic growth in U.S. metropolitan areas. J. Urban Aff.
**2020**, 43, 1355–1376. [Google Scholar] [CrossRef] - Ortega, F.; Peri, G. Openness and income: The roles of trade and migration. J. Int. Econ.
**2014**, 92, 231–251. [Google Scholar] [CrossRef] - Wang, B.; Rosenberg, M.W.; Wang, S.H.; Yang, P.; Tian, J. Multilevel and spatially heterogeneous factors influencing poor households’ income in a frontier minority area in Northeast China. Complexity
**2021**, 2021, 8834422. [Google Scholar] [CrossRef] - Pede, V.O. Diversity and regional economic growth: Evidence from US counties. J. Econ. Dev.
**2013**, 38, 111–127. [Google Scholar] [CrossRef] - Goschin, Z. Regional determinants of average wage in Romania. Procedia Econ. Financ.
**2014**, 8, 362–369. [Google Scholar] [CrossRef] [Green Version] - Martini, B.; Giannini, M. Regional wage and productivity in Italy: A spatio-temporal analysis. Spat. Econ. Anal.
**2020**, 15, 392–412. [Google Scholar] [CrossRef] - Longhi, S.; Nijkamp, P.; Poot, J. Spatial heterogeneity and the wage curve revisited. J. Reg. Sci.
**2004**, 46, 707–731. [Google Scholar] [CrossRef] [Green Version] - Candelaria, C.; Daly, M.; Hale, G. Persistence of regional wage differences in China. Pac. Econ. Rev.
**2015**, 20, 365–387. [Google Scholar] [CrossRef] - Demidova, O. Spatial effects for the eastern and western regions of Russia: A comparative analysis. Int. J. Econ. Policy Emerg. Econ.
**2015**, 8, 153–168. [Google Scholar] [CrossRef] - Ran, M.; Chen, L.; Li, W. Financial deepening, spatial spillover, and urban–rural income disparity: Evidence from China. Sustainability
**2020**, 12, 1450. [Google Scholar] [CrossRef] - Cieślik, A.; Rokicki, B. European integration and spatial wage structure in Poland. Tijdschr. Voor Econ. En Soc. Geogr.
**2015**, 107, 435–453. [Google Scholar] [CrossRef] - Egger, P.; Huber, P.; Pfaffermayr, M. A note on export openness and regional wage disparity in Central and Eastern Europe. Ann. Reg. Sci.
**2005**, 39, 63–71. [Google Scholar] [CrossRef] - Groot, S.P.; de Groota, H.L.F.; Smit, M.J. Regional wage differences in the Netherlands: Micro-evidence on agglomeration externalities. Tinbergen Inst. Discuss. Pap.
**2013**, 54, 503–523. [Google Scholar] [CrossRef] [Green Version] - Hoang, H.T.; Huynh, L.T.D.; Chen, G.S. How new economic geography explain provincial wage disparities: Generalised methods of moments approach. Ekon. Reg.
**2019**, 15, 205–215. [Google Scholar] [CrossRef] - Garson, D. Hierarchical Linear Modeling: Guide and Applications; Sage Publications: Newbury Park, CA, USA, 2013; 71p. [Google Scholar]
- Peifer, C.; Zipp, G. All at once? The effects of multitasking behavior on flow and subjective performance. Eur. J. Work. Organ. Psychol.
**2019**, 28, 682–690. [Google Scholar] [CrossRef] - Ryu, E. Effects of skewness and kurtosis on normal-theory based maximum likelihood test statistic in multilevel structural equation modeling. Behav. Res. Methods
**2011**, 43, 1066–1074. [Google Scholar] [CrossRef] [Green Version] - Pituch, K.A.; Stapleton, L.M. The Performance of Methods to Test Upper-Level Mediation in the Presence of Nonnormal Data. Multivar. Behav. Res.
**2008**, 43, 237–267. [Google Scholar] [CrossRef] - Wilcox, R.R. Understanding the practical advantages of modern ANOVA methods. J. Clin. Child Adolesc. Psychol.
**2002**, 31, 399–412. [Google Scholar] [CrossRef] - Monastiriotis, V. Inter- And intra-regional wage inequalities in the UK: An examination of the sources of UK wage inequalities and their evolution. In Proceedings of the 40th Congress of the European Regional Science Association: “European Monetary Union and Regional Policy”, Barcelona, Spain, 29 August–1 September 2000; European Regional Science Association (ERSA): Louvain-la-Neuve, Belgium, 2000. Available online: http://hdl.handle.net/10419/114861 (accessed on 10 June 2022).
- Čadil, J.; Petkovová, L.; Blatná, D. Human capital, economic structure and growth. Procedia Econ. Financ.
**2014**, 12, 85–92. [Google Scholar] [CrossRef] [Green Version] - Kuranov, G.O.; Lukyanenko, R.F. Quality and factors of economic development: Matters of evaluation and analysis. Vopr. Stat.
**2020**, 27, 26–44. (In Russian) [Google Scholar] [CrossRef]

