## 3 Materials and Methods  
This study investigates the relationship between \(CO_{2}\) emissions and economic growth, focusing on the potential existence of a cubic EKC during the period from 1990 to 2023. Table 2 presents the variables considered in the analysis, which include \(CO_{2}\) emissions per capita, foreign direct investments, GDP, renewable energy consumption, and access to electricity. The study utilizes time-series data, with the natural logarithms of the variables applied to stabilize the series and reduce fluctuations. To model the relationship between these variables, an ARDL approach is employed, followed by cointegration tests to explore long-run equilibrium relationships. Diagnostic tests ensure the robustness of the model. Table 2 in Appendix A presents the definitions of all acronyms used in this paper. Table 2 contains the variables and their sources for the period 1990-2023. This study also checks the existence of cubic EKC during the period 1990-2023. The relationship between the variables is as follows:  
\[CO_{2t}=a_{0}+a_{1}GDP_{t}+a_{2}GDP2_{t}+a_{3}GDP3_{t}+a_{4}RENC_{t}+a_{5}ACEL_{ t}+a_{6}FDI_{t}+\varepsilon_{t} \tag{1}\]  
In Equation (1), \(GDP2\) and \(GDP3\) represent the squared and cubic GDP, respectively. The time-series data, including \(CO_{2}\) emissions, were transformed using natural logarithms to address the issues of sharp fluctuations and non-stationarity often present in economic and environmental data. This transformation helps stabilize the variance of the series, which is particularly useful when data exhibit exponential growth or large variations over time. By applying the natural logarithm, the original values are compressed, reducing the scale of extreme values (both large and small) while preserving the relative relationships among the observations [29; 30]. Equation (1) is transformed into an ARDL (\(n\), \(p\), \(q\), \(r\), \(s\), \(u\), \(v\)) regression:  
\[ \begin{split}\Delta CO_{2t}=a_{0}+\sum_{k=1}^{n}a_{1}\Delta CO_{2t -k}+\sum_{k=1}^{p}a_{2}\Delta GDP_{t-k}+\sum_{k=1}^{q}a_{3}\Delta GDP2_{t-k}+ \sum_{k=1}^{r}a_{4}\Delta GDP3_{t-k}+\\ \sum_{k=1}^{s}a_{5}\Delta RENC_{t-k}+\sum_{k=1}^{u}a_{6}\Delta ACEL_{ t-k}+\sum_{k=1}^{p}a_{2}\Delta FDI_{t-k}+\lambda_{1}CO_{2t-1}+\lambda_{2} GDP_{t-1}+\lambda_{3}GDP2_{t-1}+\\ \lambda_{4}GDP3_{t-1}+\lambda_{5}RENC_{t-1}+\lambda_{6}ACEL_{t-1}+ \lambda_{7}FDI_{t-1}+\varepsilon_{t}\end{split}  \tag{2}\]  
In Equation (2), \(\Delta\) represents the first difference, and \(n\), \(p\), \(q\), \(r\), \(s\), \(u\), and \(v\) are the lag order, which will be determined later. The ARDL bounds test is applied next to check if a stable, long-run equilibrium relationship exists among the variables under study [31]. This is done by examining the presence of cointegration. In case cointegration exists, the error correction model (ECM) is used, as expressed in Equation (3):  
\[ \begin{split}\Delta CO_{2t}=a_{0}+\sum_{k=1}^{n}a_{1}\Delta CO_{2t -k}+\sum_{k=1}^{p}a_{2}\Delta GDP_{t-k}+\sum_{k=1}^{q}a_{3}\Delta GDP2_{t-k}+ \sum_{k=1}^{r}a_{4}\Delta GDP3_{t-k}+\\ \sum_{k=1}^{s}a_{5}\Delta RENC_{t-k}+\sum_{k=1}^{u}a_{6}\Delta ACEL_{ t-k}+\sum_{k=1}^{p}a_{2}\Delta FDI_{t-k}+\Gamma\text{ECM}_{t-1}+\varepsilon_{t} \end{split}  \tag{3}\]  
The error correction term (ECT) should be statistically significant and negative, with its value not exceeding \(-2\)[32]. Finally, several diagnostic tests were conducted, including \begin{table}
\begin{tabular}{c c c c} \hline
**Variable** & **Acronym** & **Measurement Unit** & **Source** \\ \hline \(CO_{2}\) emissions per capita & \(CO_{2}\) & tons & Our World in Data \\ Foreign Direct Investments, net inflows & FDI & \% of GDP & World Bank \\ Gross domestic product & GDP & constant 2015 \$USD & World Bank \\ Access to electricity & ACEL & \% of population & World Bank \\ Renewable energy consumption & RENC & \% of total final energy consumption & World Bank \\ \hline \end{tabular}
\end{table} 
Table 2: Variable specification.