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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.1 pp.59-69
DOI : https://doi.org/10.7232/iems.2020.19.1.059

# Correlation Calculation of Forming the Model Energy-Efficient Production

Murat M. Ospanov*, Assel T. Uskelenova, Kairatbek Kh. Shadiyev
Department of Finance and Accounting, University of International Business, Almaty, Republic of Kazakhstan
Department of Logistics and Management, Kazakh Academy of Transport and Communications, Almaty, Republic of Kazakhstan
Department of Jurisprudence, Abai Kazakh National Pedagogical University, Almaty, Republic of Kazakhstan
*Corresponding Author, E-mail: murat.osp@murdoch.in
November 6, 2019 January 9, 2020 January 14, 2020

## ABSTRACT

The basic principle of any industrial cooperation is first to build an effective interaction between the cost of production and the potential profit from its implementation. If the sale of products is unstable or cannot always be admitted to the market, there is often a decrease in the activity of the enterprise. At the same time, competitive advantages are gained by the production, which forms its activity on the principles of cost reduction and increase of output per unit of cost. The aim of the article is to create a model of energy efficiency and show aspects of its application in industry. To accomplish this task, the article uses the price research method. The novelty of the work is determined by the fact that as a result of research, the article shows the aspects of the formation of effective industrial cooperation based on the integration of production into the global economic environment, which in turn is aimed at increasing the competitiveness throughout the world. The practical significance of the study is determined by the fact that it allows creating a model of transition to the innovative forms of production based on energy saving.

## 1. INTRODUCTION

Scientific-methodical and practical interest are the tools of forecasting and determining the conditions of applying the developed models. The peculiarity of their use is to check for compliance with several criteria that indicate the reality of the predicted situations and missing errors, improve the quality of management decisions on planning the need for energy consumption for production, which will ensure the intensification of the development of the industrial enterprise (Pfau et al., 2017). To solve the problems of forecasting, it is expedient to conduct a detailed study of the content of the forecasts and features of building predictive models for the formation and implementation of economic decisions in the field of organizational and economic support of energy-efficient behavior of an industrial enterprise.

Implementation of economic processes in industrial enterprises aimed at the production of the desired benefits for society in the form of goods and services, is based on the consumption of several resources that are transformed into products in quantity and quality, due to the technical and technological, economic, social and institutional conditions and features of the organization of the production process. The basis for the formation of these conditions is the availability and accessibility of energy resources, which are objects of energy storage and supply, the use of which is necessary to perform several mechanical, organizational and other processes to meet the needs of consumers.

The determination and granting of the availability status for energy resources depends on the level of use of the scientific and technical progress’s achievements in the economic activity of industrial enterprises, namely – the level of innovation and progressiveness of equipment for the production of energy resources, their use for production purposes, the volume of energy consumption in the processes of consumption of products, its further recycling and recycling of waste for re-involvement in the production process. The peculiarity of energy resources in the modern world practice is their acquisition of the status of an object of both demand and supply. Energy resources lay the basis for the formation of the economy of many countries, determining the level of stability and exchange rates of national currencies that determine the level of well-being of the population through the creation of jobs, pricing processes and intensity of consumption in the production of goods and services (Dolgikh, 2018).

The peculiarity of energy is that it exists objec-tively in the environment, regardless of the knowledge and technology used to meet social needs (Bretzke and Barkawi, 2013; Abdilda et al., 2018). The evolution of technologies has allowed the company to attract various sources of energy into the economic circulation, because energy is a condition and a factor in the implementation of technological processes, and the level of its use and the degree of involvement in economic turnover is determined by the degree of progressiveness of society from the standpoint of technological criteria, the pace of scientific and technological progress and consumer needs (Zelenyak and Kostyukov, 2018). At this time, the economies of the leading developed countries, based on the extraction of energy resources and materials, their processing and sale of secondary energy resources, form the level of national welfare due to the intensity of energy consumption. Therefore, the focus of modern society on increasing energy consumption and expanding the range of real and potential sources of energy is determined by the desire to accumulate wealth, strengthening economic independence and economic sustainability, aimed at gaining control over the economies of developing countries.

