Mathematical Model on Regional Economic Vitality

Economic vitality is an important indicator of economic development. In this paper, we have created a panel data model to analyze the influencing factors of economic vitality in China. We conducted the following studies. Using the correlation analysis, the correlation among various elements is revealed to be strong, which denies the independence null hypothesis. By establishing VAR-VEC model, the long-term and short-term effects of economic policies on economic vitality in Beijing are analyzed. It is showing that the population changes and enterprise vitality have a positive impact on economic vitality with the influencing factors being 0.01 and 0.07, respectively. At least three cointegration relationships between time series exit using the ADF unit root test and Johansen cointegration test. We use Ais-Sc Criterion to determine the order of delay as the third order and OLS estimation method to get the coefficients of VEC Model. Because of experience accumulation, the economic vitality follows a W-shaped trend. Utilizing the minimum average deviation method to preprocess the index data, 9 representative indexes are obtained. We then extract two main factors by factor analysis and build an index system of economic vitality. The economic vitality of each city from 2009 to 2020 is calculated based on this index system. Beijing, Shanghai, Guangzhou and Shenzhen often rank first, while Kunming and Dongguan often rank last. The panel data model test results are like that of index system on the same data. Finally, the previous conclusions have been reviewed. The ORT development strategy to improve economic vitality is advised. Biomed

To accelerate the narrowing of the gap in local economic growth, promote the vigor of regional economic growth, and promote the coordinated growth of a regional economy is the basis and key to solving the main social contradictions in the new age, and is also the driving force of economic and social development axis. Regional economic vigor is a part of comprehensive local competitiveness.
In recent years, to improve economic vitality, some regions have introduced a lot of preferential policies to stimulate economic power, such as reducing the approval steps for investment, providing financial support for entrepreneurship, and lowering the threshold for settling down to attract talents. However, due to distinct resource endowments, these policies have distinct affects in distinct regions. How to grasp the key factors and effectively improve it is a worthy research topic.
To study how to improve the regional economic vitality, this paper takes some cities in China as an example, selects several indicators to measure it, constructs the index system, and studies the relationship model of the influencing factors of it and considers the influence of population change trend and enterprise vitality change on the regional economic vigor change. At the same time, it analyzes the short-term and long-term impact of economic policy transformation on the economic vitality of various regions. At present, scholars at home and abroad rarely make more research on economic vitality, so this paper hopes to establish a mathematical model to analyze and measure regional economic vitality, and sort DOI: 10.26717/BJSTR.2021.34.005536 the economic vitality of some cities, to better extend the economic vitality analysis model to more research fields.

