# A Random Matrix Model of Regional Economic Disparity Based on Spatial Panel Data Analysis

Authors: Jingjing Xun; Wei Li; Yunbao Xu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper adopts a random matrix approach to model spatial panel data and applies the model to the analysis of regional economic disparity. A panel data clustering method based on variable influence and response structural relationship clustering is proposed to define the structural relationship distance, cluster the panel data from the structural relationship so that the data within the class have the same structural relationship, and give the specific algorithm for determining the classification number in the clustering process based on the structural relationship. The methods of clustering panel data with the presence of linear, nonlinear, and functional relationships are considered separately to aggregate and classify the panel data in terms of the structural relationships of parameters. Using the classification modeling approach proposed in this paper, the empirical analysis is better than the estimation effect in the case of no classification; the empirical test data also show that there is a strong cross-sectional correlation between the economic and financial development data of different regions, and the estimation effect of the model with the correlation treatment is better than the estimation effect of the model without the correlation treatment. The study finds that there is a significant imbalance between financial and economic development, the phenomenon of financial exclusion is serious, and the dynamic influence relationship between financial and economic development in relatively less developed regions is different from that between financial development and economic development in relatively developed regions. In addition to the influencing factors, we should improve the efficiency of investment utilization, give better play to the geographical advantages, keep the industrial structure rationalized, develop the special economy according to the local conditions, and give full play to the spatial effect as well as improve the economic development.


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## Body

## 1. Introduction

The econometric theory is the bridge between economic theory and real data. Economists put forward corresponding economic theories based on the phenomena appearing in economic operation and need to choose suitable quantitative models to test and corroborate the economic theories; the massive data and multivariate problems in real economic problems need to choose suitable econometric models to fit the data, dig the data laws, and clarify the structural and influential relationships among the data [1]. Econometric models can provide policymakers and decision-makers with a basis for decision-making and can predict data based on data fitting. Panel data are precisely two-dimensional data that consider both time and space factors. In model construction, all data constitute an overall set, which is used to reflect the totality and cooperation of the economy, and individual differences are considered while reflecting the totality. A complete panel data model can bring great convenience to analysts of practical problems; however, more factors are considered in constructing the model, which brings new challenges to parameter estimation and model testing, and the type of model division can be more complex compared with simple time-series or cross-sectional data, which brings certain challenges to both theoretical research and empirical analysis [2]. To address the complexity of vector autoregressive modeling analysis of panel data types, this paper proposes the idea of vector autoregressive modeling based on classification. For the panel datasets that consider multiple variables at the same time, according to the specific characteristics of the data, the panel data are classified and modeled from the data relationship, so that there are relationships with the same characteristics within the class, and then PVAR modeling analysis is performed according to the specific relationship characteristics, which can effectively solve the problem of overparameterization of PVAR model, solve the problem of the complex structure of panel data, and solve the problem of correlation existing in cross-sectional space.The random matrix first appeared in the 1930s in mathematical statistics, but it did not attract enough attention at that time. These new normal regional development statuses will affect the economic development level, and the economic development level of each region will be different [3]. The eigenvalues of the stochastic matrix possess a rich mathematical structure and derive many interesting distributions and stochastic processes from it. In the process of analyzing and exploring such differences, we can further understand the relevant characteristics of regional economic development and analyze the differences in the influence of each index on Guangxi’s economy, so that we can adjust the countermeasures and measures of Guangxi’s development in a targeted way. Administrative regions have different geographical advantages, and the influence of spatial factors on economic development cannot be ignored. The basis for the exact solvability of the model is that they can be mapped into a series of disjoint path families, which, in turn, will yield deterministic point processes. Point processes with deterministic correlation functions have a rich structure, which leads to many interesting examples, and are thus presented as an interesting class of point processes [4]. Because of the difference between time variation and geographic location, a new form of data, spatial panel data, is created. This type of data can represent changes in both temporal and spatial directions. Spatial panel data cannot be applied to classical statistical models because of their spatial correlation, but spatial panel data models are not inapplicable to this problem. For spatial cross-sectional data models and spatial panel data models, the former does not represent changes in time, while the latter has a range of models that combine with the concept of time series. Neural networks train their models iteratively, with the goal of obtaining optimal model parameters. The goodness of the model parameters is evaluated using a loss function. The starting parameters w of the neural network are given randomly, and each iteration is a calculation of the loss function using the proposed better parameters w. The proposed better w is calculated by deriving the optimisation function. The whole process is in fact driven by the loss function and the optimisation function. Therefore, in addition to the choice of parameters affecting the performance of the model, the choice of the loss function and optimisation function is also a central factor.The relationship between finance and economic development has been an active research topic in the economic field, and the dynamic influence relationship between the two is studied from a regional perspective, considering both regional correlations and regional differences. Taking dualistic economic theory, dualistic financial theory, and endogenous economic growth theory as the theoretical background, the theoretical research method proposed in this paper is applied to empirically prove the relationship between finance and economy, and the specific relationship includes the argument of the uneven relationship between economic and financial development and the argument of the endogenous relationship between economic and financial development. We systematically study the imbalance caused by dualistic economic theory and dualistic financial theory, construct a financial exclusion index that responds to financial imbalance, and measure the financial exclusion index of less developed regions in China for twelve consecutive years. The empirical study finds that there is a problem of regional unevenness in the relationship between economic development and financial development, and the relationship between financial development and economic growth is different in different regions.