## 2. LITERATURE REVIEW

An important condition for strengthening the processes of energy consumption at present is the rapid growth of the population, which requires more products for consumption and leads to the intensification of industrial production. At the same time, there is an exponential relationship between the rate of population growth and the volume of energy consumption in the world. Accord-ing to economists, the global demand for energy, on the one hand, allows society to realize and expand its own needs while increasing the level of satisfaction, relative solvency, but, on the other hand, increases the uneven economic development and provokes poverty in countries that are energy dependent on foreign energy producers (Nakajima et al., 2001).

As an attempt to solve the energy needs of society there were many concepts of further development of society, the leading place among which is the concept of sustainable development, which arose in the 1970s. The central idea of this concept is to meet the current needs of society without the threat of reducing the volume of meeting the needs of future generations. As the researchers note, the need for this concept results from the closeness of the natural systems on a planetary scale, hence the impossibility of recovery of the natural resource potential of the Earth due to the circulation of energy ‘around the circle’ and thus raises the limitations and the exhaustion of natural resource potential (Wicaksono et al., 2013; Seitzhanov et al., 2018).

First, this problem concerns energy resources and materials that are dominant for performing physical work, that is, setting in motion machines and mechanisms intended for the implementation of economic tasks. The peculiarity of energy at the present stage of development of production forces and industrial relations is the use of the professional and personal practice, which determines the cost of consumption of goods and services, their quantitative volume depending on price characteristics. In the conditions of absolute growth of needs and physical quantity of consumers, there is a question of determining the limits of growth of energy consumption and the feasibility of reducing its consumption per unit of output (Shelest, 1987). In this case, the law of diminishing marginal utility will make it possible to estimate the possible and sufficient volumes of energy consumption, provided that it is used at a certain level of scientific and technological progress, which determines the level of technology development, the progressiveness of equipment and possible production under various combinations of production factors (Wang et al., 2019; Seidaliyeva et al., 2018). Therefore, it is appropriate to use a production capability curve that characterizes the possible combinations of pro-duction factors (usually two) and potential output vol-umes that will require both full and incomplete use of available resources (Redwood et al., 2008).

Within the framework of this law, it can be as-sumed that energy consumption is a variable factor, logically explained by the measure of its readiness for economic circulation. Thus, with an increase in production volumes with constant technology, an absolute increase in energy consumption to produce goods and services can be expected. In the opposite direction, as the level of progressivity and efficiency of the equipment involved increases, energy consumption may increase as production increases (Jackson, 1982). Thus, according to the law of decreasing marginal productivity, the growth of energy consumption (with unchanged consumption of other factors and resources of production) with a constant increase in production in the future can lead to a situation in which each additional unit of the energy consumed will lead to a reduction in the physical volume of production of goods and services (Medvedeva et al., 2016). Under these conditions, the desire of society to maximize the satisfaction of needs by attracting more energy is a danger of narrowing the level of satisfaction of consumer needs, and in accordance with the law of supply and demand of limited resources will cause an increase in the price of the latter, which entails an increase in prices. Therefore, the issue of optimization of energy consumption, determining an efficient strategy of energy saving, energy saving in the conditions of growing needs becomes a priority in the current business environment (Pathak et al., 2016).

First, the solution of energy saving and energy saving issues requires the stability of the terminological apparatus, which necessitates the study of the essence and economic content of energy and its derivatives or prerequisites for obtaining (Giarini and Stahel, 1993). In general, energy is the substance by which impulses are generated to perform physical work (Randolph and Masters, 2018; Uskelenova et al., 2017b). The economic importance of energy is manifested in the objective need to set in motion the machines and mechanisms that form the basis of economic processes aimed at meeting professional and personal needs, the implementation of professional functions and the organization of private life. The process of obtaining energy, its cost depends on the degree of development of technology, human knowledge of the laws of functioning and interaction of natural forces (Vagenina, 2015). If energy is abstract, it is specified through the prism of relations in the production and use of energy resources and materials that have a quantified economic value, and the conditions of consumption depend on the purpose and specifics of management (Wellmer et al., 2019).