Panel Data Model
Based on the panel data model, it collects data from various provinces and cities, performs correlation test and principal component analysis on the data. The fixed effect test and random effect test was carried out for the obtained factors. And the influence of policy and enterprise vitality on economic vigor was dissected based on the established relationship the model between each element and economic vigor.
Data Analysis and Processing: Based on the collected data has error and deficiencies, to reduce the invalid, the influence of the error data of the following model, improve the reliability of data, need to collect the data pretreatment, firstly the filtered data, remove abnormal data, secondly, proper supplement of incomplete data, finally, has correlation data linear regression analysis forecasting and slight fluctuation data using the moving average method to fill the missing value, to further improve the accuracy and the integrity of the data. a) Data Selection Principle: This paper needs to collect alien indicator data describing economic vitality and influencing economic vigor, and the following classical indicators can be obtained according to the expert method and the literature [10,11,13,14]. Dependent variable. In the existing economic vitality research and analysis, more choose gross domestic product (GDP) as a measure of it. In this paper, to measure regional economic power, main elements from the effects of the economic vigor that reflects the GDP growth rate as the level of economic development during the period of change degree of dynamic indexes, namely whether a national economic basic index of the moving and USES the linear regression analysis and panel data model analysis, the main measures for regional economic vitality. Independent variables. Based on the existing literature research outcomes and the above analysis, this paper selects nine aspects. It includes population growth rate, fiscal expenditure, and employment rate (mainly used to reflect the main influencing factors of local economic vitality and its growth trend). The employment rate is expressed by the number of unemployed; At the same time, in the establishment of the model, for the negative value of population growth rate, to reduce the error in the large number region. Dummy variables can be used instead of the original statistical samples, which are reset to zero in this paper.
Control variables. Based on the analysis of the comprehensive evaluation index system of urban economy. Considering the availability of data, this paper introduces independent innovation ability, per capita length of education, professional and technical talent inflow and other irrelevant variables as control variables.
Through analysis, the variables other than independent variables that can affect the change of dependent variables should be well controlled and regarded as constants to obtain appropriate causal relationship and attain the accurate value Table 1.  Table 3. It can be seen from Table 3 that the degree of freedom is the probability of Person chi-square, which is less than 0.05.
And the null hypothesis is rejected. The influencing factors are not independent of each other.  The number of Valid observations 309309 0 c) Correlation Analysis: Each factor in the collection is the indicator data of each city in the country, which belongs to the panel data. There may be a correlation between the data.
Considering the correlation among various elements, the linear strength relationship diagram of each element is attained based on the data as follows: As can be seen from the observation in Figure   1, there is a correlation among all factors, and the expression form and strength of the relationship among all elements. The closer the data is to 1, the stronger the correlation is. Local GDP is positively correlated with Government expenditure Gross income from international tourism Consumer price index Education funds Total corporate profits Population Unemployment and added value of the tertiary industry, and negatively correlated with the number of patent applications. SPSS was used to conduct a correlation analysis on the data and the outcomes were shown in Table 4.   Table 5. According to the above correlation analysis Table 5, there is a correlation among all factors.
And the positive correlation coefficient is distributed between 0.5 and 1 reflecting a strong correlation. According to the significance test of the correlation coefficient, the significance values are all less than 0.05. It indicates that the correlation coefficient has reached a high level of significance. Therefore, there is a strong correlation between various elements are influencing economic vitality.  The panel data model can be further divided into fixed effect model and random effect model.

a. Fixed Effect Model
The individual affect is regarded as a stable factor that does not change with time, then equation one can be expressed as a vector.

b. Random Effect Model
The individual affect ai is regarded as a random factor that changes with time. By using the random effect model, the long-term elements and short-term elements in the variance can be separated.
The setting of the model is as follows:  It can be seen from the output that the parameter estimation variance of random and fixed-effect models under this test is a positive definite matrix, which satisfies the test conditions.
Under the 95% confidence interval, the P-value is much less than 0.05. Therefore, the fixed-effect model should be opted as the A. Data stability and reliability analysis: The data of this paper comes from China National Statistical Yearbook, which includes the local government's financial expenditure, the total income of local international tourism, consumer price index, total profits of enterprises, population, unemployment, tertiary industry, total patents, and local GDP. The inconsistency of the order of magnitude of each part will cause trouble to the model fitting.
According to the statistical yearbook, the city is divided into 1-31.
The distribution of various data is shown in Figure 3. Take Figure 3 for example, standardize it first. Assume that the original data is xm, after standardization is Xm, and Xni After attaining standardized data, it is shown as follows. It can be seen from the observation Figure 4 that after the standardization, the feature expression is clearer, which is conducive to the next model inspection work.   The year (2009-2020) is the cross-section marker, the province (1-31) is the research individual. And each type of independent variable is the influencing factor. The solution is based on Stata software, and the outcomes are shown in Table 7. Among them, the F value is very close to 0, indicating that the fixed effect is very significant in this case. Among the seven independent variables, the consumer index and unemployment rate are not significant within the 95% confidence interval. Local government expenditure, total tourism income, total profits of enterprises, resident population, and tertiary industry income have statistical significance. The statistics are shown in Table 8. Among them, the third industry has the most significant impact on GDP, and the consumer the index has the least influences on GDP. We can know that all the opted indicators have positive significance for GDP growth within the statistical range. It shows that this test has adopted hypothesis one and hypothesis two for panel data. And both are true.

