## 4 Methodology and Data
### Econometric Models  
#### 4.2.1 Two-Way Fixed Effects Model and Panel Quantile Model  
We employed panel data models to study the relationship between the TFP of ports along the Yangtze River and the CO\({}_{2}\) emissions of port cities, with its simplified form as follows:  
\[Y_{i,t}=\beta_{0}+\beta_{1}TFP_{i,t}+\beta_{2}X_{i,t}+YearFE+CityFE+\varepsilon_{ i,t} \tag{6}\]  
where \(i\) and \(t\) respectively represent the port city and the year, \(Y_{i,t}\) is the dependent variable (city CO\({}_{2}\) emissions), \(X_{i,t}\) is a vector of control variables, \(\varepsilon_{i,t}\) represents the random error, \(CityFE\) denotes city fixed effects, which control for the impact of static city characteristics (such as geographical location, etc.) on CO\({}_{2}\) emissions in port cities, and \(YearFE\) represents time fixed effects, which can control for national or global trends and events affecting all port cities, such as changes in national environmental policies and fluctuations in international oil prices. When dealing with environmental data, we often encounter challenges, including distinct peaks or fat tails in the data, which implies unequal variation within the data distribution [52]. Addressing this challenge, the traditional two-way fixed effects (TWFE) ordinary least squares (OLS) estimation method may not provide a sufficient solution, as it primarily focuses on the conditional expectations of the dependent variable (i.e., the mean) [53; 54]. This is particularly relevant when analyzing the relationship between port TFP and CO\({}_{2}\) emissions from port cities, where a method capable of capturing the differences in relationships at various emission levels is deemed necessary. For this reason, we opt for panel quantile regression analysis due to the following considerations: 1. quantile regression techniques offer an effective solution by allowing coefficients to vary with different quantiles, enabling a detailed understanding of how the relationship between port TFP and CO\({}_{2}\) emissions changes with different emission levels [55]. 2. Compared to traditional OLS estimation methods, quantile regression provides a more robust estimation technique. It does not impose strict assumptions on the error term, allowing the method to handle potential issues of heteroscedasticity, outliers, and unobserved heterogeneity more effectively, which are factors that could significantly impact the accuracy of estimations [52; 56]. The form of the panel quantile model is as follows:  
\[Q^{(\tau)}_{Y_{i,t}}(\tau|X_{i,t})=\alpha_{i}+\beta_{\tau}X_{i,t}+u_{i,t}\;i=1,\;2,\;\ldots,\;N,\;t=1,2,\ldots,\;T \tag{7}\]  
where \(Y_{i,t}\) and \(X_{i,t}\) are the dependent variable (city CO\({}_{2}\) emissions) and the vector of explanatory variables (port TFP and other control variables), respectively. \(Q^{(\tau)}_{Y_{i,t}}(\tau|X_{i,t})\) represents the \(\tau^{th}\) conditional quantile of the dependent variable, with \(\tau\in(0,1)\). \(\beta_{\tau}\) represents the regression coefficient estimate of the explanatory variables at the \(\tau^{th}\) quantile, which is our primary focus. We assign the quartiles of \(\tau\) as 0.25, 0.5, and 0.75. The 25 th percentile (Q25) is used to illustrate the impact of port TFP on cities with lower CO \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & **Obs** & **Mean** & **Max** & **Min** & **Std.Dev.** \\ \hline Output Variable & & & & & \\ \hline Container throughput & 176 & 47.26675 & 330.54 & 0.4 & 66.4993 \\ \((10,000\;\mathrm{TEU})\) & 176 & 9517.479 & 33,620.35 & 554 & 6897.967 \\ \hline Input Variable & & & & & \\ \hline Number of berths used in port production & 176 & 166.8523 & 15 & 880 & 177.8984 \\ Length of berths used in port production \((\mathrm{m})\) & 176 & 16,366.36 & 76017 & 2631 & 15,666.94 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptive statistics of input and output variables.