#### Abstract

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.

**Keywords:** Panel data model; VAR-VEC
model; Factor analysis; Index system

#### Introduction

Under the background of the new age, China’s economic, social, cultural, ecological, political and other fields are coruscating new vigor and vitality. At the same time the good life is people’s increasing need to inadequate and unbalanced of the contradiction between the development of becoming the social contradiction, and the unbalanced economic growth between different regions is the concentrated reflection of unbalanced is not fully developed; 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 the economic vitality of some cities, to better extend the economic vitality analysis model to more research fields.

#### Models

### 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.

**b) Independence Test:** In the analysis of the relationship
between the factors affecting economic vitality, to fully understand
whether there is an internal relationship between the elements.
According to the processed data, this paper carries out an
independence test for each element. The data source is the national
bureau of statistics, and the independence test is executed on the
pre-processed data. See the appendix for the specific data. Make the
following assumptions about the research hypothesis:

**a. Null Hypothesis:** The factors that influence positive
energy are independent of each other.

**b. Alternative Hypothesis:** The factors influencing
economic vitality are not independent.

Firstly, a chi-square independence test was executed. SPSS was used to conduct independent test for each influencing element to observe whether there was any correlation between each factor. The test results are as follows: It can be seen from Table 2 that the cross relation between each factor and the year. The cross table shows the availability of different influencing elements which occupies a complete percentage. It indicates that the opted data are valid values with high accuracy, which can be further compared in pairs to test the independence of judgment elements. The significance analysis is used to determine whether there is independence between factors. The chi-square significance test results are shown in 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.

**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.

Correlation coefficients can quantitatively describe the closeness of linear relationships among elements. And SPSS is used for correlation analysis to attain the correlation coefficients among the influencing factors, as shown in 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.

**Establishment of Model:** This section is based on the panel
data of various factors collected from 31 provinces and cities in
China from 2009 to 2020. Considering the influence of multiple
factors on economic vitality, various methods can be used, such
as multiple linear regression and panel data model. Here, a rough
comparison is made before further model establishment. Compare
the panel data model with the multiple linear regression model, as
shown in Table 6. Based on the data and problem in this question,
the panel data model is a better choice. The panel data model
includes both the cross-section and the time dimension. Here, the
elements affecting economic vigor are taken as the cross-section.
And the year is taken as the time dimension. Where, i (i=1…8)
represents the following linear model set for the year:

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.

In the formula, AT is a column direction where all elements are 1, and the others have the same meaning as the original model.

**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:

**c. Model Determination Based on Hausman Test**

Because the missing related variables are not excluded, there will be dependent. Variable- local GDP will change with the same period correlation of random interference items. And the constraint conditions of exogenous variables are not satisfied so that the OLS estimator is biased and different. OLS is used to test the fixed-effect model. And GLS is used to test the random effect model. According to the reference [13], the difference between the random effect model and the fixed effect model is that it is difficult to try to make a high degree of distinction on the description of individuals. The stable affect will cost more degrees of freedom, while the random affect is more universal. The proposed the Hausman test can be used to distinguish them to some extent (Figure 2). Advanced random effect model test, test results are stored; Then the stable effect test is carried out on the model and the outcomes are saved at the same time. Finally, the Hausman test is performed on the outcome to obtain the final model. The method to verify that both models are satisfied is established in turn.

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 explanation model for the influence of economic vitality, while the random effect model should be opted instead.

**Model solving Process:** In this section, stability analysis is
conducted on the existing panel data, fixed-effect test and random
effect test are executed on the whole data based on the panel
mathematical model, and Hausman test is used to define the
applicable model for the panel data. Finally, the analysis outcomes
are attained based on the panel regression model.

**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.

**B. Fixed Effect Test Based on OLS:** Panel data has the
characteristics of separating long-term variables and short-term
variables, while the fixed-effect model focuses on the relationship
between variables within the group, it is necessary to test the
fixed effect model. The estimation method is OLS estimation, two
assumptions of the fixed-effect model are made.

**Hypothesis 1:**

**Hypothesis 2:**

Theξ in Hypothesis one is the independent variable interference term.

**Hypothesis 1:** Assume that the does not affect the observed
value, unobserved value, and post observed value.

**Hypothesis 2:** The general test of homogeneity. Ensure that the
model satisfies the blue estimate of OLS. And organize data into
long data types.

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.

