In this paper, the Hermitian, skew-Hermitian, Re-nonnegative definite, Re-positive definite, Re-nonnegative definite least-rank and Re-positive definite least-rank solutions of the matrix equation AXB=C are considered. The necessary and sufficient condition for the existence of such type of solution to the equation is provided and the explicit expression of the general solution is also given.
Citation: Jiao Xu, Hairui Zhang, Lina Liu, Huiting Zhang, Yongxin Yuan. A unified treatment for the restricted solutions of the matrix equation AXB=C[J]. AIMS Mathematics, 2020, 5(6): 6594-6608. doi: 10.3934/math.2020424
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Abstract
In this paper, the Hermitian, skew-Hermitian, Re-nonnegative definite, Re-positive definite, Re-nonnegative definite least-rank and Re-positive definite least-rank solutions of the matrix equation AXB=C are considered. The necessary and sufficient condition for the existence of such type of solution to the equation is provided and the explicit expression of the general solution is also given.
1.
Introduction
As environment issues have been increasingly prominent, low carbon development has gradually been a needed option for the economic development of all countries. China is striving to foster its ecological progress, seeking high-quality development, and it has been stated that China's economy has been transitioning from a phase of high-speed growth to a stage of high-quality development. Economic growth has gone from emphasizing quantity to stressing quality, which shows the intrinsic requirements of keeping a sustainable and healthy economic growth in China. Since China is at the pivotal moment of promoting ecological progress and achieving high-quality development, the exploration and grasp of the changing regularities of low carbon economy and quality of economic growth is an important guarantee for successfully achieving high-quality development. The construction of the Guangdong-Hong Kong-Macao Greater Bay Area is a key strategic plan of China. The goal is to build it into a first-class bay area and a world-class urban agglomeration and thus set it as an example of high-quality development. As a vital part of the Guangdong-Hong Kong-Macao Greater Bay Area, Guangdong Province is one of the most economically and socially developed provinces in China; therefore, it must be at the forefront of the provinces in China in exploring and achieving high-quality economic development.
There is limited research on measuring low-carbon economy and quality of economic growth from the perspective of new development concepts, and further research is needed on the regional imbalance and its changing patterns of low-carbon economy and quality of economic growth. The convergence of low-carbon economy and quality of economic growth has been a new exploration direction in recent years, with a very broad space for expansion. The major innovations of this paper include: On the one hand, we construct low-carbon economy and quality of economic growth measurement indicators from the perspective of new development concepts, and adopt the entropy method for measurement; on the other hand, we use the coefficient of variation and convergent model with fixed effects to analyze the convergences of low carbon economy and quality of economic growth.
This paper is conducive to better grasping the changing regularities of low carbon economy, quality of economic growth and their convergences, enriching related research on low carbon economy and quality of economic growth, and providing a reference for the formulation of urban and regional economic policies. This provides the realization of high-quality economic development and "double carbon goal" of China.
The Kyoto Protocol, aiming to address climate changes, promoted the formation of the concept of low carbon economy, which was first proposed in the UK's Energy White Paper in 2003 [15]. Chinese scholars have also begun to discuss low carbon economy in the context of China's economic development model [6,16,19], and since then, most studies have discussed low carbon and high-quality development at the macro level [23,44,47,50]. Some research has studied how firms can achieve low carbon development at the micro level [38].
Quality of economic growth has a definition in both narrow and broad sense. In a narrow sense, it can be analyzed from perspectives of total factor productivity [30,49], economic structure [26,45], and environment [17]. In a broad sense, which is widely recognized, scholars have begun to comprehensively analyze it from the aspects of the environment, income, welfare, education, law, and order [5,37]. The evaluation of quality of economic growth is developing in a multi-dimensional way. Also, different research needs to make the evaluation dimension subjective [27].
Some scholars put quality of economic growth into the theoretical framework of economic growth for analysis, and believed that it was an extension of the theoretical framework of economic growth and that quantitative growth and qualitative growth were two aspects of the same question in economic growth theory [33]. Ren's research team conducted in-depth discussions from this perspective [9,10,40], which had a broad impact on subsequent research. Therefore, most scholars regard in their research the quality of economic growth as a concept that is closely related to economic growth amount but has a broader connotation. In terms of measurement methods, by constructing a multi-dimensional indicator system, principal component analysis [10,21,40,48], and entropy method [41,42] are adopted to measure the quality of economic growth. The low carbon economy in economic growth quality has been a hot topic of concern in the academic community in recent years [20], with more and more related research focusing on different countries [2,12,22,28], a series of related studies have been conducted from different perspectives [51,52,53,54,55,56,57].
Early discussions on the phenomenon of economic growth convergence include the analysis of changes in per capita income within a neoclassical framework [32], and the "Yerkes-Dodson Law" in regional income convergence [43]. Since then, scholars have begun to carry out corresponding empirical research in line with local conditions [1,7]. Although some studies have also been questioned [14], they have played a great role in promoting relevant research. In the 1990s, scholars questioned the traditional empirical process and began to try new methods to empirically analyze the phenomenon of economic growth convergence [8,29,31]. Furthermore, scholars have also begun to explore the phenomenon of economic growth convergence in China [11,18,34], and influential representative results have emerged one after another, such as economic growth convergence mechanism based on the neoclassical analysis framework [3,4] and its empirical analysis [24,39], "club convergence" of China's economic growth [35]. In contrast, there are relatively few studies on the convergence of low carbon economy and quality of economic growth. In recent years, scholars have begun to try to analyze the convergence of quality of economic growth from different perspectives and with various methods [13,25,36,46], but it is at the early stage of exploration.
Most literatures have focused on the convergence of economic growth and income. Recently, scholars have begun to pay attention to the convergence of low-carbon economy and quality of economic growth, yet it is at the early stage of exploration, and there is a very broad space for expansion. As cities pay more and more attention to the quality of economic growth, how does their convergences of low carbon economy and quality of economic growth change? Based on the research status quo and the reality of urban economic development, we construct measurement indicators of low carbon economy and quality of economic growth in combination with the new development philosophy, and analyze the convergences in multiple dimensions.
2.
Measurement of low carbon economy and quality of economic growth
2.1. Construction of the indicator system
The essence of low carbon economy is to realize the green development of economy. Thus, green development indicators of quality of economic growth can be adopted to construct measurement indicators of low carbon economy (see Table 1). Therefore, the green dimension of quality of economic growth has reflected the low carbon development of economy. The secondary indicators are composed of coordination of energy efficiency and environmental pollution. Energy efficiency is shown by growth of energy consumption per unit of GRP, which is a negative indicator. Environmental pollution consists of four basic indicators, namely waste water discharge, industrial waste gas emission, industrial fume (dust) emission, and industrial solid waste production. In order to eliminate the biased error in the measurement of indicators, the indicators used here are relative indicators, which are all negative indicators. These indicators all have made energy efficiency and environmental pollution highly relative to GRP, which better shows the low carbon development of economy.
Table 1.
Indicator system of low carbon economy.
Primary indicators
Secondary indicators
Basic indicators
Calculation methods
Unit
Indicator attribute
Green development dimension (Low carbon development)
energy efficiency
energy efficiency (x11)
growth of energy consumption per unit of GRP
%
negative
environmental pollution
waste water discharge (x12)
total amount of waste water discharge/GRP
ton/ten thousand yuan
negative
industrial waste gas emission (x13)
total amount of industrial waste gas emission/GRP
M3/yuan
negative
industrial fume (dust) emission (x14)
total amount of industrial smoke (dust) emission/GRP
Many Chinese representative studies [10,33,40] have constructed indicator measurement systems of quality of economic growth, which involve some dimensions related to the new development philosophy, but few have constructed such system from the very five dimensions of the new development philosophy. Based on existing studies, we measured the quality of economic growth from the five dimensions of the new development philosophy, namely innovative, coordinated, green, open, and shared development dimensions. Quality of economic growth is first manifested in the transformation from traditional-factor-driven economic growth to innovation-driven economic growth and in the coordination within the economic system and among economic systems of different regions. Then, the external manifestation of the economic system is the coordinated development between humans and nature, as well as the linkage between domestic and international economies, whose ultimate goal is to widely share the final results. Table 2 is the indicator system of quality of economic growth with a total of 19 basic indicators.
Table 2.
Indicator system of quality of economic growth.
Primary indicators
Secondary indicators
Basic indicators
Calculation methods
Unit
Indicator attribute
Innovative development dimension
R&D input
R&D intensity (x1)
total R&D spending/GRP
%
positive
public budget on science and technology expenditure (x2)
public budget on science and technology expenditure/total public budget expenditure
%
positive
R&D result
number of licensed patent (x3)
piece
positive
Education input
public budget on education expenditure (x4)
public budget on education expenditure/total public budget expenditure
%
positive
Coordinated development dimension
coordination of industrial structure
proportion of the secondary industry (x5)
%
positive
proportion of the tertiary industry (x6)
%
positive
coordination of investment and consumption structure
investment rate(x7)
gross capital formation/expenditure-based GRP
%
positive
consumption rate (x8)
final consumption expenditure/expenditure-based GRP
%
positive
investment-to-consumption ratio(x9)
gross capital formation/final consumption expenditure
moderate
coordination of urban and rural areas
urban-rural income ratio (x10)
per capita disposable income of urban residents/per capita disposable income of rural residents
negative
Green development dimension
energy efficiency
energy efficiency (x11)
growth of energy consumption per unit of GRP
%
negative
environmental pollution
waste water discharge (x12)
total amount of waste water discharge/GRP
ton/ten thousand yuan
negative
industrial waste gas emission (x13)
total amount of industrial waste gas emission/GRP
M3/yuan
negative
industrial fume (dust) emission (x14)
total amount of industrial smoke (dust) emission/GRP
ton/one hundred million yuan
negative
industrial solid waste production (x15)
industrial solid waste production amount/GRP
ton/ten thousand yuan
negative
Open development dimension
opening structure
use of foreign capital (x16)
actual use of foreign capital/GRP
%
positive
degree of dependence on foreign trade (x17)
total import and export volume/GRP
%
positive
Shared development dimension
regional sharing
regional income sharing level (x18)
per capita GRP at prefecture level/per capita GRP at provincial level
The first dimension is the innovative development dimension. In this dimension, the secondary indicators are composed of R&D investment, R&D result, and education input. R&D investment has two basic indicators, R&D intensity and public budget on science and technology expenditure, which refer to the investment of R&D entities and the government's investment, respectively. The R&D result adopts the number of licensed patent as the basic indicator, which shows the output of R&D. Education input takes the proportion of education expenditure in the public budget as a basic indicator, which reflects the potential of R&D.