## 2. Related Work

In the first stage, the main development surface of spatial econometrics is the autocorrelation test of residual space [5]. After that, many scholars started to study how to improve the chosen model to higher-order models such as the spatial Durbin model due to the spatial correlation of the dependent variable. Zhang first proposed the maximum likelihood estimation method for spatial econometric models, although the advantage of traditional econometric model estimation dominated the maximum likelihood estimation method at that time [6]. However, Mendieta did not limit themselves to the methods available at that time and started to consider the use of Bayesian methods for estimating spatial econometric models [7]. Hao et al. explored how spatial statistics could be applied to future research on spatial data analysis [8]. Also, they considered the inclusion of spatial relations in their economic research. After some time, they started to publish papers in econometric and mainstream economics journals because the econometrics with the inclusion of spatial relations produced a series of research results. At this stage, the development of the theory of regional economics was mainly manifested in some problems of the normative derivation of small sample nature, such as estimators and test statistics [9]. With small sample sizes, there were limitations on how models should be chosen, how parameters should be estimated, how models should be tested, and how to generalize dependent variable models.Since then, economists have been applying spatial econometrics to regional economic studies and have increasingly used this approach to analyze the relationship between crime rates among regions, etc. The spatial econometric theory was assimilated from the theoretical results of the first two stages and then further developed and extended [10]. Roman et al. adopted an indicator system consisting of three aspects, namely, credit scale, credit risk, and credit accessibility, and used hierarchical analysis to construct a basic model for calculating the urban credit environment and to score the level of China’s urban credit environment comprehensively and objectively [11]. Huang and Chen measured the score of the urban credit environment index system by the entropy value method and expert scoring method, measured the coordination score by establishing the TOPSIS model and TOPSIS model based on entropy power, and finally arrived at the comprehensive value of urban credit environment by comprehensive evaluation [12]. Parameter estimation is a form of statistical inference. It is the process of estimating an unknown parameter in the overall distribution from a random sample drawn from the overall population. In terms of the form of estimation, it is distinguished between point estimation and interval estimation: in terms of the method of constructing the estimator, there are methods of moments estimation, least squares estimation, likelihood estimation, Bayesian estimation, etc. Confidence is generally expressed in terms of probability, e.g., a confidence level of 95%; precision is measured in terms of the closeness or error between the estimator and the parameter being estimated. Using spatial panel data, Medeiros et al. conducted a spatial error model analysis on the spillover effect of intellectual capital stock on total factor productivity, as well as a spatial model analysis on the determinants of regional unemployment rate variability [13].This paper uses a combination of spatial statistics and spatial modeling to analyze the issue of intraregional differences in economic development levels in recent years. Later, the gross economic product and related economic variable indicators are selected for traditional linear regression and spatial panel model analysis to analyze the impact of different factors on economic development, analyze the status of economic development, propose improvement measures, etc. The traditional use of dynamic panel econometric methods to test the relationship between economic phenomena assumes that there is no spatial correlation. In this paper, it is found that there is a strong cross-sectional correlation between economic phenomena, and this paper investigates the dynamic influence relationship between economic systems when considering the case of spatial correlation, providing new research ideas and methods for empirical studies. In the study of financial imbalance, the financial exclusion index was constructed, and the constructed index was used to measure the current situation of financial exclusion in Ningxia, a financially underdeveloped region in China. Using the relational clustering method proposed in this paper to divide the data into two categories, it was found that there is an imbalance between urban and rural financial development and financial exclusion development.