**C. 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.

**Hypothesis 1:**

**Hypothesis 2:**

**Hypothesis 3:**

**Hypothesis 4:**

**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.

**E. Analysis of Model Test Results:** Using the Hausman test,
the fixed-effect model is determined as the interpretation model of
economic vigor, and the outcomes are shown in Table 12. Among
them, the factors that have a positive influence on economic vitality
(GDP) are the local government financial expenditure, the total
annual revenue of local tourism, the total annual profit of local
enterprises, the local permanent population, and the GDP of the
tertiary industry. According to Figure 6, based on the fixed-effect
model, it can be concluded that the tertiary industry has the largest
influence on the estimated vitality. It followed by the annual income
of enterprises (enterprise vigor), the input expenditure of local
government (policy bias), the total income of local tourism, and
finally the permanent population. Among them, the influence of the
tertiary industry on economic vigor is more than seven times that
of enterprises. It indicates that the third vigor can occupy most of
the effect among the factors influencing it.

**F. Activation Scheme Proposed Based on the Fixed-
Effect Model:** According to the fixed-effect model shown in 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.

i. To increase the proportion of the tertiary industry in the overall economy, the tertiary industry plays an important role in the influencing factors. So, it is necessary to strengthen the overall proportion of the tertiary industry in the current stage of social construction. Raising the economic proportion of the tertiary industry will promote the improvement of economic vitality.

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.

### The Establishment of the VAR-VCE Dynamic Volatility Model

Based on the panel data, this section intercepts the local government expenditure of Beijing as a representation of economic policy and establishes a vector autoregressive model (VAR). Taking economic vitality as the research object, the stability of each factor is verified. Based on vector error correction model (VEC), the lag order and influence function response chart are given to describe the long-term and short-term influence of policy implementation on it.

**Data Preparation:** Based on the cross-section data of Beijing in
the panel data of the whole country, this section conducts relevant
test and Analysis on the data. Finally obtains five significant factors
that have an influence on the economic vitality: local per capita GDP,
local government expenditure, local tourism gross income, local
people’s living consumption index, and local resident population.
Among them, five kinds of data of Beijing, local per capita GDP, local
government expenditure, local tourism gross income, local people’s
living consumption index, and local resident population is given in
Figure 7 after standardization. It can be seen from Figure 7 that the
local government expenditure has an increase in each year, basically
showing a linear growth. There is no significant change in the local
tourism income, which is relatively stable compared with other
indicators. It indicates that Beijing, as the capital of the country, is
very successful in the construction of tourism culture. The resident
population gradually declined after reaching the peak from 2012 to
2013, which indicates that Beijing’s population has changed largely,
and its GDP has grown steadily. It is not obvious that GDP is affected
by the fluctuation of alien factors. To understand the dynamic
influence of various elements on GDP, we need to carry out vector
autoregression for this group of data.

**The Establishment of Vector Autoregression (VAR) Model:**
Based on the statistical properties, a function containing the lag
value of exogenous and endogenous variables are established to
construct the model, which illustrates the influence of the dynamic
changes of variables on the dependent variables. VAR model is
essentially a model of multi equation class. Based on the moving
changes of multiple variables, the interaction between alien
variables is investigated. Any endogenous variable in the equation
system is constructed to express the lag term of any variable. Its
general expression is

Where Yt is the endogenous variable vector of K dimension, Yt-i (i=1,2,…,p) is the vector of endogenous lag variable, Xt-i is the d-dimensional exogenous variable vector or lags exogenous vector. P and R are the lag orders of endogenous and exogenous variables, respectively. Ai is a k-order coefficient; Bi is a k-row-d-column coefficient matrix, these matrices need to be estimated by specific methods. The last term is a vector composed of k- dimension random error terms. According to the solution of the following figure, we can get the estimation coefficient (Figure 8). Firstly, the lag order of the AVR model is determined according to the AIC information criterion and SC criterion when the minimum value is taken. Then the lag order is substituted into the meta model, and the coefficient of the AVR model can be obtained by the OLS estimation.

**The Establishment of the VEC:** When multiple time series are
unstable, the Johansen method is used to test whether there is a
cointegration relationship. If there is co integration relationship, the
VEC model can be established to analyze the dynamic relationship
of its multi pass model.

In the formula, ECMt-1 is the error correction term. Compared with the AVR model. The error correction term is an important feature to distinguish the two. The error correction term reflects the long-term equilibrium relationship of each variable, and the deviation of long-term equilibrium can be corrected by quick shortterm adjustment. Before building the VEC model, the Johansen test is needed to define the stability and reliability of the model.