The second dimension is the coordinated development dimension. In this dimension, the secondary indicators are composed of coordination of industrial structure, coordination of investment and consumption structure, and coordination of urban and rural areas. Coordination of industrial structure consists of the proportion of the secondary industry and the proportion of the tertiary industry. Coordination of investment and consumption structure consists of investment rate and consumption rate. For the two rates, the higher is not the better. Xiang [45] believes that it is more appropriate that the investment rate is lower than 38% and the consumption rate is higher than or equal to 60%. Based on this standard, it can be inferred that investment-to-consumption ratio should be lower than 63.3%. Therefore, investment rate and consumption rate are regarded as positive indicators, and investment-to-consumption ratio, a moderate indicator whose standard is lower than 63.3%, is utilized for adjustment. Coordination of urban and rural areas takes urban-rural income ratio as its basic indicator, which is a negative indicator and shown as "per capita disposable income of urban residents/per capita disposable income of rural residents".
The third dimension is the green development dimension, whose indicators are consistent with the previous ones of low carbon economy.
The fourth dimension is the open development dimension. In this dimension, the secondary indicator is composed of an opening structure. An opening structure consists of two basic indicators, namely the use of foreign capital and the degree of dependence on foreign trade, which reflect the concept of economic openness. The use of foreign capital is shown as "actual use of foreign capital/GRP", and the degree of dependence on foreign trade is represented as "total import and export volume/GRP".
The fifth dimension is the shared development dimension. In this dimension, the secondary indicators are composed of regional sharing and welfare change. According to Wei and Li [41,42], the regional income sharing level refers to the ratio of per capita GRP at a prefecture level to per capita GRP at a provincial level, and welfare change takes per capita GRP as its basic indicator.
2.2. Methods and data
2.2.1. Measurement method
In this paper, the entropy value method and the multi-objective linear weighted function method, which are commonly used in academia, are utilized to measure the quality of economic growth of cities in the Pearl River Delta. Specific steps are as follows:
The first step is to standardize the variables. The measurement indicator xij is standardized through the range method, which can eliminate the influence of quantity variance and dimension difference on the measurement indicator.
Xij={xij−min(xij)max(xij)−min(xij)(Whenxij is a positive indicator)max(xij)−xijmax(xij)−min(xij)(When xij is a negative indicator).
(1)
In the above formulas, i represents city, j basic indicator, and xij and Xij original basic indicator and standardized basic indicator respectively.
If the indicator is a moderate one, it should be standardized in line with the following reference standards:
xij={zij−z0(when zij>z0, the more zij is greater than z0, the better)z0−zij(when zij<z0, the more zij is less than z0, the better).
(2)
In these formulas, zij is a moderate indicator, and z0 is the reference standard of a moderate indicator. The new transformed indicator should be standardized through the method of positive indicators as is shown in formula (1).
The second step is to calculate the information entropy Ej with standardized indicator Xij.
Ej=−k∑ni=1(Xij∑ni=1XijlnXij∑ni=1Xij).
(3)
In this formula, k = 1/ln(N). The data in this paper are all panel data, so N means the total number of samples.
The third step is to calculate the weight Wj of each basic indicator with the information entropy Ej.
Wj=(1−Ej)∑mj=1(1−Ej).
(4)
In this formula, 1-Ej represents the information utility value of the jth indicator.
The fourth step is to calculate the quality of economic growth indicator QEG in the multi-objective linear weighted function.
QEG=∑mj=1WjXij.
(5)
The value range of the quality of economic growth indicator QEG is (0, 1), and the larger the value, the higher the quality of economic growth of the city.
2.2.2. Data resources
According to the research needs and data availability, we selected the data of various prefecture-level cities in Guangdong Province from 2008 to 2019, which are mainly from China City Statistical Yearbook, Guangdong Statistical Yearbook, and the Statistical Yearbook of various prefecture-level cities in Guangdong. For some missing data, they were replaced by the average value (or the mean) of calculations with the series mean method, the proximal point mean method, the proximal point median method, the linear interpolation method, and the linear trend method at certain points.
(1) Urban-rural income ratio. As for the coordination of urban-rural areas, the secondary indicator in coordinated development dimension, when calculating the urban-rural income ratio, the per capita rural income was also the per capita net income of rural residents in some areas before 2013. Since there was no relevant statistics on rural residents in Shenzhen, the urban-rural income ratio in Shenzhen was replaced by the average value of Guangdong Province from 2003 to 2013. After 2014, the Pearl River Delta has collected relevant data, and then the urban-rural income ratio in Shenzhen is replaced by the mean of the data of the Pearl River Delta from 2014 to 2019.
(2) Environmental pollution. Environmental pollution, the second indicator in green development dimension, consists of four basic indicators, namely waste water discharge, industrial waste gas emission, industrial fume (dust) emission, and industrial solid waste production. When it is calculated, relative indicators are used in the measurement, that is, per unit (ten thousand yuan or yuan) GRP discharge or emission (ton or M³).
(3) The sum of actual use of foreign capital and total import and export volume. When the basic indicators in open development dimension are measured and calculated, the sum of actual use of foreign capital and total import and export volume should be converted to Chinese yuan based on the average exchange rate for that year in Guangdong Province.
3.1.
Convergence of quality of economic growth
3.1. σ convergence
The σ convergence of quality of economic growth (low carbon economy) refers to the trend that the gap of quality of economic growth (low carbon economy) between various regions gradually narrows over time, and it is usually measured by the coefficient of variation as follows①:
σrt=√[∑nri=1(QEGit−−QEGrt)2]/nr−QEGrt.
(6)
① The variable is QEG when measuring quality of economic growth and is LCE when measuring low carbon economy, the same below.
r refers to the region, and nr represents the number of prefecture-level cities in r region. Coefficient of variation is the ratio of standard deviation to the mean of quality of economic growth (low carbon economy) indicator in the prefecture-level city of the region (r).
3.2. β convergence
The β convergence of quality of economic growth (low carbon economy) refers to the trend that the gap of growth rates of economic growth quality (low carbon economy) between various regions gradually narrows over time, and the β convergence model based on standard panel data model is as follows:
ln(QEGi,t+1QEGi,t)=α+βlnQEGit+δXit+μi+γt+ϵit.
(7)
When δ = 0, it is an absolute β convergence model; when δ≠0, it is a conditional β convergence model. The government is regarded as an import factor that affects quality of economic growth (low carbon economy); therefore, it is defined as a major control variable. When β is significantly less than 0, convergence exists, and convergence rate is −ln(1+β)T. On the contrary, divergence exists.
3.3. Club convergence
The club convergence of quality of economic growth (low carbon economy) refers to the convergence of regions with similar characteristics. According to the division of economic regions in Guangdong Province in the Guangdong Statistical Yearbook, we divide economic regions in Guangdong Province into the Pearl River Delta, the East Wing, the West Wing and the mountain area. According to Opinions on Constructing the New Development Pattern of "One Core, One Belt, One Area" and Promoting Coordinated Regional Development of Guangdong Province (July, 2019) issued by Guangdong Provincial Party Committee and the People's Government of Guangdong Province, this paper divides economic regions in Guangdong Province into One Core (the Pearl River Delta), One Belt (the coastal economic belt), and One Area (the northern ecological development area). The convergences of these "clubs" are analyzed separately.
3.4. Relevant variables and data description
It is difficult to directly observe governmental influence. However, the amount of fiscal expenditure reflects the ability of the government to regulate and intervene in the economy, so fiscal expenditure is chosen as a substitute variable for governmental influence. The relevant variable data is the data of various prefecture-level cities in Guangdong Province from 2008 to 2019, mainly collected from the China City Statistical Yearbook, Guangdong Statistical Yearbook and the statistical yearbooks of various prefecture-level cities. The descriptive statistics of the main variables of the model are shown in Table 3.
Table 3.
Descriptive statistics of the main variables.
Variables
Meaning
Measured value
Mean
Standard deviation
Median
Minimum
Maximum
LCE(D3)
low carbon economy indicator (score in green development dimension)
4.