**Figure 2.**Natural breaks map of average volume of social transfers and population taxable income per capita over 2015–2019 (mln RUB).

**Figure 3.**Natural breaks map of average nominal employees’ wages of large, medium-sized and non-profit enterprises over 2015–2019 (mln. RUB).

**Figure 4.**Box plots for the volume of social transfers and the population taxable income per capita in certain regions of Russia in 2015–2019.

**Figure 5.**Box plots for average nominal employees’ wages of large, medium-sized and non-profit enterprises in certain regions of Russia in 2015–2019.

**Figure 6.**The relationships between population income and production volume at the municipal scale are under the influence of factors related to regional-level patents per capita (

**a**) and transfer share (

**b**). The relationships between population income and population size at the municipal scale are under influence of factors related to regional level Gini coefficient (

**c**). The lines describing the relationship of indicators at the municipal level were built using the OLS method, not for all groups but only for several ones with the highest and lowest values of the indicators. The influence of other factors was not taken into account.

**Figure 7.**The relationship between wage and production volume at municipal scale under the influence of regional factors: share of social transfers (

**a**) and Gini coefficient (

**b**). The lines describing the relationship of indicators at the municipal level were built using the OLS method, not for all groups but only for several ones with the highest and lowest values of the indicators. The influence of other factors was not taken into account.

Indicator | Year | Min | Max | Mean | SD | Coefficient of Variation |
---|---|---|---|---|---|---|

Income per capita (volume of social transfers and the population taxable income per capita, mln. RUB) | 2015 | 0.05 | 1.79 | 0.19 | 0.12 | 65.0 |

2016 | 0.05 | 2.49 | 0.19 | 0.12 | 65.4 | |

2017 | 0.05 | 3.15 | 0.19 | 0.14 | 70.6 | |

2018 | 0.05 | 3.22 | 0.20 | 0.15 | 72.4 | |

2019 | 0.06 | 2.52 | 0.21 | 0.16 | 74.0 | |

Average wages (average nominal employees’ wages of large, medium-sized and non-profit enterprises, thousands of RUB) | 2015 | 12.02 | 110.02 | 26.26 | 11.36 | 43.3 |

2016 | 12.15 | 115.18 | 26.38 | 11.58 | 43.9 | |

2017 | 12.59 | 124.64 | 27.75 | 12.10 | 43.6 | |

2018 | 15.32 | 128.84 | 30.02 | 12.72 | 42.4 | |

2019 | 16.42 | 130.98 | 31.10 | 13.26 | 42.6 |

Tested Variable | Df | Levene’s Test | Fligner-Killeen Test |
---|---|---|---|

Income per capita | |||

Grouping by periods | 4 | F-value = 2.48, p-value = 0.042 | χ^{2} = 12.88,p-value = 0.012 |

Grouping by regions | 74 | F-value = 42.88, p-value < 0.0001 | χ^{2} = 2116.8,p-value < 0.0001 |

Grouping by periods and regions | 374 | F-value = 9.13, p-value < 0.0001 | χ^{2} = 2213.2,p-value < 0.0001 |

Average wages | |||

Grouping by periods | 4 | F-value = 3.72, p-value = 0.005 | χ^{2} = 21.86,p-value = 0.0002 |