**Model Summary:** To determine the regression type of a group
of vectors, we need to conduct multiple tests. Finally, we can define
whether the model has a correction term. Next, we summarize the
model to construct a complete the VAR-VEC model.

**Solution of the Model:** In this section, we first judge the stability
of time series and then do the ADF test on vector series to decide
its stability. Then, the first order and second-order’s difference are
used to judge its stability. The cointegration test of the original data
is carried out. And the satisfied model type is attained. Finally, the
stability of the model is judged. And the moving influence of policy
implementation on economic vitality is obtained.

**1) The ADF Unit Root Test of Vector Sequence:** First, all
the time data are tested by the ADF test, and the distinct order is
0. The lag order is 1-2, and the test results are shown in Table 13.
It can be seen from Table 13 that under the time test of order 0
raw data, the t-values of six kinds of t-tests are greater than the
comparison data under the confidence interval of 95%, shows that
the time series of this group of data does not pass the ADF test of
the original data, and the further differential test is needed. Carry
out difference differentiation on the original data and continue the
ADF test on the data after difference. And the outcomes are shown
in Table 14. It can be seen from Table 14 that under the ADF time
series test, the T value of population t test in six species is less than
the comparison data under the confidence interval of 95%. Among
the six kinds of data, only the population has passed the first-order
difference test. The first-order difference of this group of data is not
zero. So, a further difference test is needed.

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.

Under this assumption, the trajectory value is 255.6213, which is greater than the critical value of 95.7537. If the original hypothesis is rejected, there is at least one cointegration relationship. In the case of 5% confidence level, the original assumption is that there are at least four sets of cointegration relationship. Its trajectory value is less than the critical value. And the determination of the fourth set of cointegration relationship is rejected by the assumption. There are at least three cointegration relations in the linear combination of time series with surface instability.

**Establishment and Solution of VEC Model:** When the
original data series are non-stationary time series and Johansen
cointegration test shows that there are at least three cointegration
relationships in the series. To build a proper VEC model, it is
necessary to determine the optimal lag order of the model. The
stability of the model is explained by the AR root graph and Roland
causality analysis. Finally, the impulse response chart is given, and
the long-term and short-term effects of policy implementation on
economic vitality is dissected.

**1) Determination of Lag Period Based on AIS-SC
Minimization Criterion:** When the model is not integrated and
stable, multiple VAR models with different lag periods can be
established first. According to the relationship of multiple research
variables, the values of each AIC and SC can be recorded and
compared. The optimal lag period of the model can be selected
according to the theory of reaching the minimum simultaneously.
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.

**2) Determination of VEC Model Parameters:** According
to the above analysis, through the cointegration test, there are at
least three groups of cointegration relationships between time
series. It can be used to build the EVC model. According to the AISSC
criterion, this model is a third-order lag model. And the VAR (3)
model should be established. The parameters of the model based
on the OLS estimation are shown in Table 18. Table 18 shows the
cointegration formula with the maximum log-likelihood. Thus, the
final cointegration equation can be written as

1ngdp = 0.731ntravel + 0.771ntravel −1.081nindex +0.231npop − 0.891nworklose − 4.21937

Through the cointegration relationship, we can see that the long-term equilibrium relationship between economic vitality and local government expenditure, local tourism revenue and local resident population is positive. There is a long-term negative correlation between economic vitality and residents’ living index and local unemployment rate. According to the test results (see Appendix 1), we can build the VEC model as

The specific coefficients are described as follows:

In the formula

the last remainder is

**3) Analysis of the VEC Model:** Before analyzing the model,
we need to use the AR root graph method to test the stability of
the model. According to the experimental outcomes, the impulse
response of VEC model is given. And the long-term and shortterm
effects of policy implementation on economic vitality are
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.

**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).

**1) Scientific Principle:** The selection of measurement indicators
must be based on scientific principles and can truly and
objectively reflect the influence of various factors on urban
economic vitality. The scientific comprehensive index
evaluation system of urban economic vigor is the basis of
correct analysis and evaluation of regional economic vigor.

**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
index system of local economic vitality should be an organic
combination of a series of related indexes.