Convergence analysis of economic growth quality and low carbon economy
4.1. σ convergence analysis
4.1.1. σ convergence analysis of economic regions
The σ convergence of low carbon economy divided by economic regions of Guangdong Province is shown in Figure 1. From Figure 1, we can see that the σ convergence coefficient of the low carbon economy in the province is generally declining, indicating that σ convergence appears in the data of low carbon economy in the province, and the regional imbalance of the low carbon economy in the province tends to decrease. The σ convergence coefficients of low carbon economy in the Pearl River Delta, the East Wing and the mountain areas show an overall downward trend, and the σ convergence coefficient of the low carbon economy in the West Wing increases before the year of 2016, but also tends to decline since then, indicating that σ convergence appears in the data of the regions, and the regional imbalance of low carbon economy tends to decrease.
Figure 1.σ convergences of low carbon economy divided by economic regions in Guangdong Province.
Figure 2 showed the σ convergence of quality of economic growth various economic regions of Guangdong Province. It can be seen from Figure 2 that the overall σ convergence coefficient of quality of economic growth was on the rise, indicating that there was no σ convergence in the data of the province and that its regional imbalance in quality of economic growth was obvious. The overall σ convergence coefficient of the Pearl River Delta showed an upward trend, indicating that there was no σ convergence in the data of the Pearl River Delta and that its regional imbalance in quality of economic growth was obvious. Although the σ convergence coefficients in the East Wing and the mountain area increased slightly in some years, they showed a downward trend on the whole, indicating that σ convergence occurred in the data of both the East Wing and the mountain area and that the regional imbalance of quality of economic growth in the two regions tended to be improved. The σ convergence coefficient in the data of the West Wing fluctuated greatly, and there was also a relatively large fluctuation from 2018 to 2019, indicating that there was no σ convergence in the data of the West Wing and that its regional imbalance in the quality of economic growth was obvious.
Figure 2.σ convergences of quality of economic growth various economic regions in Guangdong Province.
There was no σ convergence in the data of quality of economic growth of Guangdong Province, which meant the regional imbalance in quality of economic growth in the province was obvious. With further analysis of the σ convergences of the province's five dimensions (see Figure A1), the main reason should be that the regional imbalances in innovative, open, and shared development dimensions were quite obvious. In particular, the regional imbalance in innovative development dimension had a tendency to further expand. It can be explained that the gap between the Pearl River Delta and other regions tended to widen because the Pearl River Delta had more favorable conditions of agglomerating innovative elements than other regions. Figure A2 showed the trends of the mean of innovative development dimension in the Pearl River Delta and in the non-Pearl River Delta region and the gap between them. It is clear that the gap between the two regions continued to widen.
There was no σ convergence in the data of the Pearl River Delta, which meant the regional imbalance in quality of economic growth in the Pearl River Delta was obvious. With further analysis of the σ convergences of the five dimensions of the Pearl River Delta (see Figure A3), the main reason should be that the regional imbalances in innovative, open, and shared development dimensions were quite obvious. In particular, the regional imbalance of innovative development dimension had a trend to further expand. This may be because the gap between Guangzhou and Shenzhen and other regions in the Pearl River Delta tended to widen due to the favorable conditions of agglomerating innovative elements in Guangzhou and Shenzhen. Figure A4 showed the trends of the mean of innovative development dimension in Guangzhou and Shenzhen and in other regions in the Pearl River Delta and the gap between them. It is clear that the gap continued to widen.
Although the regional imbalance in quality of economic growth in the East Wing got slightly worse in some certain years, its overall situation tended to improve. With further analysis of the σ convergences of the five dimensions of the East Wing (see Figure A5), it can be concluded that obvious regional imbalance in quality of economic growth appeared in the innovative, open, coordinated, and shared development dimensions in the East Wing. The regional imbalance in shared development dimension had the most obvious narrowing trend; and the regional imbalance in innovative and open development dimensions fluctuated greatly.
The regional imbalance in quality of economic growth in the West Wing was obvious and fluctuated greatly. With further analysis of the σ convergences of the five dimensions of the West Wing (see Figure A6), the main reason can be concluded that the regional imbalance in open development dimension was obvious and had a relatively high fluctuation while the regional imbalance in other four dimensions was not obvious and had a downward trend.
Although the regional imbalance in quality of economic growth in the mountain area got slightly worse in some certain years, its overall situation tended to improve. With further analysis of the σ convergences of the five dimensions of the West Wing (see Figure A7), it can be concluded that obvious regional imbalance in quality of economic growth appeared in innovative and open development dimensions in the West Wing. Moreover, the regional imbalance in innovative and open development dimensions had a relatively severe fluctuation; and the regional imbalance in shared development dimension had the most obvious narrowing trend.
4.1.2. σ convergence analysis of "One Core, One Belt, One Area"
The σ convergences of low carbon economy of regions divided into "One Core, One Belt, One Area" are shown in Figure 3. It can be seen from Figure 3 that the σ convergence coefficient of each region shows an overall downward trend, indicating that there is a σ convergence in the data of each region, and the regional imbalance of the low-carbon economy tends to narrow.
Figure 3.σ convergences of low carbon economy of regions divided into "One Core, One Belt, One Area".
The σ convergences of quality of economic growth of regions divided into "One Core, One Belt, One Area" are shown in Figure 4. It can be seen from Figure 4 that the σ convergence coefficient of the coastal economic belt generally showed an upward trend from 2008 to 2019, which meant there was no σ convergence in the data of the coastal economic belt, and its regional imbalance in the quality of economic growth was obvious②.
Figure 4.σ convergences of quality of economic growth of regions divided into "One Core, One Belt, One Area".
② "One Core" and "One Area" correspond to the Pearl River Delta and the mountain area respectively, which have been analyzed in previous part, so they are not analyzed in this part.
With further analysis of the σ convergences of the five dimensions of economic growth quality of the coastal economic belt (see Figure A8), the main reason for its regional imbalance in quality of economic growth should be that the regional imbalance in innovative, open, and shared development dimensions was quite obvious. In particular, the regional imbalance in innovative development dimension had a tendency to further expand. This might have the following reasons: First, the gap between the seven coastal cities in the Pearl River Delta and the East and West Wings tended to widen in innovative development dimension. Figure A9 showed the trends of the mean of innovative development dimension of the seven coastal cities in the Pearl River Delta and the East and West Wings. Second, in terms of the seven coastal cities in the Pearl River Delta, the gap between Guangzhou, Shenzhen, and the other five coastal cities in the Pearl River Delta tended to widen in innovative development dimension. Figure A10 showed the trend of the mean of innovative development dimension in Guangzhou, Shenzhen, and the other five coastal cities in the Pearl River Delta. Third, obvious imbalance also appeared in innovative development dimension in the East and West Wings. Figure A11 showed the trends of the mean of innovative development dimension in the East and West Wings, and from the figure, it can be seen that the gap between these two regions had a fluctuation and a growing trend as well.
4.2. β convergence analysis
4.2.1. Absolute β convergence analysis
Tables 4 and 5 showed the Hausman test results of the absolute β convergence model of low carbon economy, and from the x2 value, it is more appropriate to choose the fixed effect model and reject the null hypothesis that there is no systematic difference in coefficients (i.e., ai is not correlated with xit, and random effects model should be chosen) at a significant level of 1%. Table 4 shows the estimated results of absolute β convergence of low carbon economy of different economic regions. According to the estimated results, the β coefficients of the absolute convergence model of the province and the subregions are all negative, which is significant at the 1% level, indicating that there is absolute β convergence in the data of the province and the subregions, and the growth rate of the low carbon economy is gradually converging. Table 4 shows the estimated absolute β convergence of the low carbon economy of regions divided into the "One Core, One Belt, One Area", and from the estimated results, the β coefficient of the absolute convergence model of the coastal economic belt is negative, which is significant at the 1% level, indicating that there is absolute β convergence in the data of the coastal economic belt, and the growth rate of the low carbon economy is gradually converging.
Table 4.
Estimated results of absolute β convergence model of low carbon economy (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.2329***
-0.2511***
-0.2138***
-0.3382***
-0.2304***
(-7.3841)
(-5.3896)
(-5.0191)
(-2.8374)
(-3.3772)
Constants
-0.9114***
-0.9744***
-0.8278***
-1.3220***
-0.9245***
(-7.1782)
(-5.3075)
(-4.8397)
(-2.8090)
(-3.2038)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
7.82***
sigma_u
0.0195
0.0131
0.0053
0.0268
0.0207
sigma_e
0.0622
0.0245
0.0329
0.0430
0.1157
R2
0.2069
0.2461
0.3924
0.2173
0.1888
N
231
99
44
33
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 5.
Estimated results of absolute β convergence model of low carbon economy (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.2329***
-0.2511***
-0.2316***
-0.2304***
(-7.3841)
(-5.3896)
(-7.1835)
(-3.3772)
Constants
-0.9114***
-0.9744***
-0.8991***
-0.9245***
(-7.1782)
(-5.3075)
(-7.0374)
(-3.2038)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
7.82***
sigma_u
0.0195
0.0131
0.0120
0.0207
sigma_e
0.0622
0.0245
0.0307
0.1157
R2
0.2069
0.2461
0.2707
0.1888
N
231
99
154
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the x2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Tables 6 and 7 show the Hausman test results of the absolute β convergence model of quality of economic growth, and according to the results, it is more appropriate to choose the fixed effect model. Table 6 shows the estimated results of absolute β convergence of various economic regions in Guangdong. According to the estimated results, the β coefficient of the absolute convergence model of Guangdong Province was negative and significant at the 1% level, indicating that there was absolute β convergence in the data of the province and that the growth rates of economic quality gradually converged. The β coefficient of the absolute convergence model of the Pearl River Delta was insignificantly positive, indicating that there was no absolute β convergence in the data of the Pearl River Delta. The β coefficients of the absolute convergence models of the East Wing, the West Wing, and the mountain area were all negative and significant at the 5% level, indicating that there were absolute β convergences in the data of the East Wing, the West Wing, and the mountain area, and that the growth rates of economic quality gradually converged. Table 7 showed the estimated results of absolute β convergence of regions divided into "One Core, One Belt, One Area". The results showed that the β coefficients of the absolute convergence model of the coastal economic belt are all negative and significant at the 5% level, indicating that there is absolute β convergence in the data of the coastal economic belt and that the growth rates of economic quality gradually converged.
