## 2 Index System Construction, Research Methods and Data Sources
### Research Methods

Table 1: _Cont._important, \(a_{1}=a_{2}=0.5\). Lastly, \(D\) stands for coupling coordination. The larger its value, the stronger the coupling coordination of the two subsystems. (2) Dagum Gini coefficient and its decomposition: The Gini coefficient proposed by [PERSON] (1997) is applied to test the regional difference of the coupling coordination in order to effectively avoid the sources of data overlapping and regional difference. Furthermore, the overall Gini coefficient is categorized into three parts: intra-regional difference and its contribution (\(G_{w}\)), inter-regional difference and its contribution (\(G_{nb}\)) and hypervariable density and its contribution (\(G_{t}\)) [44]. The calculation formula is the mathematical expression for estimation of the Gini coefficient, as follows:  
\[G=\frac{\sum\limits_{p=1}^{k}\sum\limits_{q=1}^{k}\sum\limits_{i=1}^{N_{p}} \sum\limits_{r=1}^{N_{q}}\left|D_{pi}-D_{qr}\right|}{2N^{2}\overline{D}} \tag{1}\]  
\(i\) and \(r\) represent the serial number of provinces in the different regions whereas \(N\) denotes the total provinces (31). Similarly, \(k\) shows the serial number of the region (total 4). Meanwhile, \(N_{p}\) and \(N_{q}\) indicate total provinces in the region \(p\) and the region \(q\). Lastly, \(D_{pi}\) and \(D_{qr}\) represent the coupling coordination level of \(i\) province in region \(p\) and \(r\) province in region \(q\), respectively. \(\overline{D}\) stands for the mean value of the coupling coordination degree between the digital economy and industrial green high-quality development. Gini coefficient and the contribution rate of intra-regional difference in the \(p\) region are as below:  
\[Gpp=\frac{\sum\limits_{i=1}^{N_{p}}\sum\limits_{r=1}^{N_{q}}\left|D_{pi}-D_{qr }\right|}{2N^{2}\overline{D}},\ G_{w}=\sum\limits_{p=1}^{N}G_{pp}h_{p}s_{p}\]  
The Gini coefficient value and the contribution rate between the \(p\) region and \(q\) region are as follows:  
\[Gpq=\frac{\sum\limits_{i=1}^{N_{p}}\sum\limits_{r=1}^{N_{q}}\left|D_{pi}-D_{qr }\right|}{N_{p}N_{q}(\overline{D}_{p}-\overline{D}_{q})},\ G_{nb}=\sum\limits_{p =2}^{N}\sum\limits_{q=1}^{p-1}G_{pq}(h_{p}s_{q}+h_{q}s_{p})F_{pq}\]  
The contribution of hypervariable density is:  
\[G_{t}=\sum\limits_{p=2}^{N}\sum\limits_{q=1}^{p-1}G_{pq}(h_{p}s_{q}+h_{q}s_{p })(1-F_{pq})\]  
Wherein, \(h_{p}=\frac{N_{p}}{N}\), \(s_{p}=\frac{N_{p}\overline{D}_{p}}{N\overline{D}_{p}}\); \(F_{pq}\) captures the relative influence of the coupling coordination degree between the digital economy and industrial green high-quality development in the \(p\) and \(q\) region, which can be expressed as follows: \(F_{pq}=\frac{(d_{pi}-h_{p})}{(d_{pi}+h_{pq})}\), \(d_{pq}\) and \(h_{pq}\) represent the mathematical expected values of the sum of all samples of \((D_{pi}-D_{qr})>0\) and \((D_{pi}-D_{qr})<0\) in regions \(p\) and \(q\), respectively. (3) B Convergence model: B Convergence is classified into absolute \(\beta\) Convergence and conditional \(\beta\) Convergence, developed by [PERSON] (2010). This paper applies the spatial Dubin model to verify the \(\beta\) Convergence process of the coupling coordination between the digital economy and industrial green high-quality development [45]. The proposed model is as follows:  
\[ \begin{array}{ll}D_{i,t+1}^{t}&=\ln(\frac{D_{i,t+1}}{D_{q}})=\alpha+\beta_ {1}\ln D_{it}+\beta_{1}Controls_{it}+\rho\omega_{ij}D_{i,t+1}^{t}+\gamma_{1} \omega_{ij}\ln D_{it}+\gamma_{2}\omega_{ij}Controls_{it}\\ &+\mu_{i}+\eta_{t}+\varepsilon_{it}\end{array}  \tag{2}\]and demonstrates the characteristic of convergence, with convergence speed as \(\
u=-\ln(1+\beta)/T\). Otherwise, it indicates divergence. Furthermore, \(\rho\) and \(\gamma\) show the spatial effect coefficient, with \(w\) as the spatial weight, and \(j\) as the \(j\)-th variable in the control variables. Subsequently, \(\mu_{i}\) reflects the individual effect, with \(\eta_{i}\) as the time effect, and \(\varepsilon_{i}\) as the random error term.