Economic and statistical modeling of the value of urban real property (using the example of Tyumen city, Russia)

The article presents the results on the derivation of an empirical formula of the relationship between the price of apartments and their area in different planning districts of the city of Tyumen, Russia. Economic and mathematical modeling was performed on the basis of analysis of variance, analysis of variational series, regression and correlation analyses. In a specific mathematical form, methods of mathematical statistics allowed answering a wide range of questions about the influence of various factors on the value of primary real property.


Introduction
The real property market is a means of redistributing land, buildings, structures and other property between owners and users by economic methods based on competitive demand and supply [1][2][3][4][5].
The main aim of the study was to derive an empirical formula for the relationship between the price of an apartment and its area (as a determining factor). The assumption of a determining factor will be axiomatic in this study, and it is from this factor that one will have to make the necessary calculations. However, in order to broaden the range of the problem under study, in our opinion, at the initial stage it is advisable to carry out a variance analysis of how the location of the studied real property impacts its value, since there is no consensus whether this factor will be significant in modeling the primary housing in the city of Tyumen (Fig. 1). The primary real property market is represented by real property that has not yet been made into ownership. As a rule, those are buildings under construction or newly constructed buildings. The primary real property market will always be formed by the supply of various developers. When it comes to the supply in the primary market, then it should be understood as all the supply in the market of new buildings. The main advantage of the primary real property market is that the apartments are "clean", that is, they have no so-called history [1][2][3][4][5][6]. It should also be borne in mind that the primary real property market involves the sale of houses, the construction of which took into account all the imperfections of design, and applied the latest technology and modern building materials.

Study areas and analyses
In our study, four urban areas are presented. At the end of 2017, the construction of new residential complexes was carried out there: Olympia Residential Complex, Vostochny-2, Zarechny-3 and Yamalsky-2 areas.
In these residential complexes, prices for one-room apartments in completed construction projects of approximately the same type of layout and area were analyzed (30 -60 m²) on the basis of economic and statistical modeling [5,7,8].

Analysis of variance
The results of the analysis of variance are presented in Table 1 (the analysis was carried out for five apartments in each area considered). The analysis took into account that the residential complexes Vostochny-2 and Zarechny-3 were in the last stages of construction, while the residential complexes Olympia and Yamalsky-2 were already built up and are characterized by a completion level of 50-60 %.
As samples, apartments which 70-80 % are suitable for living were selected (some furniture is missing, but communications are installed and sanitary equipment is fully installed, walls and ceilings are finished). We have put forward a statistical hypothesis: the location of the apartment does not have a significant impact on its value.
It was necessary to find out if the hypothesis was correct. Also, the significance level was taken as the probability of accepting an incorrect hypothesis: α = 0.05. The results of the calculations are presented in Table 2. The actually observed value of F is 6.05, which exceeds the critical value of the Fisher-Snedecor criterion at a significance level of a = 0.05 with k1 = 3 and k2 = 16 degrees of freedom F0.05; 3; 1.6 = 3.24. Since F > F0.05; 3; 1.6, we nullified the null hypothesis, that is, at a significance level of a = 0.05 (with a reliability of 0.95), it was assumed that the difference between the areas has a significant impact on the price of the primary one-room housing in Tyumen. Returning to Table 1, we concluded that apartments in Vostochny-2 and Zarechny-3 will have a higher price than apartments in Olympia and Yamalsky-2 residential complexes. This can be explained by the level of completeness of residential buildings, as well as the mass scale of development, since the built-up area is larger in two last districts, therefore, developers have the opportunity to reduce the price due to the scale of the services provided.