Table 6.
Estimated results of absolute β convergence model of quality of economic growth (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.1745***
0.0107
-0.3177***
-0.2269**
-0.3773***
(-4.3876)
(0.2027)
(-3.0155)
(-2.2221)
(-4.4361)
Constants
-0.2663***
0.0307
-0.5972***
-0.4514**
-0.7226***
(-4.1426)
(0.5089)
(-2.9565)
(-2.1130)
(-4.3814)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
20.69***
sigma_u
0.0889
0.0189
0.0328
0.0192
0.0420
sigma_e
0.0539
0.0459
0.0416
0.0695
0.0578
R2
0.0843
0.0005
0.1891
0.1455
0.2865
N
231
99
44
33
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 7.
Estimated results of absolute β convergence model of quality of economic growth (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.1745***
0.0107
-0.1202**
-0.3773***
(-4.3876)
(0.2027)
(-2.5908)
(-4.4361)
Constants
-0.2663***
0.0307
-0.1647**
-0.7226***
(-4.1426)
(0.5089)
(-2.3176)
(-4.3814)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
20.69***
sigma_u
0.0889
0.0189
0.0679
0.0420
sigma_e
0.0539
0.0459
0.0507
0.0578
R2
0.0843
0.0005
0.0461
0.2865
N
231
99
154
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
According to the estimated results of the five dimensions of quality of economic growth in Guangdong Province (see Table A1), the β coefficients of the absolute convergence model of the five dimensions were all negative and significant at the 1% level, indicating that there were absolute β convergences in the data of the five dimensions of Guangdong Province and that their growth rates of economic quality gradually converged. According to the estimated results of the five dimensions of quality of economic growth in the Pearl River Delta (see Table A2), the β coefficients of the absolute convergence model of the coordinated, green, and shared development dimensions were all negative and significant at the 1% level, and the β coefficient of the absolute convergence model of the innovative development dimension was negative and significant at the 10% level, which indicated that the growth rates of the innovative, coordinated, green, and shared development dimensions of the Pearl River Delta gradually converged. The β coefficient of the absolute convergence model of the open development dimension was insignificantly negative, indicating that there was no absolute convergence in the growth rate of the open development dimension of the Pearl River Delta.
According to the estimated results of the five dimensions of quality of economic growth in the East Wing (see Table A3), the β coefficients of the absolute convergence model of the innovative, coordinated, green (LCE), and shared development dimensions were negative and significant at the 1% level, indicating that the growth rates of economic quality of these four dimensions gradually converged. The β coefficient of the absolute convergence model of the open development dimension was insignificantly negative, indicating that there was no absolute convergence in the growth rate of the open development dimension of the East Wing. Moreover, according to the estimated results of the five dimensions of the West Wing (see Table A4), the β coefficients of the absolute convergence model of the five development dimensions were all negative. The β coefficient of the absolute convergence model was significant at the 5% level in the coordinated development dimension and at the 1% level in the other four dimensions, which indicated that the growth rates of economic quality of the five dimensions in West Wing gradually converged.
According to the estimated results of the five dimensions of quality of economic growth in the mountain area (see Table A5), the β coefficients of the absolute convergence model of the five development dimensions were all negative. The β coefficient of the absolute convergence model was significant at the 5% level in the open development dimension and at the 1% level in the other four dimensions, which indicated that the growth rates of economic quality of the five dimensions in West Wing gradually converged. Furthermore, according to the estimated results of the five dimensions of the coastal economic belt (see Table A6), the β coefficients of the absolute convergence model of the five development dimensions were all negative and significant at the 1% level, which indicated that the growth rates of economic quality of the five dimensions in the coastal economic belt gradually converged.
4.2.2. Conditional β convergence analysis
Tables 8 and 9 show the Hausman test results of the conditional β convergence model of low carbon economy. From the value of χ2, it is more appropriate to choose a fixed-effect model to reject the null hypothesis that there is no systematic difference in coefficients at a significant level of 1%. Tables 8 and 9 show the conditional β convergence estimates categorized by economic regions and "One Core, One Belt, One Area", respectively. From the estimated results, the β coefficients of the provincial and regional conditional convergence models are negative, which is significant at the 1% level, indicating that there is a conditional β convergence in the data of Guangdong Province and the regions, and that the growth rate of the low carbon economy is gradually converging. According to the convergence rate formula, the convergence rate of Guangdong Province is 0.0364, the Pearl River Delta (One Core) 0.0703, the East Wing 0.0373, the West Wing 0.0443, the mountain area (One Area) 0.0721, and the coastal economic belt 0.0357.
Table 8.
Estimated results of conditional β convergence model of low carbon economy (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.3302***
-0.5385***
-0.3366***
-0.3857**
-0.5474***
(-7.4553)
(-6.9107)
(-3.5979)
(-2.6611)
(-4.4519)
lnPFE
0.0349***
0.0383***
0.0301
0.0104
0.1599***
(3.0686)
(4.3971)
(1.4694)
(0.5898)
(3.0061)
Constants
-1.4937***
-2.3381***
-1.4676***
-1.5629**
-3.0765***
(-6.5815)
(-6.6362)
(-3.1434)
(-2.4918)
(-4.0257)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
15.72***
sigma_u
0.0303
0.0146
0.0071
0.0275
0.0848
sigma_e
0.0609
0.0223
0.0325
0.0435
0.1073
R2
0.2413
0.3819
0.4251
0.2269
0.3173
N
231
99
44
33
55
Convergence rate
0.0364
0.0703
0.0373
0.0443
0.0721
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 9.
Estimated results of conditional β convergence model of low carbon economy (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.3302***
-0.5385***
-0.3250***
-0.5474***
(-7.4553)
(-6.9107)
(-6.6772)
(-4.4519)
lnPFE
0.0349***
0.0383***
0.0186**
0.1599***
(3.0686)
(4.3971)
(2.5246)
(3.0061)
Constants
-1.4937***
-2.3381***
-1.3724***
-3.0765***
(-6.5815)
(-6.6362)
(-6.0854)
(-4.0257)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
15.72***
sigma_u
0.0303
0.0146
0.0128
0.0848
sigma_e
0.0609
0.0223
0.0301
0.1073
R2
0.2413
0.3819
0.3029
0.3173
N
231
99
154
55
Convergence rate
0.0364
0.0703
0.0357
0.0721
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Tables 10 and 11 show the Hausman test results of the conditional β convergence model of quality of economic growth. According to the results, it was more appropriate to choose the fixed-effect model. Table 10 showed the estimated results of conditional β convergence of regions divided into various economic regions. According to the estimated results, the β coefficient of the conditional convergence model of Guangdong Province was negative and significant at the 1% level, indicating that there was conditional β convergence in the data of the province and that the growth rates of economic quality gradually converged. The β coefficient of the conditional convergence model of the Pearl River Delta was insignificantly negative, indicating that there was no conditional β convergence in the data of the Pearl River Delta. The β coefficients of the conditional convergence models in the East Wing, West Wing, and the mountain area were all negative and significant at the 5% level, indicating that there was conditional β convergence in the East Wing, West Wing, and the mountain area and that their growth rates of economic quality gradually converged. According to the convergence rate formula, the convergence rates of Guangdong Province, the East Wing, the West Wing, and the mountain area were 0.0254, 0.0319, 0.0657, and 0.0500, respectively. Table 11 showed the estimated results of conditional β convergence of regions divided into "One Core, One Belt, One Area". According to the estimated results, the β coefficients of the conditional convergence model of the coastal economic belt were all negative and significant at the 1% level, indicating that there was conditional β convergence in data of the coastal economic belt and that the growth rates of economic quality gradually converged. According to the convergence rate formula, the convergence rate of the coastal economic belt was 0.0204.
Table 10.
Estimated results of the conditional β convergence model of quality of economic growth (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.2436***
-0.0694
-0.2963**
-0.5144**
-0.4231***
(-5.1577)
(-1.0082)
(-2.5859)
(-2.5889)
(-4.8011)
lnPFE
0.0219***
0.0227*
-0.0065
0.0758
0.0251
(2.6258)
(1.7799)
(-0.5041)
(1.6693)
(1.6445)
Constants
-0.4975***
-0.1975
-0.5247**
-1.4451**
-0.9393***
(-4.5851)
(-1.3974)
(-2.1022)
(-2.2926)
(-4.4961)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
30.23***
sigma_u
0.1082
0.0250
0.0307
0.0916
0.0518
sigma_e
0.0532
0.0453
0.0420
0.0674
0.0568
R2
0.1137
0.0352
0.1945
0.2228
0.3246
N
231
99
44
33
55
Convergence rate
0.0254
0.0319
0.0657
0.0500
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 11.
