## 3 Research Methodology
### Model
Conducting the cross-sectionally augmented Im-Pesaran-Shin test is critical, as it ensures the suitability of the variables for subsequent econometric modeling. The cross-sectionally augmented Im-Pesaran-Shin test, which accounts for cross-sectional dependence, enhances the robustness of the analysis by verifying that the data do not contain unit roots. The mathematical formulation of the cross-sectionally augmented Im-Pesaran-Shin test is provided in Equation (3):  
\[\Delta\text{w}_{\text{i,t}}=\text{b}_{\text{i}}+\text{b}_{\text{i}}\text{z} _{\text{i,t}-1}+\text{b}_{\text{i}}\widetilde{\text{z}_{\text{t}-1}}+\sum_{ \text{j}=0}^{\text{n}}\text{b}_{\text{ij}}\Delta\text{w}_{\text{i,t}-1}+ \text{e}_{\text{i,t}}. \tag{3}\]  
In Equation (3), \(\widetilde{\text{w}}\) represents the average value across the cross-sectional units. This parameter is integral to the analysis, as it provides a measure of the central tendency within the cross-sectional data, allowing for a more comprehensive understanding of the dataset's overall characteristics. \[\mathrm{w}_{\mathrm{i,t}}=\mathrm{b}_{\mathrm{t}}\mathrm{c}\mathrm{\widehat{a}}_{ \mathrm{i,t}}+\mathrm{b}_{\mathrm{2}}\mathrm{f}\mathrm{\widehat{e}}_{\mathrm{i,t }}^{\mathrm{c}}+\mathrm{b}_{\mathrm{3}}\mathrm{e}\mathrm{\widehat{n}}_{ \mathrm{i,t}}+\mathrm{b}_{\mathrm{4}}\mathrm{\widehat{g}}\mathrm{\widehat{r}}_{ \mathrm{i,t}}+\mathrm{b}_{\mathrm{5}}\mathrm{\widehat{n}}_{\mathrm{i,t}}+ \mathrm{b}_{\mathrm{6}}\mathrm{\widehat{f}}\mathrm{\widehat{d}}_{\mathrm{i,t}} +\mathrm{b}_{\mathrm{7}}\mathrm{\widehat{u}}\mathrm{\widehat{r}}_{\mathrm{i,t}} +\mathrm{b}_{\mathrm{8}}\mathrm{\widehat{e}}\mathrm{\widehat{c}}_{\mathrm{i,t }}. \tag{4}\]  
Consequently, the cross-sectionally augmented Im-Pesaran-Shin test is formulated as shown in Equation (4), where CADF denotes the cross-sectionally augmented Dickey-Fuller test. Additionally, previous studies have implemented the Westerlund and Edgerton (2017) to cointegration test to assess cointegration. This step is essential for selecting the appropriate econometric model. The Westerlund and Edgerton test is particularly advantageous due to its robust handling of cross-sectional dependence and its ability to accommodate structural breaks. Furthermore, it has the capability to examine regime shifts and distinguish between shifts and no-shift breaks within the structure. The test equation is presented in Equation (5):  
\[\mathrm{llog(L)}=\mathrm{c}_{0}-\frac{\sum_{\mathrm{i=1}}^{\mathrm{n}}\left[ \mathrm{tlog}\Big{(}\mathrm{\sigma}_{\mathrm{i,t}}^{2}\Big{)}}{2}-\frac{\sum_{ \mathrm{t=1}}^{\mathrm{t}}\mathrm{\epsilon}_{\mathrm{i,t}}^{2}}{\mathrm{ \sigma}_{\mathrm{i,t}}^{2}}. \tag{5}\]  
Previous studies have also utilized the cross-sectional autoregressive distributed lag approach to examine the relationships among the variables under investigation. This method is advantageous due to its ability to address endogeneity, cross-sectional dependence, and slope heterogeneity. While the autoregressive distributed lag approach is widely employed for panel data analysis, it fails to adequately account for cross-sectional dependence errors. The cross-sectional autoregressive distributed lag method, developed by [PERSON] and [PERSON] (2017), effectively addresses these issues and is particularly well-suited for handling cross-sectional dependence. It is based on stringent assumptions that enhance its robustness and reliability. The mathematical formulation of the cross-sectional autoregressive distributed lag approach is presented in Equation (6):  
\[\Delta y_{\mathrm{i,t}}=\mathrm{c}_{\mathrm{i}}+\sum_{\mathrm{j=1}}^{\mathrm{ n}}\mathrm{c}_{\mathrm{ij}}\Delta y_{\mathrm{i,t-1}}+\sum_{\mathrm{j=0}}^{\mathrm{n}} \mathrm{\widehat{c}}_{\mathrm{  
The results detailed in Table 2 reveal distinct relationships between CO\({}_{2}\) emissions and several independent variables. Renewable energy, environmental tax, green finance, and foreign direct investment are negatively correlated with carbon dioxide emissions.