## 2 Methodology and Data
### Methodology
#### 2.1.2 Spatial Correlation of Carbon Emissions  
There are similarities between economic environment or geographical environment in geographically adjacent areas. These factors may cause the regional carbon emissions to show certain correlations, that is, spatial correlations. [PERSON]'s I statistical variable is widely used as a measure of the spatial correlations between inter-regional economic variables. Indeed, the addition of local Anselin and Gestis-Ord hot spot analysis can more fully demonstrate the spatial characteristics of carbon emissions, because this part is not the focus of this paper, this part is not included in the text. The [PERSON]'s I formula is as follows.  
\[I=\frac{n\Sigma_{i=1}^{n}\Sigma_{j=1}^{n}w_{ij}(x_{i}-\overline{x})\big{(}x_{ j}-\overline{x}\big{)}}{\Big{(}\Sigma_{i=1}^{n}\Sigma_{j=1}^{n}w_{ij}\Big{)} \Sigma_{i=1}^{n}\big{(}x_{i}-\overline{x}\big{)}^{2}} \tag{2}\]  
\(\overline{x}=\frac{1}{n}\Sigma_{i=1}^{n}x_{i}\), \(x_{i}\) indicates the amount of carbon emissions in region \(i\); \(n\) is the number of regions; and, \(W\) represents the spatial weight matrix. Its value range between -1 and 1. The closer that the value of \(I\) is to 1, the stronger the positive spatial correlation of the carbon emissions between the regions is, \(I\) values closer to \(-1\), indicated stronger negative spatial correlations of carbon emissions between the regions, while those that are closer to zero indicates that there is no spatial correlation between carbon emissions.  
When measuring the spatial correlation of a region, it is first necessary to define a spatial weight matrix to describe the spatial neighbor relationship. The spatial weight matrix is generally divided into the adjacent spatial weight matrix, the geographic distance spatial weight matrix, and the economic distance spatial weight matrix. In the geographic distance spatial weight matrix, determining the Euclidean distance between two points is necessary. Therefore, the spatial distance weight matrix is not considered in the modeling analysis.  
The adjacent spatial weight matrix is mainly used to describe the geographic border relationship of regions, and it is an important weight matrix for describing relative positional relationships. In this paper, we use the first-order car adjacent to define the adjacent spatial weight matrix, as shown in Equation (3):  
\[\mathrm{W_{ij}}=\left\{\begin{array}{ll}1,&\text{when area i and area j have common borders}\\ 0,&\text{when area i and area j don't have common borders}\end{array}\right. \tag{3}\]  
The economic distance spatial weight matrix is used to describe the difference in economic development level between the two regions. It is a weight matrix that describes the absolute positional relationship between the regions from an economic perspective. The formula is as shown in Equation (4):  
\[W_{ij}=\left\{\begin{array}{ll}\frac{1}{|Y_{i}-Y_{j}|}&,i\
eq j\\ 0&,i=j\end{array}\right. \tag{4}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Spatial Scale** & **Degree of Freedom** & **The Composition of the Scale Variance** \\ \hline \(\alpha\) & \(l-1\) & \(\frac{\Sigma_{i=1}^{l}(\overline{X}_{i}-\overline{X})^{2}}{\Sigma_{i=1}^{l-1}( \overline{X}_{i}-\overline{X})^{2}}\) \\ \(\beta\) & \(\frac{l}{i=1}=(J_{i}-1)\) & \(\frac{\Sigma_{i=1}^{l}(\overline{X}_{i}-\overline{X}_{i})^{2}}{\Sigma_{i=1}^{l -1}(l-1)}\) \\ \(\gamma\) & \(\frac{l}{i=1}\Sigma_{i=1}^{l}(K_{ij}-1)\) & \(\frac{\Sigma_{i=1}^{l}\Sigma_{i=1}^{l}(X_{i}-\overline{X}_{i})^{2}}{\Sigma_{i=1 }^{l}\Sigma_{i=1}^{l}(K_{ij}-1)}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Scale variance composition at different scales.  
\(Y_{i}\) represents the economic development of region I; it is usually measured by Gross Domestic Product (GDP). This paper uses the real GDP average values from 2008 to 2015 (based on 2008), which eliminated the factor of inflation.