In general, energy materials and resources can be represented as sources of energy of mechanical, chemical or physical nature, through which the stages of production and marketing processes are implemented, ensuring the transformation of fixed assets and working capital into finished products (Ericson and Ickes, 2003). But such interpretation allows estimating more physical characteristics of participation of energy resources and materials in the production process through natural indicators and coefficients of use of materials per unit of production. Interesting is the cost characterization determined by the level of availability, resource constraints, the level of STP and R&D results. Therefore, the measure of participation of energy materials and resources depends on the specifics of the economic activity of the industrial enterprise, determining the level of cost of production and profit from sales (Costanza et al., 2017; Uskelenova et al., 2017a). Therefore, the economic meaning of attracting energy materials and resources to an industrial enterprise can be considered from the standpoint of creating a part of the added value, which shows the excess of the value of the products over the value of all the resources used for its production and marketing (Rose, 1986). This approach to the interpretation of the economic content of energy materials and resources allows assessing the economic activity of an industrial enterprise and comparing with previous periods of management or other enterprises to determine the effectiveness and results for management decisions in the field of production and marketing, pricing and determining the reserves to improve financial and economic indicators.

## 3. MATERIALS AND METHODS

Implementation of production processes requires the use of many economic resources. Volumes, types, and conditions of their consumption are determined by the specifics of technological processes, the implementation of which ensures the production of goods with certain consumer characteristics. An important characteristic is the price of the products, which is derived from the value of the two basic components – cost and profit. The value of the cost of production depends on the physical and price characteristics of the resources used, determined by the volume of production. If the amount of profit is formed mainly under the influence of market conditions of sale of the manufactured products and therefore is less sensitive to the management influences from the enter-prise, the value of the cost of production is the object of direct impact, sensitive to the economic decisions taken within the enterprises regarding the choice of suppliers, the amount of necessary resources and production. Thus, the volume of resources used, and the economic and organizational conditions of their consumption are of paramount importance in ensuring the effective operation of the enterprise.

The consumption of energy materials and re-sources in the production process can be represented as a continuous process according to the mission of the enterprise, the use of energy to manufacture the products with characteristics that correspond to the ideas of consumers about the highest degree of satisfaction of needs that allow its sale on the market. To determine the degree of efficiency of economic activity from the standpoint of energy consumption, it is advisable to divide the latter into two categories – primary and derivative. Primary energy is represented by natural resources of energy content (coal, oil, water, gas, etc.). The derived energy is the energy derived from the processing of primary energy (electrical energy, steam, gasoline, fuel oil, etc.) (Semenyutina et al., 2018). Production of primary energy is realized by enterprises of the mining industry whose mission is to form the raw material base for further pro-cessing of extracted natural resources to make a profit. Secondary energy is a product of the processing industry, the content of the mission of which can be disclosed as the production of energy resources for enterprises for which secondary energy is either the final product of production, which is released to third-party consumers or an intermediate product, acquires the properties of a resource to meet their own needs for energy resources.

In the conditions of market instability, which is manifested by the growth of competition and rapid change of consumer needs, enterprises are faced with the need to optimize the production and consumption of energy resources to reduce the cost of production while maintaining or increasing the level of profit. There is a need to make several decisions to establish sufficient and necessary volumes of use of limited energy resources, the cost of which tends to increase over time. The need for energy resources is determined by external and internal factors. That leads to the use of the price method of research.

## 4. RESULTS AND DISCUSSION

Traditionally, Gross Domestic Product (GDP) has been interpreted as the value of all final goods and services produced within a country by factors of production, whether those factors are resident or foreign. That is, when measuring GDP, the value of goods produced by enterprises, organizations, institutions within the country is considered, regardless of whether these domestic enterprises are controlled by foreign capital, and vice versa, the income received by author’s compatriots abroad is not considered in GDP.

This indicator is a dependent value, which is in-fluenced by several factors, where one of the main is the volume of industrial products sold. Therefore, in GDP as x the volume of industrial products (goods, services) is taken. To build a single-factor model, the indicators of the volume of industrial products (goods, services) sold (in millions of dollars) given below are used (Figure 1).

Using the equation of the trend line, the theoretical value of the indicator is calculated:

$y t i = b 0 + b 1 x i$
(1)

The coefficients are calculated based on the formulas (1) and (3).

$b 1 = 1 n ( ∑ i = l n x i y i ) − x ¯ y ¯ 1 n ∑ i = l n x i 2 − x ¯ 2 = 1.175$
(2)

where $y ¯ x ¯$ – average value of indicators.

