## 3 Research Design
### Measurement Model and Method  
Atmospheric self-purification capacity not only reduces environmental pollution in one region, but also affects its neighboring regions. Therefore, to verify Hypothesis 1 and Hypothesis 2, this paper considers regional correlation while conducting empirical analysis. The spatial autoregression model (SAR) includes spatial correlation between different things in a general regression model, which offsets the lack of spatial correlation of variables in a general regression model. For this reason, this paper introduces the spatial autoregression model, which focuses on analyzing the spatial correlation of the explained variables. Owing to its natural hysteresis, environmental pollution has an inter-regional effect. Accordingly, this paper designs the space lag and time lag of environmental pollution in SAR. By connecting local atmospheric self-purification capacity with neighboring regions, this paper exposes the impact of atmospheric self-purification capacity on environmental pollution more truthfully. In this paper, the estimation model of spatial correlation is expressed as follows:  
\[lnAEP_{it}=\rho\sum_{j=1}^{N}W_{it}lnAEP_{it}+\beta_{1}wnAPI_{it}+\beta_{2} liner_{it}+\beta_{3}Inden_{it}+\beta_{4}nbua_{it}+\beta_{5}lney_{it}+\beta_{6} lnin_{it}+\mu_{i}+\lambda_{i}+\varepsilon_{it} \tag{4}\]  
where \(\rho\) stands for spatial lag (autoregression) coefficient, \(W_{it}\) stands for the element in row \(i\) and column \(j\) of standardized non-negative spatial weight matrix \(W\) in \(N\)\({}^{\star}N\) dimension, and \(\mu_{i}\) and \(\lambda_{i}\) stand for spatial (individual) effect and temporal effect, respectively.  
In the spatial measurement model, the spatial Durbin model (SDM) considers spatial correlations of the explained variables and the explanatory variables respectively. The spatial Durbin model can obtain unbiased coefficient estimation, it better estimates the spillover effect generated by different observation individuals and measures the spatial spillover effect based on panel data. This paper adopts spatial Durbin model (SDM) estimation based on spatial panel estimation to better investigate the impact of relevant factors.  
\[\begin{array}{l}lnAEP_{it}=\rho\sum_{j=1}^{N}W_{it}lnAEP_{it}+\beta_{1}wnAPI_{it }+\beta_{2}lner_{it}+\beta_{3}lenden_{it}+\beta_{4}lnbua_{it}+\beta_{5}lney_{it }+\beta_{6}lnin_{it}\\ +\beta_{7}\sum_{j=1}^{N}W_{it}wAPI+\beta_{8}\sum_{j=1}^{N}W_{it}lnner_{it}+ \beta_{9}\sum_{j=1}^{N}W_{it}lnden_{it}+\beta_{10}\sum_{j=1}^{N}W_{it}lnbua_{it} +\beta_{11}\sum_{j=1}^{N}W_{it}lnvey_{it}\\ +\beta_{12}\sum_{j=1}^{N}W_{it}lnin_{it}+\mu_{i}+\lambda_{i}+\varepsilon_{it} \end{array} \tag{5}\]  
According to Hypothesis 3, environmental regulation and atmospheric self-purification capacity probably have a threshold effect. To examine the possible threshold effect, this paper introduces the threshold value of environmental regulation as an unknown variable into the measurement model, constructs the piecewise function of atmospheric self-purification capacity on environmental pollution, and tests and estimates its threshold value and threshold effect. In the case of the single threshold effect, the measurement model is expressed as follows:  
\[lnAEP_{it}=\beta_{0}+\beta_{1}lner_{it}(lner_{it}<\gamma)+\beta_{2}lner_{it}( lner_{it}\geq\gamma)+\beta_{3}wlnAPI_{it}+\beta_{4}lnden_{it}+\beta_{5}lnbua_{it}+\beta_{6} lney_{it}+\\ \beta_{7}lnin_{it}+\varepsilon_{it} \tag{6}\]  
The multiple threshold estimation equation is extended from the above equation.