Estimated results of the conditional β convergence model of quality of economic growth (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.2436***
-0.0694
-0.2008***
-0.4231***
(-5.1577)
(-1.0082)
(-3.1038)
(-4.8011)
lnPFE
0.0219***
0.0227*
0.0199*
0.0251
(2.6258)
(1.7799)
(1.7736)
(1.6445)
Constants
-0.4975***
-0.1975
-0.3989***
-0.9393***
(-4.5851)
(-1.3974)
(-2.6647)
(-4.4961)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
30.23***
sigma_u
0.1082
0.0250
0.0959
0.0518
sigma_e
0.0532
0.0453
0.0503
0.0568
R2
0.1137
0.0352
0.0673
0.3246
N
231
99
154
55
Convergence rate
0.0254
0.0204
0.0500
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
According to the estimated results of quality of economic growth in the five dimensions of Guangdong Province (see Table A7), the β coefficients of the conditional convergence model of the five dimensions were all negative and significant at the 1% level, indicating that there was conditional β convergence in the data of the five dimensions of Guangdong Province and that their growth rates of economic quality gradually converged. The convergence rates of the five dimensions were 0.0376, 0.0584, 0.0364, 0.0634, and 0.0331, respectively. According to the estimated results of quality of economic growth in the five dimensions of the Pearl River Delta (see Table A8), the β coefficients of the conditional convergence model of the innovative, coordinated, green, and shared development dimensions were all negative and significant at the 1% level, indicating that their growth rates of economic quality gradually converged. The convergence rates of these four dimensions were 0.0320, 0.0516, 0.0703, and 0.0214, respectively. The β coefficient of the conditional convergence model of the open development dimension was insignificantly negative, indicating that there was neither absolute β convergence nor conditional β convergence in quality of economic growth of the Pearl River Delta. With further analysis, the main reason why absolute and conditional β convergences did not appear in the data of the Pearl River Delta was that there was neither absolute β convergence nor conditional β convergence in the data of the open development dimension while they both appeared in the data of the other four dimensions. The advantages and conditions of opening to the outside world were quite different among the cities in the Pearl River Delta. Guangzhou and Shenzhen continued to expand their opening-up advantages by virtue of their favorable conditions. While Zhaoqing and Jiangmen also continued to expand their opening to the outside world, the speed of opening was difficult to catch up with that of Guangzhou and Shenzhen due to the limitations of basic conditions and geographical locations and some other reasons.
According to the estimated results of quality of economic growth in the five dimensions of the East Wing (see Table A9), the β coefficients of the conditional convergence model of the five dimensions were all negative and significant at the 1% level, indicating that there was conditional β convergence in the data of the East Wing and that the growth rates of economic quality of the five dimensions gradually converged. The convergence rates of the five dimensions were 0.1391, 0.0382, 0.0373, 0.1193, and 0.0404, respectively. Moreover, according to the estimated results of quality of economic growth in the five dimensions of the West Wing (see Table A10), the β coefficients of the conditional convergence model of the innovative, coordinated, green, and open development dimensions were negative and significant at the 1% level, indicating that the growth rates of economic quality of these four dimensions in West Wing gradually converged. The convergence rates of the four dimensions were 0.1533, 0.0533, 0.0443, and 0.1836, respectively. The β coefficient of the conditional convergence model of the shared development was insignificantly negative, indicating that there was no conditional convergence in the data of the shared development dimension in the West Wing.
According to the estimated results of quality of economic growth in the five dimensions of the mountain area (see Table A11), the β coefficients of the conditional convergence model of the five development dimensions were all negative and significant at the 1% level, indicating that there was conditional β convergence in the data of the mountain area and that their growth rates gradually converged. The convergence rates of the five dimensions were 0.0583, 0.0759, 0.0721, 0.0575, and 0.0430, respectively. Furthermore, according to the estimated results of quality of economic growth in the five dimensions of the coastal economic belt (see Table A12), the β coefficients of the conditional convergence model of the five development dimensions were all negative and significant at the 1% level, indicating that there was conditional β convergence in the data of the coastal economic belt and that their growth rates gradually converged. The convergence rates of the five dimensions were 0.0435, 0.0340, 0.0357, 0.1058, and 0.0332, respectively.
4.3. Analysis of club convergence
From the above analysis, some obvious differences between the "clubs" can be summarized as follows:
The first difference is in σ convergence. In terms of various economic regions, σ convergence appears in the data of both the East Wing and the mountain area, and the regional imbalance in quality of economic growth tends to be improved. On the contrary, σ convergence appears neither in the data of the Pearl River Delta nor in the data of the West Wing, and the regional imbalance in quality of economic growth in these regions is obvious. In particular, the σ convergence coefficient of the West Wing fluctuates greatly. In terms of "One Core, One Belt, One Area", σ convergence appears in the data of the northern ecological development area, and the regional imbalance of quality of economic growth tends to be improved. On the contrary, σ convergence appears neither in the data of the core region nor in the data of the coastal economic belt, and the regional imbalance of quality of economic growth is obvious. In terms of the five dimensions, each dimension varies greatly in all regions.
The second difference lies in β convergence. "Club difference" exists in convergence rates of low carbon economy of all regions mentioned. Regarding quality of economic growth, in the Pearl River Delta, there is no absolute β convergence or conditional β convergence. In the East Wing, the West Wing, and the mountain area, there are both absolute β convergence and conditional β convergence, and their growth rates of economic quality gradually converge. Their convergence rates are 0.0319, 0.0657, and 0.0500, respectively, and "club difference" shows in the convergence rates. In the coastal economic belt and the northern ecological development area, there are both absolute β convergence and conditional β convergence, and their growth rates of economic quality gradually converge. Their convergence rates are 0.0204, and 0.0500, respectively, and "club difference" shows in the convergence rates. From the perspective of the five dimensions, in the open development dimension, absolute convergence appears neither in data of the Pearl River Delta nor in the data of the East Wing, and no conditional convergence appears in the data of the Pearl River Delta; and in the shared development dimension, no conditional convergence appears in the data of the West Wing.
5.
Conclusions
Based on the samples of 21 prefecture-level cities in Guangdong Province, we adopt the entropy method to measure the quality of economic growth, and analyze the coefficient of variation and panel data convergence model with fixed effects.
Our major findings are as follows:
(1) Convergence of low carbon economy performs better than that of quality of economic growth.
(2) Low carbon economy of Guangdong Province shows a σ convergence tendency, and the regional imbalance in low carbon economy tends to be alleviated. There is no σ convergence in quality of economic growth of Guangdong Province, and the regional imbalance in quality of economic growth in the province is obvious. In terms of quality of economic growth of different regions, σ convergence appears in the data of the East Wing and the mountain area (northern ecological development area), not in the data of the Pearl River Delta (core region), the West Wing, or the coastal economic belt ("One Belt"); and especially in the West Wing, the σ convergence coefficient of the West Wing fluctuates greatly. In terms of quality of economic growth in the five dimensions, each dimension varies greatly in all regions.
(3) There are both absolute and conditional β convergences in low carbon economy and quality of economic growth of Guangdong Province. Also, the quality of economic growth in all the five dimensions appear absolute and conditional β convergences. In terms of quality of economic growth of different regions, no absolute β convergence or conditional β convergence appears in the data of the Pearl River Delta (core region), and there is no absolute or conditional β convergence in the data of the open development dimension. Absolute and conditional β convergences appear in the data of the East Wing, the West Wing, the mountain area (northern ecological development area), and the coastal economic belt. Except that there is no absolute β convergence in the data of the open development dimension of the East Wing, and no conditional β convergence in the data of the shared development dimension of the West Wing, absolute and conditional β convergences appear in the data of the other four dimensions of the East Wing, the West Wing, the mountain area (northern ecological development area), and the coastal economic belt.
(4) "Club difference" exists in convergence rates of low carbon economy of all regions mentioned. "Club difference" shows in the σ convergence, absolute β convergence, conditional β convergence, and five-dimensional convergence of quality of economic growth in various regions in Guangdong Province. This paper helps to better grasp the changing regularities of low carbon economy, quality of economic growth and their convergences, enrich related research, and provide reference for the formulation of urban and regional economic policies and the realization of high-quality economic development and "double carbon goal" of China.
The major implications are as follows: First, when seeking development, attention should be given to the overall and balanced improvement of the low carbon economy (green development) and for four dimensions, namely innovative, coordinated, open, and shared development, and importance should be attached to the balanced development of the five dimensions so as to achieve the comprehensive improvement of quality of economic growth. Second, the regional imbalance in quality of economic growth should be prevented, and the imbalance in quality of economic growth among regions should be gradually reduced in order to achieve coordinated development and enhance the overall quality and comprehensive competitiveness of the economy in Guangdong Province. Third, providing the actual situation in Guangdong Province, the gap between the Pearl River Delta and other regions in Guangdong Province needs to be narrowed, and so does the gap between the different areas within the Pearl River Delta, the East Wing, the West Wing, the mountain area (the northern ecological development area), and the coastal economic belt.
Use of AI tools declaration
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
Conflict of interest
All authors declare no conflicts of interest in this paper.
Appendix
Figure A1.σ convergences of the five dimensions of quality of economic growth of Guangdong Province.
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X. Zhang, L. Sheng, Q. Xu, A note on the real positive solutions of the operator equation AXB = C, Journal of Shanghai Normal University (Natural Sciences), 37 (2008), 454-458.
[21]
Y. Yuan, K. Zuo, The Re-nonnegative definite and Re-positive definite solutions to the matrixequation AXB = D, Appl. Math. Comput., 256 (2015), 905-912.
[22]
L. Wu, The Re-positive definite solutions to the matrix inverse problem AX = B, Linear Algebra Appl., 174 (1992), 145-151.