$b 0 = y ¯ − b 1 x ¯ = − 92603.974$
(3)

When using formula 3, the results of calculating the theoretical value of the yt index can be presented as follows (Table 1):

Graphs of actual and theoretical economic indicators are shown in Figure 2.

Let us find the density of the relationship between the dependent value of y (GDP) and independent x (volume of industrial products (goods, services)). To do this, the following correlation coefficient is used:

$r y x = 1 n ∑ i = 1 n ( x i − x ¯ ) ( y i − y ¯ ) 1 n ∑ i = 1 n ( x i − x ¯ ) 2 1 n ∑ i = 1 n ( y i − y ¯ ) 2$
(4)

The correlation coefficient is a relative measure of the relationship between the two factors. Therefore, the value of the correlation coefficient is always within $( − 1 ≤ r y x ≤ 1 )$. A positive value of the correlation coeffi-cient indicates a straight line, and a negative value indi-cates feedback between variables. The calculation of the correlation coefficient is presented in Table 2.

Based on the calculations presented in Table 3, the correlation coefficient is 0.98951. Since the correla-tion coefficient tends to an absolute value of 1, this indicates that there is a strong relationship between GDP and the volume of industrial output sold. The coefficient of determination is calculated by the following formula:

$R 2 = ∑ i = 1 n ( y t i − y ¯ ) 2 ∑ i = 1 n y i − y ¯ 2$
(5)

Calculation of the coefficient of determination is presented in Table 3.

The coefficient of determination shows the proportion of variation of the productive characteristic under the influence of the studied factors. In this case, the coefficient of determination is 0.9793. And so, about 97% of the variation of the dependent variable is considered in the model and is due to the influence of the included factors.

Checking the model for adequacy according to F-test consists of certain stages:

• 1) At the first stage, the value of so-called F-ratio is calculated:

$F ( 1 , n − 2 ) = ∑ i = 1 n ( y t i − y ¯ ) 2 / 1 ∑ i = 1 n ( y i − y t i ) 2 / n − 2$
(6)

where $n$ – sample size.

• 2) At the second stage, the level of significance λ is set.

• 3) In the third phase, by the statistical tables of F-distribution with $( 1 , n − 2 )$ degrees of freedom and significance $100 ( 1 − λ ) %$, the critical value $( F k p )$ is found.

• 4) If the value is calculated as $F > F k p$, then the hypothesis $H 0$, that $β 0$ (or that $y ^ − y ¯$), is rejected with the risk of making a mistake in no more than 5% of cases.

The F-test with $λ = 0 , 05$ is calculated. In this case, $H 0 : β 1 = 0$, alternative hypothesis $H 1 : β 1 ≠ 0$. $F ( 1 , n − 2 ) = 328.45$. Considering that $F > F k p$ (328.45>236.768), then constructed regression model is adequate to reality.

T-test for testing the significance of parameters and found according to the least-squares method. To do this, the variance estimate using the formulas (7) and (8) is calculated:

$σ ^ b 0 2 = ∑ i = 1 n e i 2 n − k ∑ i = 1 n x i 2 n ∑ i = 1 n ( x i − x ¯ ) 2$
(7)

$σ ^ b 1 2 = ∑ i = 1 n e i 2 n − k 1 n ∑ i = 1 n ( x i − x ¯ ) 2$
(8)

where $k$ – number of estimated parameters (in case of simple regression $k = 2$); $n$ – sample size; $σ ^ b 0 2 , σ ^ b 1 2$ – estimates of parameter variances $b 0$ and $b 1$ parameter variances (by regression);

In econometrics, a common form of null-hypothesis is as follows:

$N 0 : β 1 = 0$
(9)

where $β i$ – actual parameters of the entire assembly.

Opposed to the alternative one

$H 1 : β 1 ≠ 0$
(10)

This criterion is two-dimensional.

In this case, t – the statistics for the parameters have the following form:

$t * = b i σ ^ b i$
(11)

The results of the calculation of T-test at the level of significance $λ = 0.1$ for the relevant indicators are presented in the following form:

$σ ^ b 0 = 78920.76965 σ ^ b i = 0.064855736 t b 0 = − 1.173379013 t b 1 = 18.1231931 t λ / 2 = 2.36$

Based on the calculations, the following conclusion can be made. $t b 0$ is equal to (-1.173379013), which falls within the interval [-2.36; 2.36].

