# Applying the Multilevel Approach in Estimation of Income Population Differences

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## 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.

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**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 |

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## 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