## 3. Spatial Panel Data Construction

Spatial autocorrelation is a prerequisite for spatial statistical analysis. To perform spatial autocorrelation analysis, a spatial weight matrix must be established. Defining the mutual adjacency of spatial objects is also to adjust the spatial adjacency and accurately represent the spatial interactions. There are various methods to construct the spatial weight matrix, and the binary symmetric spatial weight matrixWi∗j is generally used to express the spatial adjacency of n positions, and the element Wij on the diagonal is defined as O.(1)O=0W12W1nW210W2nWn1Wn20.The rules commonly used for spatial weight matrix establishment are adjacency rules and distance rules. There are also spatial weight matrices based on the concept of proximity, which is divided into first-order proximity and higher-order proximity [14]. First-order proximity assumes that two regions will have spatial adjacency when they have a common boundary and is defined in a similar way to the binary adjacency rule, which is represented by 1 when two regions have a common boundary and 0 otherwise. When considering the use of spatial measures, it is important to determine whether there is spatial correlation first, and only with spatial correlation can further analysis be done. Spatial autocorrelation is the quantitative description of the dependence of things in space.(2)I=∑i=1n∑j=1nWijYi+Y¯Yj+Y¯S∑i=1n∑j=1nWij.Global spatial metrics are used to verify the spatial distribution state of the whole region. If a regional unit contributes more to the global spatial autocorrelation, that is, there is local spatial aggregation of observations, then local spatial correlation analysis must be applied. The different adjacency relationships of spatial units can be represented by spatial weight matrices, so it is a central problem to construct suitable spatial weight matrices to represent the relationships of adjacent spaces when studying spatial statistics. Since the construction of spatial weight matrices is a rather subjective problem, unless based on an authorized theoretical model, different spatial weight matrices may yield different results when studying other less regulated problems. Since the spatial weight matrix constructed from adjacency relations is particularly easy to compute and particularly straightforward to see the correlations between neighboring spaces, it is used very frequently in much of the current literature, yet it often does not reflect the true spatial relationships because the spatial relationships it embodies are too simple. The fundamental starting point for spatial autocorrelation is based on the first law of geography, which refers to the relationship between an attribute of a geographic thing distributed in a region and the same attribute of all other things. The basic measure of spatial autocorrelation is the spatial autocorrelation coefficient, which measures and tests whether a spatial object and its attributes are distributed high and high adjacent or high and low staggered, i.e., positive spatial correlation means that spatially distributed things have similar trends and values for their attributes, while negative spatial correlation means that spatially distributed things have opposite trends and values for their attributes.The spatial matrix constructed according to the distance relationship then focuses on the distance and mainly studies the influence of different distances, the distance of the distance on the neighboring space. The threshold spatial weight matrix is a spatial weight matrix based on the distance relationship. It means that when the distance between two regions is less than a given one, then there is a spatial interrelationship between these two regions; otherwise, there is no spatial effect between these two regions.(3)wij=0,dij≥δ,0,others.The spatial autocorrelation coefficient can be used to measure spatial autocorrelation. When studying a phenomenon, it is necessary to consider whether there is a spatial correlation between a region and its neighboring regions, which can be detected by the spatial autocorrelation coefficient. The spatial autocorrelation coefficient can be positive or negative; positive means that there is a positive spatial correlation between the region and its neighboring regions, and negative means that there is a negative spatial correlation between the region and its neighboring regions [15]. The positive spatial correlation indicates that the study phenomenon has the same trend of change in the neighborhood, while the negative spatial correlation indicates that the study phenomenon has the opposite trend of change in the neighborhood. Spatial autocorrelation analysis can also be divided into global spatial autocorrelation and local spatial autocorrelation depending on the observation angle (from the whole or local).The global spatial autocorrelation analysis focuses on the spatial distribution of an analyzed phenomenon in the whole region, which cannot specifically reflect the spatial correlation of this phenomenon in some local areas, so the local spatial autocorrelation analysis is introduced to study the heterogeneity of the distribution of a research object in different spatial units. The local spatial autocorrelation analysis is mainly used to study the spatial relationship of the attribute values of each small unit in the whole study area from a local perspective and to measure the local spatial correlation degree of the study objects in the neighboring areas.(4)ZIi=IiVarIi−EIiVarIi.The spatial lag model and spatial error model are often used to study the relationship between variables with spatial correlation. The spatial correlation generally involves two aspects, on the one hand, the increasingly close ties between different cities and frequent economic exchanges, which makes the level of economic development of neighboring cities have an obvious spillover effect; on the other hand, the error shock of neighboring cities will have an impact on the economic level of this city. The economic development level of a city will show a correlation with the economic development level of other cities. If this spatial correlation is ignored and the relationship between the economic development levels of different cities is still analyzed by the ordinary regression model, the model will be estimated incorrectly and the influence relationship between economic variables will be distorted [16]. Therefore, when building the model, the interaction relationship between regions should be introduced into the model, and the basic linear model should be corrected through the spatial weight matrix. Spatial correlation is mainly expressed in the lag and error terms of the dependent variable, and there will be two forms of spatial models at random: spatial error model and spatial lag model, as shown in Figure 1.Figure 1
Spatial econometric model corresponding to the selection of relatively significant statistics.Panel data are a mixture of cross-sectional data and time series, characterized by a large amount of information and relatively weak colinearity between variables. The panel econometric model has now become a more advanced technique for estimating econometrics, which can represent both cross-sectional and time-scale information, taking cross-sectional correlation and time correlation into account at the same time, so it can more objectively reflect the characteristics and patterns of the economic development level of the study area when subjected to spatiotemporal interaction correlation. However, the panel data model also has drawbacks. Therefore, the panel data model can be combined with spatial econometrics, and a spatial panel econometric model with time and space effects can be added on top of the ordinary panel data, i.e., a spatial panel econometric model that can consider the study variables in time and space together. Econometric models use cross-sectional data as the sample data for econometric models, and several issues should be noted. One is the issue of consistency between the sample and the parent. The parameter estimation of econometric models, mathematically speaking, is to estimate the parameters of the parent using a randomly selected sample of individuals from the parent, which then requires that the parent and the individuals must be consistent. For example, to estimate a production function model for a coal company, only data from the coal company can be used as a sample, not data from the coal industry. The cross-sectional data would then be difficult to use for the estimation of some aggregate models.For the choice of fixed and random effects, a statistical basis can generally be adopted to solve this problem. That is, observe whether the study individuals are correlated with the explanatory variables studied in this space. If correlated, then it is a fixed effect; otherwise, random effects should be used.(5)H0:h=1,h=β′β+ρρ′.As shown in Figure2, it is assumed that the weight matrix is a fixed value, i.e., it does not change over time. That is, the correlation between each spatial unit does not change with time. Based on this prerequisite, the spatial weight matrix is extended to the study of panel data. Vector autoregressive models are divided into two forms: unrestricted constrained (induced) and structural. The unrestricted constrained type requires that the periods of influence between the endogenous variables are not the same and no contemporaneous correlation is assumed between the variables. The structural vector autoregressive model, which allows for contemporaneous impact relationships between the dependent and independent variables, also captures the relationship between the error term and structural shocks. In the structural panel vector autoregressive model, conditions need to be imposed on the model for the model to be identifiable.Figure 2
Test results of urban economic efficiency.In this paper, the spatial autocorrelation of 285 prefecture-level cities is calculated using the adjacency matrix, and Moran’s I index values andP values are measured to test whether there is some spatial heterogeneity [17]. To prevent the chance of the results being calculated by using a single matrix, the K3 matrix was used to perform robustness tests on them. It is assumed that the weight matrix is a constant value, i.e., it does not change with time. That is, the correlation between each spatial unit does not change over time. Based on this prerequisite, the spatial weight matrix is extended to the study of panel data.


