## 3 Models and Methods
### Spatial Convergence Model  
To explore whether the gaps in the integrated transport efficiency between the studied Chinese provinces was widening or narrowing, this study introduced a convergence analysis. Convergence refers to the process of the difference between the related indicators of two or more research objects growing smaller over time. This is an important economic theory that was originally applied to investigate the convergence of per capita income across different regions and countries. \(\beta\)-convergence deems that a region with a low value has a higher growth rate than the region with a high value, whereby, eventually, the low value region will catch up with the high value region and a stable growth rate will be achieved across the two regions. \(\beta\)-convergence includes the absolute \(\beta\)-convergence and the conditional \(\beta\)-convergence. The absolute \(\beta\)-convergence considers that the convergence of all units is not affected by other factors, but only depends on the initial level of efficiency and will eventually reach the same stable state. The conditional \(\beta\)-convergence considers that other external factors will affect the convergence, thus making each DMU develop according to development characteristics and along the stable state of different path items. Based on imbalanced development and the significant differences between the integrated transport efficiency of China's provinces, the absolute \(\beta\)-convergence was used to analyze whether the growth rate of integrated transport efficiency in each province converges, and whether there exists a trend of regions with lagging transport efficiency catching up with other regions with more advanced efficiency. The conditional \(\beta\)-convergence takes into account the external factors affecting integrated transport in each province, and specifically analyzes the convergence of integrated transport efficiency across the provinces under the influence of external factors. Most previous research has used the traditional \(\beta\)-convergence to study convergence, ignoring the influence of the spatial unit characteristics on the convergence. In recent years, spatial econometric models have been developed and applied to many fields, as these take the spatial dependence and the spatial correlation between regions fully into account in a developing society. Therefore, we explored the convergence characteristics of integrated transport efficiency in China's provinces by considering spatial effect, constructing a spatial absolute \(\beta\)-convergence model and a conditional \(\beta\)-convergence model for the purposes of this study. The model is described as follows:  
\[Ln(ITE_{i/ITE_{i/1}})=\alpha+\beta LnITE_{i,t}+\lambda Ln(ITE_{i/ITE_{i/1}})+ \varepsilon\quad\varepsilon\,\sim\,(0,\sigma^{2}) \tag{6}\]  
\[Ln(ITE_{i/ITE_{i/1}})=\alpha+\beta LnITE_{i,t}+\lambda Ln(ITE_{i/ITE_{i/1}})+C _{k}LnX_{k}+\varepsilon\quad\varepsilon\,\sim\,(0,\sigma^{2}) \tag{7}\]  
\[Ln(ITE_{i/ITE_{i/1}})=\alpha+\beta LnITE_{i,t}+\gamma W\varepsilon_{i,t-1}+ \omega\quad\omega\,\sim\,(0,\sigma^{2}I) \tag{8}\]\[Ln(^{TE_{i,t}}/\tau_{TE_{i,t-1}})=\alpha+\beta LnTE_{i,t}+C_{k}LnX_{k}+\gamma W \epsilon_{i,t-1}+\omega\quad\omega\,\sim\,(0,\sigma^{2}I) \tag{9}\]  
where \(ITE_{i,t}\) and \(ITE_{i,t-1}\) represent the values of integrated transport efficiency of the \(t\)th year and the \((t-1)\)th year for the \(i\)th province, respectively. \(\lambda\) represents the spatial autoregressive coefficient, reflecting the size of the spatial interactions in each region. \(\lambda Ln(^{ITE_{i,t}}/ITE_{i,t-1})\) represents the spatial lag term of the interpreted variable. \(W\) represents the spatial weight matrix. \(\epsilon\) and \(\epsilon_{i,t-1}\) are random error terms. \(Y\) and \(\omega\) are random variables that reflect the correlation between regression residual space and spatial correlation, respectively. \(X_{k}\) is the control variable of the conditional convergence analysis.