With probability $( 1 − λ ) = ( 1 − 0.1 ) = 0.9$, it can be concluded that the estimate $b 0$ is not statistically determined, so it is possible to ignore it. $t b 1$ does not fall within this interval, so the coefficient $b 1$ is statistically significant.

To determine how the parameters $b 0$ and $b 1$ are connected with $β 1 , β 2$ it is necessary to construct confidence intervals for the parameters of the generalized regression model, i.e. such intervals that cover their value with a given probability. The confidence interval is calculated using the following formula:

$b i − t λ / 2 σ ^ b i < β i < b i + t λ / 2 σ ^ b i$
(12)

The results of the confidence interval calculation are presented as follows:

For coefficient b0 within $b 0 − t λ / 2 σ ^ b 0$, the value is –278856.9912; within $b 0 + t λ / 2 σ ^ b 0$, it is 93649.04159.

For coefficient $b 1$ within $b 1 + t λ / 2 σ ^ b 1$ the value is 1.022333497, and within $b 1 + t λ / 2 σ ^ b 1$ it is 1.328452573.

Coefficient b0 according to the above calculations is insignificant. This conclusion is also confirmed by the fact that the trust interval includes zero.

Confidence interval [1.0223; 1.3284] for coefficient $b 1$ with probability 0.9 covers the unknown parameter $β 1$ of the entire assembly.

A multi-factor dependence of the volume of industrial products (goods, services) $( y )$ will be built from the use of fuel and energy resources for production and operational needs $( x 1 )$, the degree of depreciation of fixed assets of the enterprises $( x 2 )$ and capital invest-ment in industry $− ( x 3 )$. This dependence can be repre-sented as follows:

$y t = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3$

To do this, the data presented in Table 4 is used.

To find the coefficients of the trend line equation, the least-squares method is used. The calculations are presented in Table 5.

According to the results of mathematical operations, the coefficients of the trend line equation are represented by the following series: {1647354; -9387.8; 1054.407; 10.10861}.

Considering the obtained data, the trend line equation has the following form:

$y t = 1647354 − 9387.8 x 1 + 1054.407 x 2 + 10.10861 x 3$
(14)

The correlation coefficient for the multifactor model is calculated as follows:

$R = ∑ i = i n ( y i − y ¯ ) ( y ¯ i − y ^ ¯ ) ∑ i = 1 n ( y i − y ¯ ) 2 ∑ i = i n ( y ^ i − y ^ ¯ ) 2$
(15)

This coefficient is equal to $R = 0.94284$, which indicates a high density of the relationship between the coefficients. The coefficient of determination calculated by the formula (14) was $R 2 = 0.88895$. That is, about 88% of the variation of the dependent variable is considered in the model and is due to the influence of the included factors. To check the adequacy of the model, F-test is used. In this case, the null hypothesis is generalized and has the following form: $H 0 : β 1 = β 2 = … = β p = 0$ opposed to the alternative hypothesis H1: at least one value $β 1$ other than zero. To test the hypothesis $H 1$, $F$-Fisher statistics is calculated with $( n − p − 1 )$ degrees of freedom $( n = 9 , p = 3 )$:

$F p , n − p − 1 = R 2 / p ( 1 − R 2 ) / ( n − p − 1 )$
(16)

According to the Fisher tables, critical value is found as 9.01. And here F-test at $λ = 0.05$ is equal to 13.341. As $F p , n − p − 1 > F к р$ (13.341>9.01), then constructed regression model is adequate to reality. The T-test is calculated from the following formulas (17) and (18):

$t b i = b i σ b i$
(17)

$σ b i = σ e b i i$
(18)

where $b i i$ – diagonal element of the matrix $( X T X ) − 1$.

$σ e 2 = ∑ i = 1 n e i 2 n − p − 1$
(19)

Null-hypothesis $H 0 : β i = 0$ is opposed to the alternative one $H 1 : β i ≠ 0$. The critical value is found from the corresponding tables $( t k p m = 2 , 5706 )$. An equal level of significance 0.05 is taken. Thus, $t b 0 = 3.08254 , t b 1 = − 4.1243 , t b 3 = 3.14084$. This means that the corresponding coefficients are statistically significant $( t b 0 > t k p m , t b 1 > t k p m , t b 3 > t k p m )$. But $t b 2 = 0.18897$. Since this figure is less than $t k p m = 2.5706$, then with probability 0.9, coefficient b2 may be considered as insignificant.