## 4. Random Matrix Regional Economic Variability Analysis Model

The simulation results for several scenarios with different data lengths and spatial widths, classified and unclassified, are simulated separately. The significance level of the model fit test statistic increases significantly, and within each class, the difference between the estimated and true parameters gradually decreases as the number of cross-sections and the length of time increase. The design of a theoretical model consists of three main tasks, namely, selecting the variables, determining the mathematical relationships between the variables, and formulating the range of values of the parameters to be estimated in the model. Determining the variables to be included in the model, in a single equation model, the variables are divided into two categories. The variables that are the object of study, i.e., the “effect” of the causal relationship, such as the quantity of output in the production function, are the explanatory variables in the model, and the variables that are the “causes,” such as capital, labour, and technology in the production function, are the explanatory variables in the model. Determining the variables included in the model is essentially a matter of identifying the explanatory variables. The following types of variables can be used as explanatory variables: exogenous economic variables, exogenous conditional variables, exogenous policy variables, and lagged explanatory variables. Some of these variables, such as policy variables and condition variables, often appear as dummy variables.The simulated data are considered for different data lengths,T = 10, T = 15, and T = 20, and section widths, N = 12, N = 32, and N = 48, respectively. According to the simulation data generated by the cycle, the parametric data are divided into four classes according to the parameter relationships. In general, the data classification is beneficial to reduce the correlation of cross-sectional residual terms when the time length and the number of cross-sections are constant, and the effect of classification is more significant when the number of cross-sections is larger; when the time length is constant, the correlation of residuals tends to increase with the increase of the number of cross-sections, and no trend pattern is found in the correlation of residuals between classified and unclassified cases when the number of cross-sections is certain. A tessellation of Aztec squares consisting of dominoes can be represented by a family of disjoint paths, which can be obtained by drawing paths on different types of dominoes, as shown in Figure 3.Figure 3
Boundary process.Although mean-variance theory provides a convenient mathematical tool for investors, however, in practice, especially when the portfolio contains multiple assets, there are many troubles in the operation process. This is true not only for the mean-variance theory but also for most mathematical models applied to actual financial operations because of the complexity of the factors at play compared to the idealized theoretical model. However, it is the consideration of these realistic factors that enable the development and improvement of the corresponding theories and the closer integration of mathematics with real applications.The factors affecting the economic development of different cities are complicated, and many scholars, when studying the regional economic differences of provinces or counties, use only the indicator of GDP per capita to reflect the economic development level of different regions, but few scholars use a comprehensive indicator to study the economic differences of different regions [18]. Therefore, to reflect the economic development level of each region more comprehensively, this paper adopts a multiindicator integrated analysis of the economic level. The reasons for the obvious differences in regional economic development are manifold, and the combined effect of multiple factors has led to an increasing gap in regional development. On the one hand, it is the result of government policy guidance. On the other hand, this is the result of regional protectionism, driven by regional interests. Local governments are increasing their support for the region, insisting on trade protectionism and unilaterally supporting the development of the region in an overt or covert manner. The ineffective flow of resources and technology between regions and the problem of monopolisation of commodity industries are prominent. As a result, factors of production such as commodities do not flow freely between regions, the advantages of regions and industries are not balanced, and resources are not optimally allocated, leading to the fragmentation of the domestic market and affecting the overall coordinated development of the regional economy.Since each principal component indicator is related to the eigenvalues of the correlation coefficient matrix and its corresponding eigenvector, it is more common to select several principal components to be retained by the size of the eigenvalues obtained [19]. The first principal component indicator is related to the largest eigenvalue, the second principal component indicator is determined by the second-largest feature, and so on. Therefore, in this paper, we first retain the principal components with eigenvalues greater than 1 and discard those with eigenvalues less than 1 according to the eigenvalue criterion.The eigenvalues of the correlation matrix are obtained byR programming. From the eigenvalues derived in Figure 4, we can see that the eigen roots of the first and second three principal components are 5.6114, 2.5277, and 1.4061, respectively, which are all greater than 1. The eigen root of the next principal component, i.e., the fourth principal component, is only 0.2495, less than 1. Therefore, it is more appropriate to select the first three principal components.Figure 4
Gravel test.In addition to selecting the appropriate principal component with a characteristic root greater than 1, the gravel test can also plot a graph of each principal component with its corresponding eigenvalue, from which the change in the curve can be seen, and basically, we keep the principal component above the point where the slope changes the most [20]. In addition, the ready test can also take the method of parallel analysis to model and select the appropriate number of principal components. In this paper, the number of principal components is determined by three discriminative methods, i.e., eigenvalue, gravel test, and 1 parallel analysis (dashed line), as shown in Figure 4. It can be seen from the figure that retaining 3 principal components can retain most of the information of the dataset.For a better interpretation of the results, when multiple components are extracted, we rotate them to better interpret the principal component loading matrix [21]. In this paper, we choose orthogonal rotation, which keeps the 3 selected principal components uncorrelated and allows each component to be interpreted by only a limited set of variables. The actual interpretation of the three principal components can be derived from the principal component loadings matrix and the rotated image after the orthogonal rotation (great variance rotation).We can also observe from the graphs and tables that the three principal components become uncorrelated after rotation, and the degree of explanation of the original data remains the same [22]. In addition, the explanation of the cumulative variance of the three principal components after rotation remains unchanged at 95%; what has changed is only the explanation of the variance of each principal component [23]. The explanatory degree of variance of each component after rotation tends to be the same, so they are defined as components.


