## 3 Methods and Data Sources
### Construction of the Spatial Econometric Model  
#### 3.4.1 Model Specification  
IPAT (I = Human Impact, P = Population, A = Affluence, T = Technology) is the original model used to study the effects of economic and social activities on environmental changes [40]. Since then, [PERSON] and [PERSON] proposed an extensible random environmental effect assessment model STIRPAT (Stochastic Impacts by Regression on Population, Affluence, and Technology), and revealed that the number of people, economic development, and technological level are the main variables of environmental effects [41; 42]. It has become an empirical model widely used by academia. The formula is as follows: In the formula, P represents the population. A represents the economic development. T representsthe technical level. \(b\), \(c\), and \(d\) are the exponential terms. \(a\) is a constant term, and \(e\) is a random error term. Formula (2) is obtained by taking the natural logarithm of both sides on the basis of Formula (1).  
\[I=aP^{b}A^{c}T^{d}e \tag{6}\]  
\[\ln I=\ln a+b\left(\ln P\right)+c(\ln A)+d(\ln T)+e \tag{7}\]  
Considering the spatial correlation between variables, this article chooses the most general SDM (Spatial Durbin Model) [43; 44]. The model is as follows: In the above formula, ice represents carbon emissions, and ser represents the agglomeration index of producer service industry, which is calculated by location entropy. W represents the spatial weight matrix, which is used to reflect the spatial relationship between 30 units. In order to improve the accuracy and robustness of the results, this paper sets up four kinds of spatial weight matrices, namely the contiguity weight (IW), the inverse distance weight (IW1), the economic distance weight (IW2), and the minimum proximity weight (IW3) [45; 46; 47]. \(X^{\prime}_{it}\) represents the control variable matrix, \(\phi\) represents the coefficient vector, and the subscripts \(i\) and \(t\) represent the provincial unit and year, respectively. In addition, \(a_{i}\) and \(\lambda_{t}\) represent individual fixed effects and time fixed effects, and \(u_{it}\) represents a random disturbance term.  
\[\ln ce_{it}=\rho W\ln ce_{it}+\gamma\ln ser_{it}+\xi W\ln ser_{it}+\phi X^{ \prime}_{it}+\xi WX^{\prime}_{it}+a_{i}+\lambda_{t}+u_{it} \tag{8}\]  
In the analysis of regional heterogeneity, it is necessary to systematically study the regional differences about the effect of the agglomeration of the producer service industry on carbon emissions [48; 49]. The model settings are as follows: In the above formula, geoi represents the geographic location, which does not change with time. \(\beta\) indicates the coefficient of the interaction term, and the remaining variables are consistent with model (1). This article sets the dummy variable east, which means that the unit in the eastern region is set to 1, and the unit in the remaining regions is set to 0. In addition, this article sets the dummy variable as ns, which means that the unit in the northern region is set to 1, and the unit in the remaining regions is set to 0. Further, the dummy variable is _hhy_, which means that the unit in southeast area is set to 0, and the unit in the northwest is set to 0.  
\[\ln ce_{it}=\rho W\ln ce_{it}+\beta geo_{i}\times\ln ser_{it}+\xi Wgeo_{i} \times\ln ser_{it}+\phi X^{\prime}_{it}+\xi WX^{\prime}_{it}+a_{i}+\lambda_{t}+ u_{it} \tag{9}\]