Confidence intervals are calculated by formula (10). The intervals of the coefficients $b 0 , b 1 , b 2 , b 3$ are as follows: [273591.2; 3021117.2], [-15238.98; -3536.6], [-13288.6; 15397.4], [1.835; 18.381] respectively. With probability 0.95 these intervals cover the unknown parameter of the entire assembly $β 0 , β 1 , β 2 , β 3$ respectively. Coefficient $b 2$ is negligible, therefore the corresponding confidence interval includes zero. The pair correlation coefficient between the variables $x 1$ and $x 3$ is calculated to test the model for multicollinearity:

$r x 1 x 3 = 1 n ∑ i = 1 n ( x i 1 − x 1 ¯ ) ( x i 3 − x 3 ¯ ) 1 n ∑ i = 1 n ( x i 1 − x 1 ¯ ) 2 1 n ∑ i = 1 n ( x i 3 − x 3 ¯ ) 2 r x 1 x 3 = − 682570.977 4120.915 × 2.679 × 10 9 × 8.939 × 10 11 = − 0.20544$
(20)

As $| r x 1 x 3 | = 0.20544 < 0.7$, then multicollinearity between variables $x 1$ and $x 3$ is absent.

The significance of pair correlation coefficients is calculated by T-test:

$t t a b l e = r ( 1 − r 2 ) n − p − 1 t t a b l e ( r x 1 x 3 ) = − 0.75951 ( 1 − 0.75951 2 ) n − p − 1 = 0.082$
(21)

For the level of significance $λ = 0 , 1$ from the corresponding tables $t k p τ = 1.94$. The found value of $t t a b l ( r x 1 x 3 ) = − 0.082$ is within [-1.94; 1.94]. Therefore, it can be concluded that the null hypothesis of the absence of multicollinearity between variables $x 1$ and $x 3$ is accepted.

It is impossible to check heteroscedasticity using Goldfeld-Quandt test, as there is little data. So, the Spearman test is conducted considering the following stages:

• 1) The sample is ordered by factor x. The ranks of x are calculated.

• 2) Using the regression equation:

$y i = a 0 + a 1 x 1 i + a 2 x 2 i + ∑ ​ i$
(22)
the remainders are calculated:

$e i = Y i − a 0 − a 1 x 1 i − a 2 x 2 i$
(23)

• 4) The Spearman rank correlation coefficient between the ranks of factors $x$ and the remainders $e$ is calculated:

$r x , e = 1 − 6 ∑ ​ D i 2 n ( n 2 − 1 ) r x 1 , e = 1 − 6 × 98 9 ( 81 − 1 ) = 0.1833 r x 3 , e = 1 − 6 × 134 9 ( 81 − 1 ) = − 0.1167$
(24)

where $D i$ – difference between ranks $x$ and $e$.

• 5) The statistics is calculated

$z = r n − 1 z x 1 , e = 0.1833 × 8 = 0.518 z x 3 , e = − 0.1167 × 8 = − 0.33$
(25)

This statistic is subject to distribution $x 2$. The corresponding tables provide $x k p m 2 = 15.51$. I.e. $| z x 1 , e |$ and $| z x 3 , e |$ are less than $x k p m 2$. Therefore, the null hypothesis of the absence of heteroscedasticity is accepted. To check the presence of autocorrelation, the Durbin-Watson model is used, which is calculated as follows:

$D W = ∑ i = 1 n ( e i − e i − 1 ) 2 ∑ i = 1 n e i 2 D W = 99266484154 2 , 66845 × 10 11 = 2.68817$
(26)