## 5. Analysis of Results

### 5.1. Clustering Results of Panel Data with Spatial Correlations

Now, to discuss the panel data with significant spatial correlation, on the one hand, to consider the extraction of spatial public information, so on the other hand, to reflect the dynamic endogenous influence relationship betweenm endogenous variables, the processing idea of this paper is to first extract all the public information in the spatial data because the extracted public information already contains most of the information in all cross-sectional data, the public information extracted from the space can represent most of the panel data information, and then the extracted public information is used to further measure the dynamic influence relationship between variables. By setting a set of original panel data, there are significant correlations between different cross-sections i and j. By starting from the correlation matrix or variance-covariance matrix of the data, the public information reflecting the common development trend of the space is extracted by applying the principal component idea.The data in Figure5 show that, in the case of strong cross-sectional correlation between cross-sections, but the degree of linear correlation is not very significant, the estimation effect of parameters with strong correlation is improved by applying the theoretical method of this paper, but the improvement is not high, and for the same time length and cross-sectional width data, the simulation results show that the estimated residuals of the parameter estimation results are smaller than those without correlation treatment. With the increase in time length and spatial number, the parameter relationship of the data with correlation treatment is closer to the true value parameters set by the model. Regional accessibility refers to the possibility of exchange of labour, products, capital, information and technology, etc. Spatial interaction is positively related to regional accessibility. Regional accessibility is influenced by four main factors: one is the relationship between transport distance and time, i.e., the shorter the transport distance and the less time, the easier it is to link economic activities and the better the accessibility, and conversely, the accessibility is poor.Figure 5
Simulation of data with strong spatial correlation.In general, when the data have a strong spatial correlation, the estimation results are slightly better using the method proposed in this paper. Both the PVAR model without correlation and the PVAR model dealing with correlation converge as the number of samples and the length of time increase, and the fitted data parametric estimates converge to the true value parameters. In the spatially correlated case, the VAR regression model using serial averaging outperforms the conventional PVAR model, and the model estimation effect using the vector autoregressive model with correlation handled outperforms the model estimation effect without correlation handled, and the parameters converge faster as the sample size and time length increase.The model fitting data show that the model parameters are also well estimated by using Pesaran’s sequential averaging method for data with high cross-sectional correlation in different cases, and relatively speaking, the test statistics of the parameters are more significant and closer to the true parameters by using the estimation method based on the main spatial information proposed in this paper. Fixing the time length constant and increasing the spatial widthN = 20, the parameter estimation effect under this scale is simulated to be more satisfactory. It is reasonable to use the first factor to build a dynamic VAR model between variables, and the estimation results show that the model estimation results of class one and class two are close to the true values, with small estimated residuals and significant t-test statistics. The parameter estimation results of the vector autoregressive model using the first spatial factor are ideal, and the parameter estimation results of both classes are close to the true value parameters. With the increase in the number of cross-sections and the increase in time length, the superiority of the regression results obtained by applying the first spatial factor is more obvious than those obtained by the sequential time averaging method.Figure6 shows the information extracted from the main space when the cross-sectional correlation is very strong. It is found that, in general, the spatial information extraction rate of the first spatial factor of class I and class II reaches more than 99%, and the spatial information extraction rate does not decrease with the increase of the number of cross-sections, and the extraction rate of class I and class II reaches more than 99%. The results also show that the parameter estimation results in the classified case are more satisfactory than those in the unclassified case and the information extraction rate in the classified case is higher than that in the unclassified case. Financial development and economic growth are interdependent and inextricably linked. Under the influence of China’s economic system reform and government policy guidance, China’s financial market has made great progress. However, in the context of economic globalisation, China’s financial development is facing certain challenges. There is a long-term equilibrium relationship between national interest rates and GDP; exchange rate movements can lead to a decline in the money supply and ultimately drive up prices, and both capital flows and foreign exchange inflows can contribute to financial development.Figure 6
Spatial information extraction rate at classification.Under the conditions of the same time length and spatial width, the extracted first principal spatial information rate tends to decrease as the overall spatial correlation decreases; under the conditions of the same time length and spatial width, as the overall spatial correlation decreases, the parameter estimation effect obtained by using the sequential averaging method becomes worse, and the parameter estimation effect obtained by using the principal spatial information method has better stability. There is no significant change in the parameter estimation effect.The study in this chapter found that it is necessary to do a preliminary judgmental analysis of the data before modeling and analysis, especially for the more complex panel data. Each quantitative model has its conditions of applicability, and it is under certain assumptions that the conclusions can be valid and the effects can be significant. When empirical analysis is carried out, the preliminary characteristics of the data should be judged and the appropriate model should be selected according to the different characteristics of the data so that the model can better serve the problem and can accurately explain the problem.