**Data Filtering:** The minimum mean square deviation method
is used to screen the preliminary data. The observation value is
xij, where i is the number of evaluation objects, i.e., the number
of cities, j is the number of evaluation indexes, there are 19 cities,
each city has 14 indexes. First, the average value and mean square
deviation of index j are calculated.

Then the minimum mean square deviation of all indexes is calculated, such as:

If the minimum mean square deviation is close to 0, then the index be eliminated and calculated in turn.

xj corresponding to Sj can. Finally, nine indexes meeting the requirements can be opted from 14 indexes, namely, local GDP, financial expenditure, the added value of the primary industry, the added value of the tertiary industry, real estate investment, number of college students, population, per capita wage and road traffic noise level.

**Factor Analysis:** Using the factor analysis method, the extracted
nine indicators, including 190 sample data from 19 cities in 2009-
2020 are dimensioned down. And then the coefficient matrix is
multiplied by the standardized element to calculate the score and
find out the factors that have the greatest impact on economic
vitality.

Where Fi 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 the component transformation matrix, the value of component one has changed, and the value of component two has also changed. It is necessary to extract the component matrix of the factor load matrix.

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.

**Comparative Analysis of Factors Affecting Economic
Vitality:** In the above, according to the factor analysis method, two
main elements that affect economic vitality are social production
and security elements and government regulation factors, which
have a positive correlation with economic vitalities. According to the
nine influencing factors opted above, the secondary industry, house
price, total retail sales of social goods, number of hospitals, and
number of post offices all have a positive impact on the economic
vitality. The comparative analysis is made on each element to see if
there is any difference.

**1) Model Establishment:** To test the accuracy of the index
system established to measure economic vitality, considering that
the individual effect of each index is not observable, and the time
effect is not observable, a panel data model is established to test it.
And the following model is established:

In the formula, ecoit is a comprehensive index system to measure economic vigor, xit is an independent variable of N rows and K columns. The factors affecting economic vitality can be divided into

a. Social security system: Number of hospitals and Post offices

b. Processing and production: The secondary industry

c. Consumption level: house price, total retail sales of social goods

**2) Descriptive Statistics:** To analyze 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. 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.

**3) Correlation Analysis:** 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).

From the correlation analysis outcomes of Table 26, it can be concluded that the correlation coefficients between all explanatory variables and the interpreted variables are significant, and there is no strong correlation between the explanatory variables. Therefore, there is no multicollinearity between the explanatory variables. To further study the collinearity among the validation variables. The model was validated by using the VIF test. And the outcomes are shown in Table 27. It can be seen from the table that the VIF value of the explanatory variable and the control variable is less than 5. That is to say, the multicollinearity among the variables is low, which will not have a great impact on the outcomes of the model.

Therefore, the following modeling and regression analysis can be continued. Continue with the residual analysis (Table 28). From the analysis of variance, the F value is far greater than 1, which shows that the differences among the factors are statistically significant. The interaction effect among the factors is more significant. Fixed effect model and mixed model are tested by F test, random effect model and mixed model are tested by BP test, stable effect model and random effect model are tested by the Hausman test, and they are compared (Table 29): According to the model test results, if the p-value corresponding to the F test is 0, less than 0.05, it means that the fixed effect model is due to the mixed model. If the p-value 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.

If the p-value corresponding to the Hausman test is 0, it means that the fixed effect model tends to be selected. See Table 30 for regression outcomes of fixed effect model to be opted after inspection. From Table 30, we can see the regression results of the model. And the fixed effect model is opted after the test. The correlation coefficients of the first principal component, the second principal component, the second industry, the house price, the retail sales of social goods, the number of hospitals, and the number of post offices are all positively correlated with Eco. These indicate that they all have a positive impact on economic vitality. From the perspective of economic vigor, the secondary industry, house price, retail sales of social goods, and hospitals, and number of post offices are all positively related to economic vigor. In these variables, when one variable changes, the other variables remain unchanged. Then the economic vigor changes in the same direction. 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 the index system of urban economic vitality into indicators of economic growth, indicators of attractiveness to capital and production factors, indicators of employment and residents’ quality of life, indicators of innovation capacity, and indicators of intellectual property protection. Which chose the per capital GDP, fiscal revenue, education and human capital, income levels, employment, innovation, and intellectual property rights protection for data collection, processing, modeling and analysis, and it can provide indicators and economic vitality all remain positive correlation. 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 outcomes of economic growth, but also the average quality of life of the people, such as the level of education and health standards, as well as the overall progress of the economic structure and social structure (Figure 16). Given the excessive factors affecting economic vitality, we roughly divide the policy into three parts, including optimizing industrial structure (O), rationally controlling investment intensity (R) and technological innovation (T), namely ORT development strategy.