Grouping by regions | 74 | F-value = 28.56, p-value < 0.0001 | χ^{2} = 1917.9,p-value < 0.0001 |

Grouping by periods and regions | 374 | F-value = 5.56, p-value < 0.0001 | χ^{2} = 2108.4,p-value < 0.0001 |

Variable | Model 1i | Model 2i | Model 3i | Model 4i |
---|---|---|---|---|

Constant, ${\theta}_{00}$ | 0.22 *** (0.02) | 0.044 *** (0.01) | −0.196 *** (0.051) | −0.211 *** (0.042) |

Municipal level | ||||

Working population per capita, β_{1ijt} | 0.814 *** (0.011) | 0.785 *** (0.011) | 0.757 *** (0.012) | |

PopulationSize, β_{2ijt} | 0.049 *** (0.005) | 0.038 *** (0.004) | 0.174 ** (0.058) | |

PopulationSize * Gini, γ_{21jt} | −0.327 * (0.141) | |||

Production per capita, β_{3ijt} | 8.718 *** (0.502) | 8.396 *** (0.494) | 43.59 *** (2.38) | |

Production per capita * Patent, γ_{31jt} | 3.625 *** (0.885) | |||

Production per capita * Transfers, γ_{32jt} | −2.166 *** (0.129) | |||

WIncome per capita, β_{4ijt} | 0.238 *** (0.012) | 0.243 *** (0.012) | ||

Regional level | ||||

EmpAgricultura, γ_{01jt} | −0.001 ˠ (0.001) | −0.001 * (0.001) | ||

EmpMining, γ_{02jt} | 0.002 * (0.001) | 0.002 * (0.001) | ||

EmpManufacturing, γ_{03jt} | −0.001 (0.001) | |||

EmpConstruction, γ_{04jt} | 0.0003 (0.001) | |||

EmpCateringHotels, γ_{05jt} | 0.001 (0.003) | |||

Entrepreneurship, γ_{06jt} | 0.012 *** (0.003) | 0.01 *** (0.003) | ||

High school, γ_{07jt} | 0.00002 (0.0003) | |||

College education, γ_{08jt} | 0.001 ˠ (0) | 0.001 * (0.0003) | ||

Elementary school, γ_{09jt} | 0.001 (0.001) | |||

Patent, γ_{010jt} | 0.006 *** (0.002) | 0.005 ** (0.001) | ||

Female, γ_{011jt} | 0.001 *** (0.000) | 0.001 *** (0.0003) | ||

Gini, γ_{012jt} | 0.148 * (0.072) | 0.194 ** (0.07) | ||

Transfers, γ_{013jt} | 0.001 (0.001) | 0.001 ˠ (0.001) | ||

Random effects | ||||

σ^{2} | 0.001 | 0.001 | 0.001 | 0.001 |

τ^{2} | 0.008 | 0.002 | 0.002 | 0.002 |

e_{j}^{2} | 0.024 | 0.005 | 0.001 | 0.001 |

e_{t}^{2} | 0.0001 | 0.0001 | 0.0002 | 0.0002 |

Estimation of model quality | ||||

Log likelihood | 15,385 | 17,519 | 17,791 | 17,519 |

AIC | −30,760 | −35,023 | −35,538 | −35,896 |

BIC | −30,724 | −34,965 | −35,379 | −35,752 |

DIC | −30,770 | −35,039 | −35,582 | −35,936 |

Variable | Model 1w | Model 2w | Model 3w | Model 4w |
---|---|---|---|---|

$\mathrm{Constant},{\theta}_{00}$ | 31.06 *** (1.95) | 23.01 *** (1.683) | −2.082 (2.753) | −9.902 *** (2.612) |

Municipal level | ||||

Urban/Rural, β_{1ijt} | 0.049 *** (0.003) | 0.042 *** (0.003) | 0.041 *** (0.003) | |

Working population per capita, β_{2ijt} | 27.17 *** (0.883) | 22.86 *** (0.824) | 21.24 *** (0.82) | |

Productivity, β_{3ijt} | 213.5 *** (13.54) | 159 *** (12.59) | 5208 *** (609.6) | |