### 5.2. Results of Regional Economic Variability Analysis of Random Matrix

In the panel data model analysis, the spatial autocorrelation test is only a preliminary test for the existence of spatial effects, and a formal spatial panel data model must be built for in-depth testing. Before conducting regression analysis, it is necessary to conduct a smoothness test to check whether the series is smooth or not. EViews software was used to perform unit root tests on the eight variables. For the series of nonstationary variables, the first-order difference is processed directly and then the unit root test is performed.As shown in Figure7, the P value of the explanatory variable FGDP in the model is 0.001, which is a good fit and the model is significant. FGDP, GGDP, and KFD in the model are significant at the 5% level, and GGDP and KFD have a significant negative effect on LNGDP, while the coefficients of CZ/ZR and KGDP on LNGDP are not significant, which may be due to the explanatory variables themselves or the problem of model setting, such as the absence of spatial variables. However, according to the previous analysis, Guangxi’s economy will be influenced by the negative correlation of space, so the spatial factor should be added for analysis.Figure 7
Results of panel regression analysis of economic development level.The spatial autoregressive coefficient is 0.0265118, which is significant at the 5% level, so there is a spatial autoregressive effect. The value of the likelihood ratio test statistic for the spatial autoregressive coefficient is calculated to be 22.539 with aP value of 0.001 and the value of the Lagrangian test statistic is 19.698 with a P value of 0.001.From the estimation results of the spatially lagged panel random-effects model, first, the significance of the explanatory variables is estimated; FGDP and GGDP have significant effects on LNGDP at the 5% level, indicating that economic structure and government support have significant effects on the regional economy; specifically, the coefficient of FGDP is 14.994119, which is the highest coefficient among the five explanatory variables. The coefficient of GGDP is −4.329679, which is the smallest among the five explanatory variables, indicating that government consumption expenditure has a significant hindering effect on the regional economy of Guangxi. There is no significant effect of urbanization level, fixed investment, and foreign opening level on the regional economy of Guangxi. Second, the estimated parameter is 0.595967 and theP-value is 0.001. That is, the spatial lag variable is significant at the level of 1%, which indicates that the economic development of one region will have a significant impact on the surrounding neighboring regions, as shown in Figure 8.Figure 8
Results of model analysis.The statistic is significantly positive, which means that the regional economic development level has a positive spatial correlation; that is, in general, the economic development of the neighboring regions can drive the economic development of the region; on the contrary, if the Moran index is significantly negative, it indicates that the backward economy of the neighboring regions will affect the economic development of the region. Also, the index increases year by year, which indicates that the influence between municipalities is gradually increasing. However, the global Moran index only represents the overall spatial correlation of economic development level in a general way, and the type and degree of spatial correlation of each city cannot be expressed. To be able to describe the spatial correlation between each city specifically and not limited to the overall level, we use Moran scatter diagram to reflect the correlation effect of spatial individuals. The significance of local spatial autocorrelation analysis, as opposed to global spatial autocorrelation, is to find the location of local spatial autocorrelation that may be masked when global spatial autocorrelation does not exist. When global spatial autocorrelation exists, the analysis is explored for the presence of spatial heterogeneity. The location of spatial outliers or strong influence points is determined by looking for locations of local spatial autocorrelation that may exist that are inconsistent with the findings of global spatial autocorrelation, and the existence of spatial locations with a small amount of negative local spatial autocorrelation when the global spatial autocorrelation analysis concludes that there is positive global spatial autocorrelation.After the global spatial autocorrelation analysis, it is found that the economic development level of each city has autocorrelation and only generalizes the overall spatial correlation of the economic development level of Shandong province, but the type and degree of correlation of each city spatial correlation cannot be expressed; to reflect the difference of the local regional economic development level, the local spatial autocorrelation method is used to analyze the local relationship of regional economic development. In this section, the Moran scatter diagram and Lisa clustering diagram are used to reflect the correlation effect of spatial individuals. The least-squares methods are first used to estimate ordinary standard panel data without spatial lags, and the Lagrangian statistic LM test is used to determine whether there is a correlation between the spatial lag term and the spatial error term.