Analysis of variational series
Analysis of variance performed above suggests that apartments in residential complexes Zarechny-3 and Vostochny-2 are more expensive because of the economically inexpedient distribution of one-room apartments by their area. This fact led the study to its main phase: the derivation of an empirical formula of the relationship between the price of an apartment and its area. For this purpose, it was necessary at the initial stage to apply the main method of descriptive statistics -the method of analysis of variational series.
For the study, 100 apartments in Olympia and Yamalsky-2 residential complexes were selected by this method and were distributed across the average area (interval -5 m²) in order to determine for which of the given average areas the demand is greatest (since developers are developing areas of apartments under construction according to the level of demand for each of the given area intervals). The data are presented in Table 3.  The table shows that in the considered area interval, there is a gradual increase in the number of apartments to 45 m² and its decrease after passing through this area. It is also obvious that apartments with an area of 45 m² have the largest share of 26 % in the existing structure, which demonstrates a large percentage of the accumulation of specific weight in the interval from 40 to 50 m² (from 40 % to 80 %). For a more convenient presentation of the data of Table 3, a frequency polygon is constructed (Fig. 2). The polyline of the constructed polygon shows the presence of a distinct peak -26 apartments out of 100 considered (more than 40 %) have an average area of 45 m². For the interval grouping of the available data, it was necessary to determine the number of the group interval and the step size of the interval. Our calculations show that these figures will be equal to six intervals, respectively, and the interval spacing corresponds to 5 m². When analyzing the interval series constructed in the form of Table 4, it becomes obvious that apartments with a smaller area are still more popular than apartments with a higher price (more than 45 m²) that may be explained by the desire of potential buyers to save on amenities, due to the lower price. The second stage of the variational analysis is the calculation of the numerical characteristics obtained in Tables 3 and 4. Characteristics are divided into two main groups: average values and indices of variation. The results of the calculation of average values are given in Table 5. The value of the sample average 44.3 m² shows that the average area of the proposed apartments is very close to 45 m². The same fact is confirmed by the same (also 45 m²) modal and median values. The second group of calculated characteristics is the variation indicators; the results of the calculations are presented in Table 6. The values of the indicators allow us to conclude that: firstly, the empirical distribution of apartments by their area is almost symmetrical (the asymmetry factor is only 3 %, which indicates a slight advantage in demand for apartments up to 45 m²); secondly, the value of excess less than 0 (-0.64) shows that this series has no evident peak, i.e. mode does not have much advantage over the other variants of the series. And finally, the value of the variation coefficient of 18 % allows to make a key conclusion that the distribution of the number of new one-room apartment buildings of Tyumen, depending on their area, is subject to a negative-parabolic law. That in turn created the prerequisite for the regression analysis.

Regression analysis
At the initial stage of regression analysis, one should take the first two lines of a discrete variational series of distribution of the number of proposed new one-room apartment buildings in Vostochny-2 and Zarechny-3 residential complexes in Tyumen as the initial tabular dependence between the considered parameters (Table 7). Frequency polygon built according to the above table in Fig. 2 suggests that there is a dependence of the parabolic shape between the values under study. In order to check the hypothesis put forward, it is necessary to use the method of least squares. The calculations are given in the form of Table 8 (the area of apartments is taken for x, and the number of apartments for y).   The final stage of the regression analysis was the comparison of empirical and theoretical curves of the relationship between the studied characteristics. It is shown in Figure 3.
Comparison of the curves presented in Figure 3 confirms that the distribution of the proposed apartments in Zarechny-3 and Vostochny-2 complexes has a parabolic dependence on their area.

Correlation analysis
This analysis involves a comparison of the influence of two main factors on the value of an apartment: price and area. An initial correlation table was compiled, where various options for price and area are given, as well as the number of matching pair frequencies, for each of the possible options. These frequencies were the number of apartments, each of which corresponds to some combination of price and area.
Apartments are located in similar areas (new buildings) and have identical physical characteristics. The area range of apartments is set from 30 to 60 m 2 , the price range is from 1,800 to 2,800 million rubles. We present these data in Table 9. Table 9. Comparison of the number of apartments with their price and area.
The basis for the study was 100 apartments, which made it possible to comprehensively consider this issue, eliminating the possible error associated with the sample size. In this table, there is a tendency: the larger area apartments have, the more expensive they are. The task was to identify, on the basis of this obvious fact, the extent to which the indicators under consideration are interrelated.
To test the hypothesis about the close relationship between area and price, you can calculate the correlation coefficient (Table 10). The obtained correlation coefficient is 0.57, which indicates that the degree of relationship between these indicators is characterized as average.

Conclusion
1. The hypothesis put forward at the beginning of the study is rejected on the basis that the actually observed value of F exceeds the critical value of the Fisher-Snedecor criterion.
2. The location of apartments has a significant impact on the value of apartments, as evidenced by the data in Table 1. As a conclusion, it follows that that apartments in Vostochny-2 and Zarechny-3 districts will have a higher price than apartments in Olympia and Yamal-2 residential complexes.
3. Methods of mathematical statistics allowed in a specific mathematical form to answer a wide range of questions about the influence of various factors on the value of primary real property: location, urban planning solutions, the presence and condition of communication networks, the technical condition of the building, environmental factors [2,3,5].