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L. Wu, B. Cain, The Re-nonnegative definite solutions to the matrix inverse problem AX = B, Linear Algebra Appl., 236 (1996), 137-146. doi: 10.1016/0024-3795(94)00142-1
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J. Groß, Explicit solutions to the matrix inverse problem AX = B, Linear Algebra Appl., 289 (1999), 131-134. doi: 10.1016/S0024-3795(97)10008-8
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X. Liu, Comments on "The common Re-nnd and Re-pd solutions to the matrix equations AX = Cand XB = D", Appl. Math. Comput., 236 (2014), 663-668.
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A. Ben-Israel, T. N. E. Greville, Generalized Inverses: Theory and Applications (second edition), Springer, New York, 2003.
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H. W. Braden, The equationsA⊤X±X⊤A=B, SIAM J. Matrix Anal. Appl., 20 (1998), 295-302.
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Y. Yuan, On the symmetric solutions of a class of linear matrix equation, Chinese J. Eng. Math., 15 (1998), 25-29.
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L. Mihályffy, An alternative representation of the generalized inverse of partitioned matrices, Linear Algebra Appl., 4 (1971), 95-100. doi: 10.1016/0024-3795(71)90031-0
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A. Albert, Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. Appl. Math., 17 (1969), 434-440. doi: 10.1137/0117041
[31]
Y. Tian, H. Wang, Relations between least-squares and least-rank solutions of the matrix equationAXB = C, Appl. Math. Comput., 219 (2013), 10293-10301.
Table 4.
Estimated results of absolute β convergence model of low carbon economy (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.2329***
-0.2511***
-0.2138***
-0.3382***
-0.2304***
(-7.3841)
(-5.3896)
(-5.0191)
(-2.8374)
(-3.3772)
Constants
-0.9114***
-0.9744***
-0.8278***
-1.3220***
-0.9245***
(-7.1782)
(-5.3075)
(-4.8397)
(-2.8090)
(-3.2038)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
7.82***
sigma_u
0.0195
0.0131
0.0053
0.0268
0.0207
sigma_e
0.0622
0.0245
0.0329
0.0430
0.1157
R2
0.2069
0.2461
0.3924
0.2173
0.1888
N
231
99
44
33
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 5.
Estimated results of absolute β convergence model of low carbon economy (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.2329***
-0.2511***
-0.2316***
-0.2304***
(-7.3841)
(-5.3896)
(-7.1835)
(-3.3772)
Constants
-0.9114***
-0.9744***
-0.8991***
-0.9245***
(-7.1782)
(-5.3075)
(-7.0374)
(-3.2038)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
7.82***
sigma_u
0.0195
0.0131
0.0120
0.0207
sigma_e
0.0622
0.0245
0.0307
0.1157
R2
0.2069
0.2461
0.2707
0.1888
N
231
99
154
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the x2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 6.
Estimated results of absolute β convergence model of quality of economic growth (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.1745***
0.0107
-0.3177***
-0.2269**
-0.3773***
(-4.3876)
(0.2027)
(-3.0155)
(-2.2221)
(-4.4361)
Constants
-0.2663***
0.0307
-0.5972***
-0.4514**
-0.7226***
(-4.1426)
(0.5089)
(-2.9565)
(-2.1130)
(-4.3814)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
20.69***
sigma_u
0.0889
0.0189
0.0328
0.0192
0.0420
sigma_e
0.0539
0.0459
0.0416
0.0695
0.0578
R2
0.0843
0.0005
0.1891
0.1455
0.2865
N
231
99
44
33
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 7.
Estimated results of absolute β convergence model of quality of economic growth (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.1745***
0.0107
-0.1202**
-0.3773***
(-4.3876)
(0.2027)
(-2.5908)
(-4.4361)
Constants
-0.2663***
0.0307
-0.1647**
-0.7226***
(-4.1426)
(0.5089)
(-2.3176)
(-4.3814)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
20.69***
sigma_u
0.0889
0.0189
0.0679
0.0420
sigma_e
0.0539
0.0459
0.0507
0.0578
R2
0.0843
0.0005
0.0461
0.2865
N
231
99
154
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 8.
Estimated results of conditional β convergence model of low carbon economy (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.3302***
-0.5385***
-0.3366***
-0.3857**
-0.5474***
(-7.4553)
(-6.9107)
(-3.5979)
(-2.6611)
(-4.4519)
lnPFE
0.0349***
0.0383***
0.0301
0.0104
0.1599***
(3.0686)
(4.3971)
(1.4694)
(0.5898)
(3.0061)
Constants
-1.4937***
-2.3381***
-1.4676***
-1.5629**
-3.0765***
(-6.5815)
(-6.6362)
(-3.1434)
(-2.4918)
(-4.0257)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
15.72***
sigma_u
0.0303
0.0146
0.0071
0.0275
0.0848
sigma_e
0.0609
0.0223
0.0325
0.0435
0.1073
R2
0.2413
0.3819
0.4251
0.2269
0.3173
N
231
99
44
33
55
Convergence rate
0.0364
0.0703
0.0373
0.0443
0.0721
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 9.
Estimated results of conditional β convergence model of low carbon economy (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.3302***
-0.5385***
-0.3250***
-0.5474***
(-7.4553)
(-6.9107)
(-6.6772)
(-4.4519)
lnPFE
0.0349***
0.0383***
0.0186**
0.1599***
(3.0686)
(4.3971)
(2.5246)
(3.0061)
Constants
-1.4937***
-2.3381***
-1.3724***
-3.0765***
(-6.5815)
(-6.6362)
(-6.0854)
(-4.0257)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
15.72***
sigma_u
0.0303
0.0146
0.0128
0.0848
sigma_e
0.0609
0.0223
0.0301
0.1073
R2
0.2413
0.3819
0.3029
0.3173
N
231
99
154
55
Convergence rate
0.0364
0.0703
0.0357
0.0721
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 10.
Estimated results of the conditional β convergence model of quality of economic growth (of various economic regions).
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.2436***
-0.0694
-0.2963**
-0.5144**
-0.4231***
(-5.1577)
(-1.0082)
(-2.5859)
(-2.5889)
(-4.8011)
lnPFE
0.0219***
0.0227*
-0.0065
0.0758
0.0251
(2.6258)
(1.7799)
(-0.5041)
(1.6693)
(1.6445)
Constants
-0.4975***
-0.1975
-0.5247**
-1.4451**
-0.9393***
(-4.5851)
(-1.3974)
(-2.1022)
(-2.2926)
(-4.4961)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
30.23***
sigma_u
0.1082
0.0250
0.0307
0.0916
0.0518
sigma_e
0.0532
0.0453
0.0420
0.0674
0.0568
R2
0.1137
0.0352
0.1945
0.2228
0.3246
N
231
99
44
33
55
Convergence rate
0.0254
0.0319
0.0657
0.0500
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Table 11.
Estimated results of the conditional β convergence model of quality of economic growth (of "One Core, One Belt, One Area").
