## 3 Coupling and Coordination Modeling
### Comprehensive Evaluation Index  
The comprehensive evaluation index \(U\) quantifies how well the subsystem is developed based on the setup predictors. More specifically, \(U\) is defined by Equation (1):  
\[U=\sum_{i_{1}}^{n}w_{i}x_{i} \tag{1}\]  
where \(x_{i}\) (\(i=1,2,\ldots,n\)) is the \(i\)-th normalized predictor for the subsystem, \(w_{i}\) is the weight for predictor \(x_{i}\), and \(n\) is the total number of predictors for the subsystem. As such, \(U\) is in the range of 0 and 1, with 0 being the worst and 1 being the best. A key step is to estimate the weight for each predictor. The most dominating method is the entropy weighting method [44]. The first step in calculating the weight for each predictor is to normalize \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Subsystem** & **Category** & **Predictors** & **Unit** \\ \hline \multirow{8}{*}{Technological Innovation} & \multirow{4}{*}{Environment (\(X_{1}\))} & Gross Domestic Product (GDP) per capita (\(X_{11}\)) & \multirow{2}{*}{\
\begin{tabular}{c} Yuan/person \\ \end{tabular} } \\  & & Average number of students enrolled in higher education institutions per & \\  & & 100,000 people (\(X_{12}\)) & \\  & & Number of Internet broadband access (\(X_{13}\)) & \\  & & Number of high-tech enterprises in each region (\(X_{14}\)) & \\  & & Number of research and development (R\&D) institutions by region (\(X_{15}\)) & Item \\ \cline{2-4}  & \multirow{4}{*}{Investment (\(X_{2}\))} & Local fiscal expenditure on science and technology (\(X_{21}\)) & \(10^{8}\) Yuan \\  & & R\&D funding intensity (\(Y_{22}\)) & \% \\  & & Internal expenditure on R\&D (\(X_{23}\)) & Yuan \\  & & Full-time equivalent of R\&D personnel by region (\(X_{24}\)) & Person \\ \cline{2-4}  & \multirow{2}{*}{Output (\(X_{3}\))} & Technology market transaction amount (\(X_{31}\)) & \(10^{4}\) Yuan \\  & & Number of patents granted (\(X_{32}\)) & Item \\  & & Number of scientific papers indexed by major international publication & Item \\  & & repositories by region (\(X_{33}\)) & \\ \hline \multirow{8}{*}{Financial Development} & \multirow{4}{*}{Scale (\(Y_{1}\))} & Proportion of financial practitioners (\(Y_{11}\)) & \% \\  & & Bank balance (\(Y_{12}\)) & \(10^{8}\) Yuan \\  & & Bank loan balance (\(Y_{13}\)) & \(10^{8}\) Yuan \\  & & Total market value of stocks (\(Y_{14}\)) & \(10^{8}\) Yuan \\  & & Premium income (\(Y_{15}\)) & \(10^{8}\) Yuan \\ \cline{2-4}  & \multirow{4}{*}{Structure (\(Y_{2}\))} & Savings per capita (\(Y_{21}\)) & \multirow{2}{*}{
\begin{tabular}{c} Yuan/person \\ \end{tabular} } \\  & & Number of securities industry institutions (\(Y_{22}\)) & Item \\  & & Insurance depth (\(Y_{23}\)) & \% \\  & & Insurance density (\(Y_{24}\)) & Yuan/person \\ \cline{2-4}  & \multirow{2}{*}{Efficiency (\(Y_{3}\))} & Loan-to-deposti ratio (\(Y_{31}\)) & \% \\  & & Securitization rate (\(Y_{32}\)) & \% \\  & & Insurance claims ratio (\(Y_{33}\)) & \% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Categories and predictors used for the two subsystems in our model. the data across all the years in the dataset. Several normalization methods could be used. In the context of our study, the min-max normalization is used, where each data point is normalized into the range of \([0,1]\) as defined in Equation (2):  
\[p_{ij}=\frac{x_{ij}-min(x_{j})}{max(x_{j})-min(x_{j})} \tag{2}\]  
where \(x_{ij}\) is the \(i\)-th value for the \(j\)-th predictor, \(p_{ij}\) is the corresponding normalized value, and \(x_{j}\) is the vector for the \(j\)-th predictor. The second step is to calculate the entropy for each predictor according to Equation (3):  
\[e_{j}=-k\sum_{i=1}^{m}p_{ij}ln(p_{ij}) \tag{3}\]  
where \(k=1/ln(m)\) is a constant to scale \(e_{j}\) to the range of \([0,1]\), \(m\) is the number of values for each predictor, and \(e_{j}\) is the entropy of the \(j\)-th predictor.