Random Effect Model Test Based on GLS Estimation
The number of indexes (N) is 10 and the period (T) is ten years.
In this case, it is also possible to meet the random effect model; the further test of the random effect model is needed.
Cov a χ = Hypothesis 5: According to the above assumption, suppose that the distribution of each independent variable is constrained in a specific case. And the effect of each independent variable obeys the mean value of 0. The second is the description of random interference, which is not correlated with explanatory variables. The third term makes the two coefficients independent of each other. Based on the above description, the GLS estimation method can be used to obtain whether the panel data model conforms to the random effect test when the collected variables are close to the period. Organize data into long data types. The year (2009-2020) was used as the cross-section marker, the province (1-31) as the study individual, and each type of independent variable as the influencing factor. Use Stata software to solve the problem and get the outcomes, as shown in Table 9. In 95% confidence interval, P value is 0 five hypotheses are passed in this case. This case is suitable for the random effect model (Table 10). The third industry has the most significant impact on GDP, and the resident population has the least influence on GDP. We can know that all the selected indicators have positive significance for GDP growth within the statistical range. At the same time, it shows that the test has passed all the hypotheses of panel data and satisfies the random effect ( Figure 5). The number of indexes (N) selected in this paper is 10 and the time span (T) is 10 years. In this case, both the fixed effect model and the random effect model are satisfied, and the further model test is needed. At this time, the Hausman test should be taken.

D. Model Determination Based on Hausman Test:
According to the reference [13], the difference between the random effect model and the fixed effect model is that it is very difficult to distinguish them to a high degree in the description of individuals.
The stable effect will consume a large degree of freedom. At meanwhile, the random affect is more universal on this basis. The proposed Hausman test can be used to distinguish them to some extent. The test of the advanced random effect model will store the test results, then test the fixed effect of the model and save the outcome. The Hausman test is used to get the final model. Then the method to test the two models simultaneously is built (Table   11). It is known from the output that the variance of parameter estimation of random and fixed effect models under this test is a positive definite matrix, which satisfies the test conditions. Under 95% confidence interval, P-value is far less than 0.05. Therefore, we should choose the fixed-effect model as the explanation model of economic vitality.    Figure   4, the explanation degree of each factor to economic vitality has been given. And the following Suggestions are given according to the influence degree. ii.
In the process of development, the region should combine its resource endowment and industrial foundation to find the optimal ratio of enterprise structure. And it will complete the adjustment of enterprise structure as soon as possible and develop appropriate leading industries to promote economic growth. It will be conducive to a steady increase in economic vitality.

iii.
Local government expenditure has an impact on economic vitality. The government needs to be tightly managed to make its spending transparent. We will increase government support for enterprises.

iv.
Entrepreneurship is encouraged. The government takes the lead in encouraging entrepreneurship and social practices are carried out to transform enterprises.
According to the influence of individual factors on economic vitality attained from the fixed model, the influence law of elements is summarized, among which policy adjustment (government expenditure) and enterprise vigor (total annual profit of enterprises) have a positive influence on economic power, and the implementation of policies in this respect should also be intensified.

G. The Influence of Changing Trends of Population and Enterprise Vigor on Economic Vitality
Seven variables were opted, GDP was taken as the expression of economic vitality, and the fixed-effect model in the panel data model was used to draw the following conclusions: The growth rate of the permanent resident population has a positive impact on economic vitality. The increase of permanent resident population will increase economic vigor in a small extent. If the population grows too fast, it will increase the rate of job competition and lead to the rise of unemployment, which will hurt economic vitality.
However, the growth decline of enterprise vigor directly affects economic power and presents a positive correlation change.        Table 13.

The Establishment of the VAR-VCE Dynamic Volatility Model
It can be seen from    Carry out the second-order difference differentiation on original data and continue the ADF test on the data after the difference and the outcomes are shown in Table 15. It can be seen from Table 15 that all the data after the second-order difference have passed the ADF test this group of data is zero in the second order, and then the inter group cointegration test is carried out. Figure 9 shows the visual information of three points of each variable under three tests. The confidence intervals of the middle three levels are 1%, 5%, and 10% respectively. After the first-order difference, the only population passed the test. After the second-order difference, all the data pass the test. And the group of data is the second-order zero integer data.