## 5.1. Clustering Results of Panel Data with Spatial Correlations

Now, to discuss the panel data with significant spatial correlation, on the one hand, to consider the extraction of spatial public information, so on the other hand, to reflect the dynamic endogenous influence relationship betweenm endogenous variables, the processing idea of this paper is to first extract all the public information in the spatial data because the extracted public information already contains most of the information in all cross-sectional data, the public information extracted from the space can represent most of the panel data information, and then the extracted public information is used to further measure the dynamic influence relationship between variables. By setting a set of original panel data, there are significant correlations between different cross-sections i and j. By starting from the correlation matrix or variance-covariance matrix of the data, the public information reflecting the common development trend of the space is extracted by applying the principal component idea.The data in Figure5 show that, in the case of strong cross-sectional correlation between cross-sections, but the degree of linear correlation is not very significant, the estimation effect of parameters with strong correlation is improved by applying the theoretical method of this paper, but the improvement is not high, and for the same time length and cross-sectional width data, the simulation results show that the estimated residuals of the parameter estimation results are smaller than those without correlation treatment. With the increase in time length and spatial number, the parameter relationship of the data with correlation treatment is closer to the true value parameters set by the model. Regional accessibility refers to the possibility of exchange of labour, products, capital, information and technology, etc. Spatial interaction is positively related to regional accessibility. Regional accessibility is influenced by four main factors: one is the relationship between transport distance and time, i.e., the shorter the transport distance and the less time, the easier it is to link economic activities and the better the accessibility, and conversely, the accessibility is poor.Figure 5
Simulation of data with strong spatial correlation.In general, when the data have a strong spatial correlation, the estimation results are slightly better using the method proposed in this paper. Both the PVAR model without correlation and the PVAR model dealing with correlation converge as the number of samples and the length of time increase, and the fitted data parametric estimates converge to the true value parameters. In the spatially correlated case, the VAR regression model using serial averaging outperforms the conventional PVAR model, and the model estimation effect using the vector autoregressive model with correlation handled outperforms the model estimation effect without correlation handled, and the parameters converge faster as the sample size and time length increase.The model fitting data show that the model parameters are also well estimated by using Pesaran’s sequential averaging method for data with high cross-sectional correlation in different cases, and relatively speaking, the test statistics of the parameters are more significant and closer to the true parameters by using the estimation method based on the main spatial information proposed in this paper. Fixing the time length constant and increasing the spatial widthN = 20, the parameter estimation effect under this scale is simulated to be more satisfactory. It is reasonable to use the first factor to build a dynamic VAR model between variables, and the estimation results show that the model estimation results of class one and class two are close to the true values, with small estimated residuals and significant t-test statistics. The parameter estimation results of the vector autoregressive model using the first spatial factor are ideal, and the parameter estimation results of both classes are close to the true value parameters. With the increase in the number of cross-sections and the increase in time length, the superiority of the regression results obtained by applying the first spatial factor is more obvious than those obtained by the sequential time averaging method.Figure6 shows the information extracted from the main space when the cross-sectional correlation is very strong. It is found that, in general, the spatial information extraction rate of the first spatial factor of class I and class II reaches more than 99%, and the spatial information extraction rate does not decrease with the increase of the number of cross-sections, and the extraction rate of class I and class II reaches more than 99%. The results also show that the parameter estimation results in the classified case are more satisfactory than those in the unclassified case and the information extraction rate in the classified case is higher than that in the unclassified case. Financial development and economic growth are interdependent and inextricably linked. Under the influence of China’s economic system reform and government policy guidance, China’s financial market has made great progress. However, in the context of economic globalisation, China’s financial development is facing certain challenges. There is a long-term equilibrium relationship between national interest rates and GDP; exchange rate movements can lead to a decline in the money supply and ultimately drive up prices, and both capital flows and foreign exchange inflows can contribute to financial development.Figure 6
Spatial information extraction rate at classification.Under the conditions of the same time length and spatial width, the extracted first principal spatial information rate tends to decrease as the overall spatial correlation decreases; under the conditions of the same time length and spatial width, as the overall spatial correlation decreases, the parameter estimation effect obtained by using the sequential averaging method becomes worse, and the parameter estimation effect obtained by using the principal spatial information method has better stability. There is no significant change in the parameter estimation effect.The study in this chapter found that it is necessary to do a preliminary judgmental analysis of the data before modeling and analysis, especially for the more complex panel data. Each quantitative model has its conditions of applicability, and it is under certain assumptions that the conclusions can be valid and the effects can be significant. When empirical analysis is carried out, the preliminary characteristics of the data should be judged and the appropriate model should be selected according to the different characteristics of the data so that the model can better serve the problem and can accurately explain the problem.