Productivity*Gini, γ_{31jt} | −11210 *** (1171) | |||

Productivity*EmpMining, γ_{32jt} | 22.97 *** (3.973) | |||

Productivity*EmpAgricultura, γ_{33jt} | 40.27 *** (9.084) | |||

Productivity*EmpManufacturing, γ_{34jt} | 12.02 * (5.3) | |||

Productivity*Transfers, γ_{35jt} | −44.65 *** (7.663) | |||

WWage, β_{4ijt} | 0.447 *** (0.014) | 0.446 *** (0.013) | ||

Regional level | ||||

Unemployment, γ_{01jt} | −0.07 ˠ (0.041) | |||

WUnemployment, γ_{02jt} | −0.012 (0.076) | |||

EmpAgriculture, γ_{03jt} | −0.078 ˠ (0.041) | −0.122 ** (0.037) | ||

EmpMinning, γ_{04jt} | 0.501 *** (0.085) | 0.471 *** (0.088) | ||

EmpManufacturing, γ_{05jt} | −0.07 (0.044) | −0.079 ˠ (0.044) | ||

EmpConstruction, γ_{06jt} | 0.192 *** (0.047) | 0.173 *** (0.046) | ||

EmpCateringHotels, γ_{07jt} | −0.024 (0.162) | |||

Entrepreneurship, γ_{08jt} | 0.275 (0.168) | |||

EmpSmallBusiness, γ_{09jt} | −22.35 ** (6.827) | −21.67 *** (6.436) | ||

High school, γ_{010jt} | 0.035 * (0.016) | 0.044 ** (0.014) | ||

College education, γ_{011jt} | −0.024 (0.15) | |||

Patent, γ_{012jt} | 0.076 (0.079) | |||

Female, γ_{013jt} | 0.11 *** (0.016) | 0.11 *** (0.015) | ||

Younger, γ_{014jt} | −0.243 (0.153) | |||

Older, γ_{015jt} | −0.078 ** (0.029) | −0.08 ** (0.027) | ||

Harmful, γ_{016jt} | 0.02 (0.013) | |||

Trade union, γ_{017jt} | 0.002 *** (0.001) | 0.002 *** (0.001) | ||

Gini, γ_{018jt} | 15.45 *** (3.678) | 31.78 *** (3.911) | ||

Transfers, γ_{019jt} | 0.06 ˠ (0.034) | 0.139 *** (0.035) | ||

Export, γ_{020jt} | 0.132 *** (0.016) | 0.127 *** (0.015) | ||

WGRP, γ_{021jt} | −0.002 (0.002) | |||

Random effects | ||||

σ^{2} | 3.15 | 2.99 | 2.54 | 2.49 |

τ^{2} | 42.27 | 25.44 | 21.69 | 21.04 |

e_{j}^{2} | 210.77 | 136.27 | 11.03 | 12.5 |

e_{t}^{2} | 4.72 | 4.74 | 2.47 | 2.39 |

Estimation of model quality | ||||

Log-likelihood | −24,538 | −23,827 | −22,910 | −22,807 |

AIC | 49,087 | 47,669 | 45,880 | 45,667 |

BIC | 49,123 | 47,727 | 46,097 | 45,854 |

DIC | 49,077 | 47,653 | 45,820 | 45,615 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Timiryanova, V.; Krasnoselskaya, D.; Kuzminykh, N.
Applying the Multilevel Approach in Estimation of Income Population Differences. *Stats* **2023**, *6*, 67-98.
https://doi.org/10.3390/stats6010005

**AMA Style**

Timiryanova V, Krasnoselskaya D, Kuzminykh N.
Applying the Multilevel Approach in Estimation of Income Population Differences. *Stats*. 2023; 6(1):67-98.
https://doi.org/10.3390/stats6010005

**Chicago/Turabian Style**

Timiryanova, Venera, Dina Krasnoselskaya, and Natalia Kuzminykh.
2023. "Applying the Multilevel Approach in Estimation of Income Population Differences" *Stats* 6, no. 1: 67-98.
https://doi.org/10.3390/stats6010005