## 4 Key Indicators Measurement and Data Description
### The Industrial Collaborative Agglomeration Index  
To calculate the collaborative agglomeration index, it is essential to first assess the agglomeration index of manufacturing and producer services. There are several main indicators to assess the level of industrial agglomeration, including location entropy, the Gini coefficient, the Herfindahl-Hirschman Index, and employment density. [PERSON] et al. [42] stated that the location entropy eradicates the differences in regional scale and additionally exposes the spatial allocation of geographic elements. Referring to their approach, this article employs location entropy to estimate the degree of specialized agglomeration ofmanufacturing and producer service industries in prefecture-level cities. The equation is shown below:  
\[\text{LocalEntropy}_{\text{ir}}=\frac{\text{e}_{\text{ir}}/\text{e}_{\text{i}}}{ \text{e}_{\text{r}}/\text{e}}\]  
where \(\text{e}_{\text{ir}}\) indicates the output value or the number of employees of industry r in city i; e\({}_{\text{i}}\) represents the output value or the number of employees of all industries in the city i; e\({}_{\text{r}}\) is the output value or the number of employees of industry r in all cities; e means the output value or the number of employees of all industries in all cities. Generally, a large location entropy index means a high level of industrial agglomeration in an area. Due to the availability of data, for this paper, we finally decided to use the number of employees in manufacturing and eight types of productive service industries to calculate the degree of agglomeration. It should be noted that the eight types of productive service industries generally include \"Traffic, Transport, Storage and Post\", \"Information Transmission, Computer Services and Software\", \"Financial Intermediation\", \"Real Estate\", \"Leasing and Business Services\", \"Scientific Research and Technical Service\", \"Services to Households and Other Services\", and \"Education\" [28]. Then, according to the method of [PERSON] [57], the calculation of the industrial collaborative agglomeration index of manufacturing and producer service industries (COAGGit) is as follows:  
\[\text{COAGGit}_{\text{it}}=\left(1-\frac{\left|\text{MAGGit}_{\text{it}}- \text{SAGGit}_{\text{it}}\right|}{\text{MAGGit}_{\text{it}}+\text{SAGGit}_{ \text{it}}}\right)+\left(\text{MAGGit}_{\text{it}}+\text{SAGGit}_{\text{it}}\right)\]  
where COAGGit is the industrial collaborative agglomeration index of manufacturing and producer services in city i in year t; MAGGit describes the manufacturing agglomeration index of city i in year t; and SAGGit denotes the agglomeration index of the producer service industry in city i in year t. Specifically, \(\left(1-\frac{\left|\text{MAGGit}_{\text{it}}-\text{SAGGit}_{\text{it}}\right|}{ \text{MAGGit}_{\text{it}}+\text{SAGGit}_{\text{it}}}\right)\) stands for the quality of the collaborative agglomeration index while \(\left(\text{MAGGit}_{\text{it}}+\text{SAGGit}_{\text{it}}\right)\) means the depth. A large industrial collaborative agglomeration index signifies that the quality of collaborative agglomeration is great and the depth is deep.