### Optimizing the Industrial Structure

**a) Vigorously Developing the Tertiary Industry:** The
third industry is an important indicator of a country’s economic
development. And the tertiary industry has the characteristics
of less investment, short cycle, quick effect, and high wages of
employees. Vigorously developing the tertiary industry can
rapidly expand employment fields and jobs, avoid labor surplus,
and improve residents’ income. For modern cities, residents not
only have material needs but also pursue spiritual level. This
growth trend promotes the region to continuously develop new
industries to meet the needs of the people, to improve residents
and to improve the quality of life. Therefore, we vigorously develop
the tertiary industry, which has a significant role in promoting the
sustainable development of economic vitality.

**b) Strengthening the Development of Primary and
Secondary Industries:** For the adjustment of Beijing’s economic
structure and the promotion of its regional competitiveness, it is
necessary to develop the tertiary industry while strengthening the
primary industry and expanding the scale of the secondary industry.
The first industry is the basic trade of the national economy,
strengthening the first industry, and laying the foundation for the
development of the second industry and the third industry.

**Reasonable Control of Investment:** Investment is an
important part of GDP but also an element of economic vitality. The
growth of investment is of great significance to the promotion of it.
In China, investment is mainly divided into a private investment,
government investment, and foreign investment. From the
perspective of Beijing, as a first-tier city, many foreign enterprises
and state- owned enterprises invest in Beijing in various forms.
However, the unreasonable investment may cause the princess
of resources, leading to an imbalance of social and economic
development. Therefore, Beijing should control the scope of
investment and improve the efficiency of investment.

**Technology innovation:** Science and technology are the
primary productive forces, and innovation is a force that cannot be
ignored to drive economic development. Innovation is conducive
to the optimization and transformation of China’s economic
growth mode. Economic vitality comes from the sound growth
of the economy. We should strengthen the policy support for the
investment in science and technology of enterprises, guide the flow
of resources such as projects, funds, and talents to enterprises, and
establish an innovation support system with enterprises as the
main body. Cultivate and develop the next generation Internet, new
generation mobile communication, Internet of things, navigation
and location services, biomedicine, and other high-tech strategic
emerging industries. Accelerate the construction of high-end
talents gathering special zone, and actively introduce and cultivate
high-end talents.

#### Model Evaluations

### Advantages

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

**1) Factor Analysis:** Through dimensionality reduction of
a variety of influence indicators, the main elements are extracted
from the complex elements, and a few factors are used to describe
the relationship between many indicators that is, several closely
related indicators are classified into the same category, each
category of indicators becomes a factor and the economic vitality
is measured by a few factors. It simplifies the problem, measures
most of the information of it, and gets more scientific and accurate
information at the same time.

**2) Panel Data Model:** Compared with the traditional time
series model, the panel data model can provide more data points, increase the degree of freedom of data and reduce the degree
of collinearity between explanatory variables, thus improving
the effectiveness and accuracy of model estimation; Panel data
model is more conducive to reflect the randomness of the gap
between individuals. In this paper, panel data model can not only
report the information between the given factors, but also report
the information of a certain factor through the study of other
influencing factors.

### 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 (fixed effect model and random effect model), var-vec model and principal component index system construction model. The var-vec model has great generalization. The applicable fields are

a. VaR Based Research on agricultural economic development. Because of its unique ability of testing dynamic fluctuation, AVR can solve the research difficulty of agricultural economy which is greatly affected by seasonal.

b. Oil price evaluation. International oil has been affected by many factors, among which the fluctuation of international political content has a greater impact. Using VAR model can highlight the impact of oil price changes in the short term and make adjustments at any time.

c. The new decline of real estate has an important impact on the national economy. Choosing appropriate research methods will provide a new solution to this kind of problem.

d. Var can also be used for monetary policy analysis to study the long-term and short-term impact of changes in money supply and interest rate on economic fluctuations and their contribution.

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.

#### Acknowledgement

This work was supported by the Philosophical and Social Sciences Research Project of Hubei Education Department (19Y049) and the Staring Research Foundation for the Ph.D. of Hubei University of Technology (BSQD2019054), Hubei Province, China.

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