2) Johansen co Integration Test of Variables:
According to the ADF test, the original variable is a second-order zero integer sequence; that is to say, the original variable is an unstable sequence. First, the Johansen co integration test is carried out to find out whether there is a co integration relationship between its combinations. The test method is to calculate the trace statistics trace and the maximum eigenvalue Max eige value. Using the cyclic statistical hypothesis, the existence of a cointegration logarithm is assumed. Table 16 shows the Johansen co integration test results.
From the trace statistics trace in Table 16, it is assumed that none is the sequence without cointegration. The outcomes in Table 17 are calculated by Eviews software ( Figure   10). It can be seen from Table 17 that the AIS value decreases with the increase of VAR (N) lag period, presenting a monotonic decreasing state. The SC has a minimum at VAR (3). According to the AIS information standard and SC standard, the optimal lag time is opted as the third-order lag time.   Table 18.
In the formula given under certain circumstances. Figure 11 is the AR root test. The absolute value of the root is less than one. All the roots are in the plane of the unit circle, and the stability test of the model is passed. The impulse function is applied to the model to observe the long-term and short-term effects of economic policies on economic vitality. It can be seen from Figure 12 that the promotion effect of economic policies on economic vitality gradually declines after 1-3 periods. And it has increased since the third period. Because the experience of implementation after the implementation of economic policies can be applied, which has a secondary effect.
After the fourth period, the promoting the effect gradually decreased. The decreasing trend was relatively slow. And the longterm positive correlation effect continued. Figure 12: Impulse response.

Principal Component Index System Model:
To build a measurement model to measure the vitality of regional economy.
Firstly, we establish a scientific, economic vitality index system as the standard of data selection. Secondly, the minimum average difference method is used to screen the data. And the index is initially extracted. Further, the element analysis the method is used to select the main influencing factors. Finally, the comprehensive score of each factor is weighted to give the ranking of urban economic vigor.

The Construction Principle of Index System of Economic
Vitality: To select effective data to measure the economic vigor of each city, the following five theory s are given in this paper. The general process is as follows: (Figure 13).

2) Principle of Practicability:
The construction of evaluation index system is mainly theoretical analysis, which will be affected by the data sources of each index in practical application. Therefore, the availability and reliability of data sources should be ensured in reselecting indicators.

3) Systematic Principle: There should be a certain logical
relationship between indicators, which should not only report economic vitality from different aspects.

4) Principle of Comparability:
The data of each city should conform to comparability, so the data of each city can be compared horizontally and vertically.

5) Principle of relevance:
The comprehensive evaluation Where F i is the score of the i factor; x1, x2 , xp is the standardized value of the index; the corresponding coefficient is the component score coefficient; The total element score is equal to the weighted arithmetic mean of the scores of each factor, 10 that is Where is the total factor score, Fi is the score of the first influencing element; Bi is the contribution of the first element, and factor contribution = variance contribution rate/total variance interpretation after the element rotation.