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.2436***
-0.0694
-0.2008***
-0.4231***
(-5.1577)
(-1.0082)
(-3.1038)
(-4.8011)
lnPFE
0.0219***
0.0227*
0.0199*
0.0251
(2.6258)
(1.7799)
(1.7736)
(1.6445)
Constants
-0.4975***
-0.1975
-0.3989***
-0.9393***
(-4.5851)
(-1.3974)
(-2.6647)
(-4.4961)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
30.23***
sigma_u
0.1082
0.0250
0.0959
0.0518
sigma_e
0.0532
0.0453
0.0503
0.0568
R2
0.1137
0.0352
0.0673
0.3246
N
231
99
154
55
Convergence rate
0.0254
0.0204
0.0500
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Green development dimension (Low carbon development)
energy efficiency
energy efficiency (x11)
growth of energy consumption per unit of GRP
%
negative
environmental pollution
waste water discharge (x12)
total amount of waste water discharge/GRP
ton/ten thousand yuan
negative
industrial waste gas emission (x13)
total amount of industrial waste gas emission/GRP
M3/yuan
negative
industrial fume (dust) emission (x14)
total amount of industrial smoke (dust) emission/GRP
ton/one hundred million yuan
negative
industrial solid waste production (x15)
industrial solid waste production amount/GRP
ton/ten thousand yuan
negative
Primary indicators
Secondary indicators
Basic indicators
Calculation methods
Unit
Indicator attribute
Innovative development dimension
R&D input
R&D intensity (x1)
total R&D spending/GRP
%
positive
public budget on science and technology expenditure (x2)
public budget on science and technology expenditure/total public budget expenditure
%
positive
R&D result
number of licensed patent (x3)
piece
positive
Education input
public budget on education expenditure (x4)
public budget on education expenditure/total public budget expenditure
%
positive
Coordinated development dimension
coordination of industrial structure
proportion of the secondary industry (x5)
%
positive
proportion of the tertiary industry (x6)
%
positive
coordination of investment and consumption structure
investment rate(x7)
gross capital formation/expenditure-based GRP
%
positive
consumption rate (x8)
final consumption expenditure/expenditure-based GRP
%
positive
investment-to-consumption ratio(x9)
gross capital formation/final consumption expenditure
moderate
coordination of urban and rural areas
urban-rural income ratio (x10)
per capita disposable income of urban residents/per capita disposable income of rural residents
negative
Green development dimension
energy efficiency
energy efficiency (x11)
growth of energy consumption per unit of GRP
%
negative
environmental pollution
waste water discharge (x12)
total amount of waste water discharge/GRP
ton/ten thousand yuan
negative
industrial waste gas emission (x13)
total amount of industrial waste gas emission/GRP
M3/yuan
negative
industrial fume (dust) emission (x14)
total amount of industrial smoke (dust) emission/GRP
ton/one hundred million yuan
negative
industrial solid waste production (x15)
industrial solid waste production amount/GRP
ton/ten thousand yuan
negative
Open development dimension
opening structure
use of foreign capital (x16)
actual use of foreign capital/GRP
%
positive
degree of dependence on foreign trade (x17)
total import and export volume/GRP
%
positive
Shared development dimension
regional sharing
regional income sharing level (x18)
per capita GRP at prefecture level/per capita GRP at provincial level
%
positive
welfare change
per capita GRP (x19)
yuan
positive
Variables
Meaning
Measured value
Mean
Standard deviation
Median
Minimum
Maximum
LCE(D3)
low carbon economy indicator (score in green development dimension)
231
0.0182
0.0029
0.0188
0.0063
0.0230
lnLCE
logarithm of low carbon economy indicator
231
-4.0229
0.1960
-3.9734
-5.0654
-3.7714
ln(LCEi,t+1LCEi,t)
growth rate of low carbon economy
231
0.0257
0.0690
0.0157
-0.1870
0.4242
QEG
growth quality indicator
231
0.2242
0.1175
0.1650
0.0893
0.6832
lnQEG
logarithm of growth quality indicator
231
-1.6132
0.4716
-1.8015
-2.4155
-0.3810
ln(QEGi,t+1QEGi,t)
growth rate of economic quality
231
0.0153
0.0564
0.0189
-0.2408
0.2447
D1
score in innovative development dimension
231
0.0536
0.0503
0.0317
0.0122
0.3601
D2
score in coordinated development dimension
231
0.0633
0.0125
0.0618
0.0368
0.1077
D4
score in open development dimension
231
0.0415
0.0371
0.0294
0.0005
0.1968
D5
score in shared development dimension
231
0.0475
0.0479
0.0225
0.0010
0.1869
lnPFE
public fiscal expenditure
231
5.4648
0.9156
5.3735
3.6225
8.4325
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.2329***
-0.2511***
-0.2138***
-0.3382***
-0.2304***
(-7.3841)
(-5.3896)
(-5.0191)
(-2.8374)
(-3.3772)
Constants
-0.9114***
-0.9744***
-0.8278***
-1.3220***
-0.9245***
(-7.1782)
(-5.3075)
(-4.8397)
(-2.8090)
(-3.2038)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
7.82***
sigma_u
0.0195
0.0131
0.0053
0.0268
0.0207
sigma_e
0.0622
0.0245
0.0329
0.0430
0.1157
R2
0.2069
0.2461
0.3924
0.2173
0.1888
N
231
99
44
33
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.2329***
-0.2511***
-0.2316***
-0.2304***
(-7.3841)
(-5.3896)
(-7.1835)
(-3.3772)
Constants
-0.9114***
-0.9744***
-0.8991***
-0.9245***
(-7.1782)
(-5.3075)
(-7.0374)
(-3.2038)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
7.82***
sigma_u
0.0195
0.0131
0.0120
0.0207
sigma_e
0.0622
0.0245
0.0307
0.1157
R2
0.2069
0.2461
0.2707
0.1888
N
231
99
154
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the x2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.1745***
0.0107
-0.3177***
-0.2269**
-0.3773***
(-4.3876)
(0.2027)
(-3.0155)
(-2.2221)
(-4.4361)
Constants
-0.2663***
0.0307
-0.5972***
-0.4514**
-0.7226***
(-4.1426)
(0.5089)
(-2.9565)
(-2.1130)
(-4.3814)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
20.69***
sigma_u
0.0889
0.0189
0.0328
0.0192
0.0420
sigma_e
0.0539
0.0459
0.0416
0.0695
0.0578
R2
0.0843
0.0005
0.1891
0.1455
0.2865
N
231
99
44
33
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.1745***
0.0107
-0.1202**
-0.3773***
(-4.3876)
(0.2027)
(-2.5908)
(-4.4361)
Constants
-0.2663***
0.0307
-0.1647**
-0.7226***
(-4.1426)
(0.5089)
(-2.3176)
(-4.3814)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
20.69***
sigma_u
0.0889
0.0189
0.0679
0.0420
sigma_e
0.0539
0.0459
0.0507
0.0578
R2
0.0843
0.0005
0.0461
0.2865
N
231
99
154
55
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.3302***
-0.5385***
-0.3366***
-0.3857**
-0.5474***
(-7.4553)
(-6.9107)
(-3.5979)
(-2.6611)
(-4.4519)
lnPFE
0.0349***
0.0383***
0.0301
0.0104
0.1599***
(3.0686)
(4.3971)
(1.4694)
(0.5898)
(3.0061)
Constants
-1.4937***
-2.3381***
-1.4676***
-1.5629**
-3.0765***
(-6.5815)
(-6.6362)
(-3.1434)
(-2.4918)
(-4.0257)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
15.72***
sigma_u
0.0303
0.0146
0.0071
0.0275
0.0848
sigma_e
0.0609
0.0223
0.0325
0.0435
0.1073
R2
0.2413
0.3819
0.4251
0.2269
0.3173
N
231
99
44
33
55
Convergence rate
0.0364
0.0703
0.0373
0.0443
0.0721
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.3302***
-0.5385***
-0.3250***
-0.5474***
(-7.4553)
(-6.9107)
(-6.6772)
(-4.4519)
lnPFE
0.0349***
0.0383***
0.0186**
0.1599***
(3.0686)
(4.3971)
(2.5246)
(3.0061)
Constants
-1.4937***
-2.3381***
-1.3724***
-3.0765***
(-6.5815)
(-6.6362)
(-6.0854)
(-4.0257)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
15.72***
sigma_u
0.0303
0.0146
0.0128
0.0848
sigma_e
0.0609
0.0223
0.0301
0.1073
R2
0.2413
0.3819
0.3029
0.3173
N
231
99
154
55
Convergence rate
0.0364
0.0703
0.0357
0.0721
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Pearl River Delta
East Wing
West Wing
Mountain area
β
-0.2436***
-0.0694
-0.2963**
-0.5144**
-0.4231***
(-5.1577)
(-1.0082)
(-2.5859)
(-2.5889)
(-4.8011)
lnPFE
0.0219***
0.0227*
-0.0065
0.0758
0.0251
(2.6258)
(1.7799)
(-0.5041)
(1.6693)
(1.6445)
Constants
-0.4975***
-0.1975
-0.5247**
-1.4451**
-0.9393***
(-4.5851)
(-1.3974)
(-2.1022)
(-2.2926)
(-4.4961)
Individual effects
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Hausman test
30.23***
sigma_u
0.1082
0.0250
0.0307
0.0916
0.0518
sigma_e
0.0532
0.0453
0.0420
0.0674
0.0568
R2
0.1137
0.0352
0.1945
0.2228
0.3246
N
231
99
44
33
55
Convergence rate
0.0254
0.0319
0.0657
0.0500
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Pearl River Delta
Coastal economic belt
Northern ecological development area
β
-0.2436***
-0.0694
-0.2008***
-0.4231***
(-5.1577)
(-1.0082)
(-3.1038)
(-4.8011)
lnPFE
0.0219***
0.0227*
0.0199*
0.0251
(2.6258)
(1.7799)
(1.7736)
(1.6445)
Constants
-0.4975***
-0.1975
-0.3989***
-0.9393***
(-4.5851)
(-1.3974)
(-2.6647)
(-4.4961)
Individual effects
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Hausman test