## 5.2. Results of Regional Economic Variability Analysis of Random Matrix

In the panel data model analysis, the spatial autocorrelation test is only a preliminary test for the existence of spatial effects, and a formal spatial panel data model must be built for in-depth testing. Before conducting regression analysis, it is necessary to conduct a smoothness test to check whether the series is smooth or not. EViews software was used to perform unit root tests on the eight variables. For the series of nonstationary variables, the first-order difference is processed directly and then the unit root test is performed.As shown in Figure7, the P value of the explanatory variable FGDP in the model is 0.001, which is a good fit and the model is significant. FGDP, GGDP, and KFD in the model are significant at the 5% level, and GGDP and KFD have a significant negative effect on LNGDP, while the coefficients of CZ/ZR and KGDP on LNGDP are not significant, which may be due to the explanatory variables themselves or the problem of model setting, such as the absence of spatial variables. However, according to the previous analysis, Guangxi’s economy will be influenced by the negative correlation of space, so the spatial factor should be added for analysis.Figure 7
Results of panel regression analysis of economic development level.The spatial autoregressive coefficient is 0.0265118, which is significant at the 5% level, so there is a spatial autoregressive effect. The value of the likelihood ratio test statistic for the spatial autoregressive coefficient is calculated to be 22.539 with aP value of 0.001 and the value of the Lagrangian test statistic is 19.698 with a P value of 0.001.From the estimation results of the spatially lagged panel random-effects model, first, the significance of the explanatory variables is estimated; FGDP and GGDP have significant effects on LNGDP at the 5% level, indicating that economic structure and government support have significant effects on the regional economy; specifically, the coefficient of FGDP is 14.994119, which is the highest coefficient among the five explanatory variables. The coefficient of GGDP is −4.329679, which is the smallest among the five explanatory variables, indicating that government consumption expenditure has a significant hindering effect on the regional economy of Guangxi. There is no significant effect of urbanization level, fixed investment, and foreign opening level on the regional economy of Guangxi. Second, the estimated parameter is 0.595967 and theP-value is 0.001. That is, the spatial lag variable is significant at the level of 1%, which indicates that the economic development of one region will have a significant impact on the surrounding neighboring regions, as shown in Figure 8.Figure 8
Results of model analysis.The statistic is significantly positive, which means that the regional economic development level has a positive spatial correlation; that is, in general, the economic development of the neighboring regions can drive the economic development of the region; on the contrary, if the Moran index is significantly negative, it indicates that the backward economy of the neighboring regions will affect the economic development of the region. Also, the index increases year by year, which indicates that the influence between municipalities is gradually increasing. However, the global Moran index only represents the overall spatial correlation of economic development level in a general way, and the type and degree of spatial correlation of each city cannot be expressed. To be able to describe the spatial correlation between each city specifically and not limited to the overall level, we use Moran scatter diagram to reflect the correlation effect of spatial individuals. The significance of local spatial autocorrelation analysis, as opposed to global spatial autocorrelation, is to find the location of local spatial autocorrelation that may be masked when global spatial autocorrelation does not exist. When global spatial autocorrelation exists, the analysis is explored for the presence of spatial heterogeneity. The location of spatial outliers or strong influence points is determined by looking for locations of local spatial autocorrelation that may exist that are inconsistent with the findings of global spatial autocorrelation, and the existence of spatial locations with a small amount of negative local spatial autocorrelation when the global spatial autocorrelation analysis concludes that there is positive global spatial autocorrelation.After the global spatial autocorrelation analysis, it is found that the economic development level of each city has autocorrelation and only generalizes the overall spatial correlation of the economic development level of Shandong province, but the type and degree of correlation of each city spatial correlation cannot be expressed; to reflect the difference of the local regional economic development level, the local spatial autocorrelation method is used to analyze the local relationship of regional economic development. In this section, the Moran scatter diagram and Lisa clustering diagram are used to reflect the correlation effect of spatial individuals. The least-squares methods are first used to estimate ordinary standard panel data without spatial lags, and the Lagrangian statistic LM test is used to determine whether there is a correlation between the spatial lag term and the spatial error term.


## 6. Conclusion

In this paper, a systematic study of vector autoregressive models based on panel data has been conducted. Combined with the principle of panel data clustering, the similarity metrics of the panel data model are constructed for the relationship characteristics between panel data variables, including the similarity metrics of linear structural relationships, the similarity metrics of nonlinear structural relationships, and the similarity metrics based on trajectory characteristics. After completing the similarity metrics for measuring the relationships between model variables, the clustering process for different variable relationship structures is given. The results of PVAR empirical analysis without considering the spatial correlation are not satisfactory; testing the spatial correlation, we found that China’s economic development and financial development data are very strong spatial correlation problems. The principal component principle is used to extract the main spatial information carrying the spatial information, and a vector autoregressive model based on the main spatial factors is established to analyze the spatial correlation of regional economic and financial data in China. Through the empirical research of this paper, it is found that economic development and financial development are dynamically influenced relationships. Among the factors affecting regional economic development, economic structure and government support have a significant influence on regional economic development in Guangxi, with economic structure as a positive influence and government expenditure as a negative influence; the level of urbanization, fixed investment, and the level of opening to the outside world have no significant influence on the regional economy in Guangxi. In addition, the spatial spillover effect is significant at the level of 1%, indicating that the surrounding adjacent areas have a significant influence on the GDP of this region. Combined with the spatial correlation analysis, the significant spatial effect, as well as the spatial distribution characteristics, should be reasonably exerted to promote the growth of GDP in the neighboring areas.



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*Source: 7993818-2022-08-19.xml*