Measurement of the Economic Vitality of Regional Cities:
Before measuring the economic vigor of each city, the relationship between variables and element analysis is further verified through the variance of common factors (Table 19). The common factor variance can effectively reflect the strength of its interpretation ability. The larger the common element variance extracted between variables, the stronger the ability to be interpreted by the common factor. Most of the variable factors proposed by the extracted common element variance are explained to a higher degree than 70%. Therefore, the extraction effect is better, the information of the original data loss is less, and the data extracted is more reliable. For the element, whose characteristic root is greater than 1, data analysis is carried out based on SPSS software and two factors are finally obtained, as shown in the table below, with the explanation of total variance. From Table 20, the cumulative variance contribution rate is 73.174%, indicating that the first two factors contain 73.174% of all indicator information (Figure 14).
And the extracted information is large and highly representative.
Therefore, element analysis is effective in extracting original variable information. From Table 20, the cumulative variance contribution rate is 73.174%, indicating that the first two factors contain 73.174% of all indicator information. And the extracted information is large and highly representative. Therefore, element analysis is effective in extracting original variable information. It can also be seen from the gravel map that the information contributed by the first two factors in the overall influence elements represent that the broken line is relatively steep. And the slope of the broken line is relatively gentle after that, so it can be considered that the two factors extracted are relatively reasonable (Table 21).  It can be seen that the primary industry, tertiary industry, college students, population, and road traffic noise level are factor 1, which reports the level of social production and security. Therefore, element 1 can be named as social production and security factor; local GDP, financial expenditure, real estate investment, and per capita wage are element 2, which reflects the government regulation and control. Therefore, the element two is called the government regulation element. The contribution rate of elements is analyzed by the method of normal maximization variance. And the conversion correlation coefficient is obtained, which shows the correlation of two factors.
It can be seen from Table 22 that in   According to the component score coefficient matrix, local GDP, fiscal expenditure, tertiary industry, tertiary industry, and real estate investment have a positive impact on the ranking; the primary industry hurt the ranking. The expression of each influence factor is given according to Table 23. Taking the variance contribution rate of each factor as the weight, the weighted analysis is carried out. After the weighted average, the growth index scores are as follows: The final weight value of each influencing factor is attained by the factor analysis. And the comprehensive score of each element is obtained by element score weighting function. The element ranking and comprehensive element ranking of each city are shown in Table   24. It can be seen from the ranking table that the cities such as Beijing, Shanghai, and Guangzhou rank the second, third and fourth respectively in the ranking, which indicates that the central economic zone of the country has high stability and is not easy to change. The highest ranking is Chongqing. Shenyang is ranked next, and the transfer of its industrial center may be one of the reasons for this outcome. It can be seen from Figure 15 that the ranking of Kunming and Ningbo fluctuates greatly. Considering that the local industrial structure is not obvious enough, it is necessary to strengthen the industrial structure adjustment to improve its economic vitality. Shenyang's ranking is declining year by year, which may also be related to local policies and development strategies. So, it needs to be noticed in time.  Table   25. To dissect the regional economic vitality more specifically, it is necessary to understand the distribution characteristics of each data. Through descriptive statistical analysis of the data, the basic information of each variable (Including sample number, mean value, standard deviation, minimum value and maximum value) is obtained as shown in Table 25. It can be seen from Table 25 that the average value of eco is close to 0, indicating that the statistical effect is very good. The fluctuation of the house price is large, which is in line with China's national conditions. The number of hospitals is quite different, which deserves the attention of local government.
The number of post offices is on the high side in some areas, resulting in wastes of resources. Table 25 is the basic situation of the data. After the description and statistics of the data, the correlation analysis of the data is carried out. If the correlation of some indicators is too low, it may lead to the low chi-square significance value, which needs to be screened. Then, the Pearson correlation coefficient is opted to measure the correlation between the variables. If the correlation between the illustrated variables and the explained variables is high, the study of the model is intentional. However, if the correlation between explanatory variables is too high, it may lead to collinearity among variables, which may affect the outcomes of the model. The following studies the correlation between the two variables analyze the correlation between the two variables. And the tests are significant (Figure 16). is to say, the multicollinearity among the variables is low, which will not have a great impact on the outcomes of the model.  corresponding to the BP test is also 0, less than 0.05, it means that the random effect model is better than the mixed model.   Therefore, it can be further proved that the economic vigor index system constructed in this paper can accurately measure it.

A Development Plan Based on the Perspective of the Decision-Maker
This section first reviews questions 1 to 3 above and obtains the general universality of the model established in this paper.
Finally, according to the outcomes, it proposes measures conducive to improving economic vitality and promoting economic growth.

Conclusion Review
From question 1 to question 3, we can roughly divide Therefore, we can dissect from the perspective of the above and advise the sustainable development of the economic vigor of benign and stronger regional competitiveness.

Suggestions on the Benign Sustainable Development of Beijing's Economic Vitality
Economic vitality includes not only the speed, stability, and

Advantages
The advantages and disadvantages of model factor analysis and panel data model are analyzed.

Improvements Needed
When using the panel data model to study influence factors, there are some difficulties in variable design and data collection, some errors in factor prediction, and selection difficulties in influence factors; panel data analysis of time series of factors is short, which can only reflect the data characteristics in the short term, not the long-term changes of factors.

Conclusions and Recommendations
This paper establishes three models, namely panel data model Especially in the context of the global epidemic, the economic growth situation and economic growth potential of various countries are issues that need special attention. For a country, the epidemic outbreak may present a point like outbreak trend, and the relationship between the regional epidemic development trend and the regional economy is also inseparable. In the future epidemic prevention and control, we can establish the relevant panel data model and VAR-VEC model and use the relevant local economic indicators to model and analyze how to stabilize the local economy and promote the national economy.

Conflict of Interest
We have no conflict of interests to disclose and the manuscript has been read and approved by all named authors.