*3.6. Econometric Modeling*

In this research, we applied a contemporary economic and mathematical instrument that uses complex variables. It has been successfully used by some researchers to solve different economic problems [61]. The key principle of this tool is to combine two economic indicators into one model variable. This approach allows different aspects of a phenomenon to be addressed and its influence on parameters to be analyzed, which, in turn, could be a complex variable. Thus, we applied basic complex-valued model (1) with regard to migration processes.

$$y\_{rt} + iy\_{it} = (a\_0 + ia\_1) + (b\_0 + ib\_1)(x\_{rt} + ix\_{it})\_\prime \tag{1}$$

where *yr* and *yi* are components of an endogenous complex variable; *xr* and *xi* are components of an exogenous complex variable; and *a*0 + *ia*1 and *b0 + ib1* are model coefficients.

Correlation between two complex indicators is evaluated as follows [61]:

$$r\_{XY} = \frac{\sum \left( y\_{rt} + i y\_{it} \right) \left( \mathbf{x}\_{rt} + i \mathbf{x}\_{it} \right)}{\sqrt{\sum \left( \mathbf{x}\_{rt} + i \mathbf{x}\_{it} \right)^2 \sum \left( y\_{rt} + i y\_{it} \right)^2}} \tag{2}$$

If the real part of *rXY* is close to 1, then the endogenous variable is linearly dependent on the exogenous one, while the imaginary part reflects the plot scatter of the regression model.

Econometric modeling of migration processes was carried out using data published by state statistics bodies using the Murmansk region as an example. We used indicators representing migration processes—the number of arrivals and departures from the region in the period—as the dependent complex variable in all econometric models. As factors that determine the variability of the considered indicators of migration, we chose seven socio-economic indicators that have high linear correlations with the dependent variables (Table 2). Then, we formed complex variables by grouping a pair of particular indicators that describe one process or phenomenon from two different aspects. The time series included seven observations, namely, the real values of the indicators from 2011 to 2017. Using the developed models, we predicted the values of migration inflows and outflows in the Murmansk region for the year 2018. The actual values in 2018 were used as benchmarks to assess the adequacy of the obtained forecasts.

Taking the above into account, we studied simple linear complex-valued models of type (1), as follows:


Model 1 includes social and economic indicators as factors. Income per capita is the real part of the complex factor, and its imaginary part is the indicator of the volume of paid services provided to residents in the Arctic regions. This indicator reflects the degree to which high-level personal needs are satisfied in contrast to the basic ones.

The profit indicator is normally the main appeal to potential labor migrants to the region [71]. Additionally, high-level needs require appropriate social infrastructure. The amount of money spent indirectly indicates the availability of various social opportunities [61]. Therefore, we deemed the volume of paid services rendered to be a crucial factor in judging regional attractiveness and incorporated it as an imaginary component.

Model 2 is a simple linear regression model in which migration processes are the result of variation in two economic indicators that characterize the average expenses of residents of the Arctic regions. Consumer expenditure per capita and the cost of a fixed set of consumer goods and services both describe the process of spending personal funds but from different angles. In fact, they represent the average amount of money that the consumer possesses after all compulsory monthly expenses. This value is of interest to potential migrants to the Arctic regions, since it characterizes their material security if they move there.

Model 3 is a simple linear regression model in which the values of migration indicators depend on the social components of the attractiveness of the region; these components represent the provision of families with preschool educational institutions and housing. Model 4 is a modification of Model 3 and incorporates the social indicator of natural population growth instead of the indicator "Commissioning of residential buildings".

Figures 6–9 show graphs of the actual and calculated values of arrivals and departures in the Murmansk region. The calculated values were obtained using the above econometric complex-valued models.

**Figure 6.** Actual and calculated values of migration indicators in the Murmansk region: Model 1.

**Figure 7.** Actual and calculated values of migration indicators in the Murmansk region: Model 2.

**Figure 8.** Actual and calculated values of migration indicators in the Murmansk region: Model 3.

**Figure 9.** Actual and calculated values of migration indicators in the Murmansk region: Model 4.

The figures show that the predictions of all four econometric models closely match the initial data and are suitable for forecasting. In addition to the four models presented above, we performed a linear extrapolation of trends of migration inflows and outflows based on their dynamics from 2011 to 2017 in order to calculate forecast values for 2018. Table 3 shows the results of an analysis that compares different forecasts of arrivals and departures in the Murmansk region.


**Table 3.** Forecast and actual values of migration indicators in the Murmansk region for 2018.

As indicated in the table, the forecasts of the arrivals and departures derived from the four studied complex-valued models are closer to the actual data than the results of a linear extrapolation of their trends for 2018.

Furthermore, the best forecasting results are obtained with model 2 for the indicator "arrivals" and model 4 for the indicator "departures". The fourth model has the highest complex R-squared. Thus, we can conclude that linear complex-valued models are suitable for predicting migration processes in the Arctic regions of Russia, despite their simplicity and inability to consider many factors.

#### *3.7. Integral Estimation of the Attractiveness of Arctic Regions*

As mentioned above, a large number of domestic studies have been devoted to assessing the attractiveness of territories. However, the literature contains little research assessing the attractiveness of the Arctic regions to labor migrants. However, it should be noted that living and working conditions in the regions of the Russian Arctic differ significantly. In this regard, one of the objectives of the study was to conduct a comparative analysis of four Arctic regions that are entirely included in the Arctic zone of the Russian Federation (Murmansk region and Nenets, Yamalo-Nenets, and Chukotka Autonomous

districts) according to a number of key indicators. For this purpose, we calculated two integral indicators characterizing the economic and social attractiveness of the territories of the Russian Arctic using state statistics data.