Null-hypothesis $H 0$: there is no first-order au-tocorrelation between the remainders of the regression model and opposed to alternative one $H 1$: first order autocorrelation between the remainders of the regression model takes place. Tables provide the upper $d 2$ and lower $d 2$ of the critical point. As $D W > d 2$ or 2.68817>2.5, then null-hypothesis about the absence of first-order autocorrelation between the residuals of the regression model is accepted. As the function and the factors are most often measured in different units, then to eliminate the size variation and to estimate the correlation between each factor and the studied index by using the relative values the coefficient of elasticity is determined. The latter shows how much the function will change with the change of a certain factor by 1% with a fixed (average) value of other factors. The elasticity coefficients are calculated as follows:

$E x i = ∂ f ( x 1 , x 2 , … , x n ) ∂ x i x ¯ i y ¯ E y x 1 = − 9387.8 ( 1305.7 9 ) 10578127 9 = − 1.1587 E y x 3 = 10.1086 ( 727453 / 9 ) 10578127 / 9 = 0.6951$
(27)

Thus, if the indicator ‘use of fuel and energy re-sources for production and operational needs’ changes by 1%, the presented function will change by 1.1587%. And with the change in capital investment in industry (1%), the function will change by 0.6951%. Thus, the change in the first indicator affects the volume of industrial products (goods, services) sold more than the second one, since 1.1587%>0.6951%.

## 5. CONCLUSION

According to the results of the trend analysis, it can be concluded that the gross domestic product is a dependent value, this indicator is influenced by several factors, one of the main is the volume of industrial products (goods, services) sold. Based on the presented indicators, a single-factor model is built. The calculation of the close relationship between the dependent value y and the independent x (the correlation coefficient is 0.98951) indicates that there is a strong relationship between GDP and the volume of industrial output sold. The coefficient of determination is 0.9793. And so, about 97% of the variation of the dependent variable is considered in the model and is due to the influence of the included factors. F-test $λ = 0.05$ is equal $F ( 1 , n − 2 ) = 328.45$. As $F > F k p$ (328.45> 236.768), then constructed regression model is adequate to reality. In the calculation of T-test, value $t b 0$ is equal to (1.173379013), which falls within [-2.36; 2.36]. Therefore, with probability $( 1 − λ ) = ( 1 − 0.1 ) = 0.9$, one may conclude that estimate $b 0$ is not statistically defined, so it is possible to ignore it. Value $t b 1$ does not fall within this interval, so the coefficient $b 1$ is statistically significant. The calculation of the confidence interval for coefficient $b 1$ was equal to [1.0223; 1.3284], i.e. with probability 0.9 it covers an unknown parameter $β 1$ of the entire assembly.

To build a multi-factor regression model, the following indicators were used: the volume of industrial products (goods, services) sold as a dependent value $( y )$, and the use of fuel and energy resources for production and operational needs, the degree of depreciation of fixed assets of enterprises and capital investments in industry is independent $( x )$. The correlation coefficient was $R = 0.94284$, which indicates a high density of the connection between the coefficients. The coefficient of determination indicates that about 88% of the variation of the dependent variable is considered in the model and is due to the influence of the included factors. T-test showed that the coefficients $t b 0 , t b 1 , t b 3$ are statistically significant, except for $t b 2$. The calculation of even coefficients gave the value $| r x 1 x 3 | = 0.20544 < 0.7$, which indicates that there is no multicollinearity between the variables $x 1$ and $x 3$. It is impossible to check heteroscedasticity using Goldfeld-Quandt test, as there is little data. So, the Spearman test shows the absence of heteroscedasticity.

To check the presence of autocorrelation, the Durbin-Watson model is used. As $D W > d 2$ or 2.68817 >2.5, then the basic hypothesis of no first order autocorrelation between the residuals of the regression model is accepted. The coefficient of elasticity showed that with a change in the indicator ‘use of fuel and energy resources for production and operational needs’ by 1%, the presented function will change by 1.1587%. And if the capital investment in industry changes (by 1%), the function will change by 0.6951%. Thus, the change in the first indicator affects the volume of industrial products (goods, services) sold more than the second one, since 1.1587%>0.6951%.

## Figure

Realized Industrial Products within the Framework of Industrial Enterprises in the Akmola Region, mln.

Graphs of theoretical and current gdp dependence on the volume of industrial output sold.

## Table

Calculation of the industrial products’ potential sales volume for the akmola region, million dollars

Correlation coefficient

Coefficient of determination

Key Indicators for Constructing Multivariate Regression

Transpose of Matrix (XT)

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