30.23***
sigma_u
0.1082
0.0250
0.0959
0.0518
sigma_e
0.0532
0.0453
0.0503
0.0568
R2
0.1137
0.0352
0.0673
0.3246
N
231
99
154
55
Convergence rate
0.0254
0.0204
0.0500
Note: Figures in parentheses are t statistics; Hausman test here refers to the χ2 value of Hausman test; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.1745***
-0.1539***
-0.3196***
-0.2329***
-0.2114***
-0.2170***
(-4.3876)
(-4.5696)
(-8.3474)
(-7.3841)
(-4.6537)
(-14.7338)
Constants
-0.2663***
-0.4330***
-0.8663***
-0.9114***
-0.8870***
-0.7071***
(-4.1426)
(-3.9510)
(-8.1408)
(-7.1782)
(-5.3113)
(-13.3941)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0889
0.1399
0.0544
0.0195
0.2260
0.1868
sigma_e
0.0539
0.1318
0.0612
0.0622
0.3260
0.0673
R2
0.0843
0.0908
0.2500
0.2069
0.0939
0.5095
N
231
231
231
231
231
231
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Pearl River Delta
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
0.0107
-0.0598*
-0.2292***
-0.2511***
-0.0089
-0.1262***
(0.2027)
(-1.7876)
(-5.3152)
(-5.3896)
(-0.2210)
(-5.0324)
Constants
0.0307
-0.0554
-0.6278***
-0.9744***
-0.1173
-0.2938***
(0.5089)
(-0.6505)
(-5.1436)
(-5.3075)
(-1.0428)
(-4.5587)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0189
0.0485
0.0334
0.0131
0.0526
0.0648
sigma_e
0.0459
0.1044
0.0383
0.0245
0.1468
0.0355
R2
0.0005
0.0347
0.2410
0.2461
0.0005
0.2215
N
99
99
99
99
99
99
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
East Wing
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.3177***
-0.3258***
-0.3167***
-0.2138***
-0.0484
-0.2575***
(-3.0155)
(-3.1007)
(-4.4393)
(-5.0191)
(-0.6290)
(-11.2578)
Constants
-0.5972***
-1.1375***
-0.8423***
-0.8278***
-0.3302
-1.0162***
(-2.9565)
(-2.9681)
(-4.3558)
(-4.8397)
(-1.0599)
(-10.0187)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0328
0.0842
0.0493
0.0053
0.0228
0.0429
sigma_e
0.0416
0.1568
0.0272
0.0329
0.2637
0.0673
R2
0.1891
0.1978
0.3357
0.3924
0.0100
0.7647
N
44
44
44
44
44
44
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
West Wing
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.2269**
-0.3010***
-0.1706**
-0.3382***
-0.7793***
-0.1987***
(-2.2221)
(-2.7719)
(-2.7374)
(-2.8374)
(-4.1944)
(-4.2849)
Constants
-0.4514**
-1.1439**
-0.4770**
-1.3220***
-4.1057***
-0.6946***
(-2.1130)
(-2.6701)
(-2.5937)
(-2.8090)
(-4.2913)
(-3.9068)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0192
0.0238
0.0300
0.0268
0.8482
0.0425
sigma_e
0.0695
0.1120
0.0436
0.0430
0.6164
0.0677
R2
0.1455
0.2094
0.2053
0.2173
0.3776
0.3877
N
33
33
33
33
33
33
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Mountain area
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.3773***
-0.3214***
-0.4474***
-0.2304***
-0.1048**
-0.2009***
(-4.4361)
(-3.2295)
(-4.5658)
(-3.3772)
(-2.0520)
(-5.7055)
Constants
-0.7226***
-1.1767***
-1.1556***
-0.9245***
-0.5669***
-0.8200***
(-4.3814)
(-3.1150)
(-4.4656)
(-3.2038)
(-2.7146)
(-5.0883)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0420
0.0789
0.0539
0.0207
0.0396
0.0789
sigma_e
0.0578
0.1498
0.1042
0.1157
0.2012
0.1001
R2
0.2865
0.1755
0.2985
0.1888
0.0791
0.3992
N
55
55
55
55
55
55
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Coastal economic belt
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.1202**
-0.1447***
-0.2278***
-0.2316***
-0.3411***
-0.2286***
(-2.5908)
(-3.8652)
(-6.8822)
(-7.1835)
(-5.0351)
(-13.9911)
Constants
-0.1647**
-0.3751***
-0.6193***
-0.8991***
-1.3186***
-0.6950***
(-2.3176)
(-3.2207)
(-6.6534)
(-7.0374)
(-5.4158)
(-12.8137)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0679
0.1406
0.0370
0.0120
0.4151
0.1869
sigma_e
0.0507
0.1293
0.0369
0.0307
0.3579
0.0565
R2
0.0461
0.0970
0.2541
0.2707
0.1543
0.5848
N
154
154
154
154
154
154
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Guangdong Province
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.2436***
-0.3384***
-0.4740***
-0.3302***
-0.5023***
-0.3051***
(-5.1577)
(-6.1530)
(-8.7327)
(-7.4553)
(-7.9999)
(-13.3076)
lnPFE
0.0219***
0.1166***
0.0440***
0.0349***
-0.3650***
0.0663***
(2.6258)
(4.1540)
(3.8933)
(3.0686)
(-6.2114)
(4.8526)
Constants
-0.4975***
-1.6683***
-1.5361***
-1.4937***
0.0471
-1.3841***
(-4.5851)
(-5.2870)
(-7.6615)
(-6.5815)
(0.2191)
(-9.3360)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.1082
0.2079
0.1015
0.0303
0.6955
0.2431
sigma_e
0.0532
0.1270
0.0592
0.0609
0.3002
0.0639
r2_w
0.1137
0.1605
0.3010
0.2413
0.2357
0.5594
N
231
231
231
231
231
231
Convergence rate
0.0254
0.0376
0.0584
0.0364
0.0634
0.0331
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Pearl River Delta
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.0694
-0.2971***
-0.4331***
-0.5385***
-0.0652
-0.2098***
(-1.0082)
(-3.5375)
(-4.8539)
(-6.9107)
(-1.0868)
(-5.2699)
lnPFE
0.0227*
0.1706***
0.0436**
0.0383***
-0.0588
0.0317***
(1.7799)
(3.0559)
(2.5867)
(4.3971)
(-1.2639)
(2.6485)
Constants
-0.1975
-1.6853***
-1.4677***
-2.3381***
0.0815
-0.6996***
(-1.3974)
(-3.1236)
(-4.2471)
(-6.6362)
(0.4221)
(-4.2287)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0250
0.0910
0.0886
0.0146
0.0955
0.0966
sigma_e
0.0453
0.0998
0.0371
0.0223
0.1463
0.0343
R2
0.0352
0.1273
0.2946
0.3819
0.0184
0.2790
N
99
99
99
99
99
99
Convergence rate
0.0320
0.0516
0.0703
0.0214
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
East Wing
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.2963**
-0.7835***
-0.3432***
-0.3366***
-0.7307***
-0.3587***
(-2.5859)
(-4.5828)
(-3.4461)
(-3.5979)
(-5.1721)
(-8.7568)
lnPFE
-0.0065
0.2317***
0.0042
0.0301
-0.7279***
0.0980***
(-0.5041)
(3.2111)
(0.3864)
(1.4694)
(-5.3161)
(2.8763)
Constants
-0.5247**
-3.9314***
-0.9345***
-1.4676***
0.4750
-1.9395***
(-2.1022)
(-4.2013)
(-3.0298)
(-3.1434)
(1.6791)
(-5.8026)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0307
0.1624
0.0541
0.0071
0.3033
0.0865
sigma_e
0.0420
0.1409
0.0276
0.0325
0.2023
0.0618
R2
0.1945
0.3690
0.3383
0.4251
0.4323
0.8068
N
44
44
44
44
44
44
Convergence rate
0.0319
0.1391
0.0382
0.0373
0.1193
0.0404
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
West Wing
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.5144**
-0.8149***
-0.4438***
-0.3857**
-0.8672***
-0.1544
(-2.5889)
(-4.8703)
(-2.8907)
(-2.6611)
(-4.4591)
(-1.4413)
lnPFE
0.0758
0.2121***
0.0697*
0.0104
-0.2923
-0.0242
(1.6693)
(3.6584)
(1.9309)
(0.5898)
(-1.3485)
(-0.4604)
Constants
-1.4451**
-4.2697***
-1.6446**
-1.5629**
-3.0394**
-0.3996
(-2.2926)
(-4.6078)
(-2.6117)
(-2.4918)
(-2.4689)
(-0.6006)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0916
0.1416
0.0856
0.0275
0.8703
0.0244
sigma_e
0.0674
0.0937
0.0417
0.0435
0.6079
0.0686
R2
0.2228
0.4651
0.2987
0.2269
0.4155
0.3923
N
33
33
33
33
33
33
Convergence rate
0.0657
0.1533
0.0533
0.0443
0.1836
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Mountain area
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.4231***
-0.4736***
-0.5659***
-0.5474***
-0.4689***
-0.3767***
(-4.8011)
(-3.8243)
(-4.7944)
(-4.4519)
(-4.2717)
(-5.8174)
lnPFE
0.0251
0.0935*
0.0553*
0.1599***
-0.4019***
0.1472***
(1.6445)
(1.9680)
(1.7277)
(3.0061)
(-3.6474)
(3.1364)
Constants
-0.9393***
-2.2291***
-1.7493***
-3.0765***
0.0021
-2.3706***
(-4.4961)
(-3.4365)
(-4.0955)
(-4.0257)
(0.0087)
(-4.5926)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0518
0.1309
0.0769
0.0848
0.2140
0.2036
sigma_e
0.0568
0.1456
0.1022
0.1073
0.1799
0.0921
R2
0.3246
0.2371
0.3395
0.3173
0.2790
0.5014
N
55
55
55
55
55
55
Convergence rate
0.0500
0.0583
0.0759
0.0721
0.0575
0.0430
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
Variables
Coastal economic belt
Innovative development
Coordinated development
Green development (LCE)
Open development
Shared development
β
-0.2008***
-0.3802***
-0.3123***
-0.3250***
-0.6877***
-0.3056***
(-3.1038)
(-5.4462)
(-5.3474)
(-6.6772)
(-8.0891)
(-12.6408)
lnPFE
0.0199*
0.1502***
0.0181*
0.0186**
-0.4166***
0.0550***
(1.7736)
(3.9232)
(1.7501)
(2.5246)
(-5.8409)
(4.1457)
Constants
-0.3989***
-1.9408***
-0.9575***
-1.3724***
-0.2366
-1.2561***
(-2.6647)
(-4.6856)
(-4.4699)
(-6.0854)
(-0.8251)
(-8.6773)
Individual effects
Yes
Yes
Yes
Yes
Yes
Yes
Time effect
Yes
Yes
Yes
Yes
Yes
Yes
sigma_u
0.0959
0.2283
0.0596
0.0128
1.0412
0.2264
sigma_e
0.0503
0.1231
0.0366
0.0301
0.3217
0.0534
r2_w
0.0673
0.1877
0.2703
0.3029
0.3219
0.6308
N
154
154
154
154
154
154
Convergence rate
0.0204
0.0435
0.0340
0.0357
0.1058
0.0332
Note: Figures in parentheses are t statistics; *, **, and *** refer to the significant level of 10%, 5%, and 1% respectively.
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