To calculate the integral indicators of economic and social attractiveness of the Arctic regions, the authors used the values of six economic and six social indicators presented above for the period 2010–2018. To this end, four source data tables were generated for each region under consideration, one of which is shown below as an example (Table 4). The source data series were converted into values on a relative scale according to the "maximum–minimum" method. Its minimum and maximum values correspond to 0 and 1, respectively. The following Equations were used:

$$\frac{X\_i - X\_{\min}}{X\_{\max} - X\_{\min}},$$

$$1 - \frac{X\_i - X\_{\min}}{X\_{\max} - X\_{\min}}\tag{4}$$

where Х*i* is the value of the indicator for region *i*, *Xmax* is the maximum value of the indicator among the regions in the year under review, and *Xmin* is the minimum value of the indicator among the regions in the year under review.

**Table 4.** Initial data for the Murmansk region for calculating integral indicators of attractiveness.


The authors used the "maximum –minimum" method because it avoids the excessive influence of each individual indicator on the integral one. Equation (3) is used when higher values of the indicator are preferred, and Equation (4) is applied otherwise.

The integral indicators of economic and social attractiveness were calculated separately for each year of the studied period according to Equations (5)–(7):

$$EA\_{\dot{j}} = \sum\_{i=1}^{n} a\_{i\dot{j}} w\_{i\prime} \tag{5}$$

$$SA\_j = \sum\_{i=1}^{m} c\_{ij} z\_{i\star} \tag{6}$$

$$\sum\_{i=1}^{n} w\_i = 1; \sum\_{i=1}^{m} z\_i = 1,\tag{7}$$

where *EAj* is the economic attractiveness of the region in year *j; aij, cij* are values of the *i*-th economic and social indicator in year *j*, respectively; *Saj* is the social attractiveness of the region in year *j*; *wi*, *zi* are the *i*-th economic and social indicators' weights, respectively; *n* and *m* are the number of economic and social indicators, respectively.

Table 5 presents the values obtained for each of the four regions, which were calculated based on 12 socio-economic indicators. We assume that the weight coefficients of all indicators are equal in order to avoid the excessive influence of any of the studied indicators on the integral assessment.

**Table 5.** Integral evaluation of economic and social attractiveness of the Arctic regions in 2010–2018.


The data in this table show that the Yamalo-Nenets Autonomous district had the most economic stimulation in the 9 studied years. The economic activity is related to the high income per capita in the region, which, in turn, is conditioned by the presence of oil and gas extraction companies with good salaries and a small population. This region has had the best values of average consumer expenditure per capita, share of the population with incomes lower than the subsistence minimum in the total population, and unemployment rate. The Murmansk region has the lowest economic appeal. This is due to low incomes and consumer spending and high unemployment rates.

The Chukotka Autonomous district also has low economic attractiveness. It ranks last in terms of "Cost of a fixed set of consumer goods and services", which is significantly higher in the Chukotka district than in the other regions, and "Average consumer expenditure per capita". Although the "Unemployment rate" and "Share of population with incomes below the subsistence minimum in the total population" are quite low, these variables have had little effect on its attractiveness. "Income per capita" is also quite good.

However, the Chukotka district does not have a leading position in any of the indicators for the studied period, which determined its relatively low attractiveness.

In terms of social attractiveness, the Chukotka Autonomous district is highlighted by the set of studied indicators. When analyzing the individual values of indicators, it was found that this region was characterized by the lowest crime rate per person for almost all 9 years under review. Moreover, the region has a large number of square meters of commissioned residential buildings per person, which is several times higher compared to the Nenets Autonomous district, which has almost the same population size.

In addition, this region has consistently had the leading position among the four regions in terms of "Gross enrollment rate in preschool education" since 2010. In 2013, the "Volume of paid services per capita" in the Chukotka Autonomous district almost doubled compared to its value in 2012.

As a result, the region took the leading position for this indicator until 2018. In terms of "Birth rate", the Chukotka Autonomous district is significantly behind the Yamalo-Nenets Autonomous district, which has a population that is about 10 times higher.

Despite having the largest population of the four Arctic regions studied, which seems to be an attractive factor for young people, the Murmansk region ranks only third after the Chukotka and Yamalo-Nenets Autonomous districts according to the integral indicator of social attractiveness. This is due to its low (or even minimum) values of indicators such as "Commissioning of residential buildings per person", "Volume of paid services per capita", and "Birth rate", which is actually associated with a natural population decline. In addition, the Chukotka, Nenets, and particularly Yamalo-Nenets Autonomous districts show stable natural population growth.

The Nenets Autonomous district ranks last in terms of social attractiveness, primarily because it has the lowest values of "Population size" and "Volume of paid services per capita", as well as poor values of all the other indicators in comparison with other regions.

In the course of the study, the authors attempted to analyze the relationship between migration growth and the integral indicators of the attractiveness of the region. We hypothesized that changes in indicators of social and economic attractiveness have linear effects on the number of arrivals and departures in the region. However, the calculation of the linear complex correlation coefficient did not reveal a simple linear relationship between them, and this hypothesis was not confirmed. Further research can aim to identify non-linear models that reflect the relationship between migration flows in the regions of the Russian Arctic and their attractiveness to labor migrants.
