## 3 Variables and Model
### Model  
In analyzing the long-term interaction between trade policy, energy efficiency, and carbon dioxide emissions in South Korea, it is critical to account for both the economic and environmental dimensions that have underpinned the country's development trajectory. South Korea's export-oriented industrialization, heavily dependent on energy-intensive sectors such as manufacturing and petrochemicals, has historically contributed to rising CO2 emissions ([PERSON] et al. [52]). Nevertheless, recent empirical research indicates that advancements in energy efficiency, catalyzed by domestic regulatory frameworks and international trade agreements, have the potential to alleviate the environmental pressures associated with industrial growth ([PERSON] and [PERSON] [53]). The integration of energy efficiency measures within trade policies is particularly important for decoupling economic growth from environmental degradation. This suggests that increased trade openness, when aligned with robust energy efficiency standards, can support long-term reductions in CO2 emissions ([PERSON] and [PERSON] [54]; [PERSON] et al. [55]). Building on these insights, this paper introduces a long-term empirical model to quantify the effects of trade policy and energy efficiency on CO2 emissions, offering a comprehensive framework for understanding the pathways to sustainable economic growth. The baseline model is outlined as follows:  
\[\text{CO}_{2,t}=\text{a}_{0}+\text{a}_{1}\text{tp}_{\text{t}}+\text{a}_{2} \text{ee}_{\text{t}}+\text{a}_{3}\text{eg}_{\text{t}}+\text{a}_{4}\text{ur}_{ \text{t}}+\text{a}_{5}\text{in}_{\text{t}}+\text{a}_{6}\text{fdi}_{\text{t}} +\text{a}_{7}\text{gt}_{\text{t}}+\epsilon_{\text{t}}. \tag{1}\]  
In Equation (1), t represents the time variable, denoting the year under consideration. \(\text{a}_{0}\) refers to the constant term, which captures the baseline level of the dependent variable when all other explanatory variables are zero. The terms \([\text{a}_{1},\text{a}_{7}]\) correspond to the coefficients to be estimated, reflecting the magnitude and direction of the relationships between the independent variables and the dependent variable. Finally, \(\epsilon\) denotes the white noise \begin{table}
\begin{tabular}{c c c c} \hline
**Variable** & **Form** & **Definition** & **Source** \\ \hline Carbon dioxide emissions & CO2 & Metric tons per capita in log & World Bank \\ Trade policy & tp & Trade policy index & FRED Economic Database \\ energy efficiency & ee & GDP/energy consumption & World Bank \\ Economic growth & eg & GDP (constant 2015 USD, and unit: billion & World Bank \\ USD) in log & & World Bank \\ Urbanization & ur & Share of the urban population in the total & World Bank \\ Industrialization & in & Ratio of industry to GDP & World Bank \\ Foreign direct investment & fdi & Foreign direct investment, net inflows (\% of GDP) & World Bank \\ Green technology innovation & gt & Values for patents (\% of total) in environment-related technologies & Organization for Economic Cooperation and Development \\ \hline \end{tabular}
\end{table} 
Table 1: Description of variables. error term, which accounts for the random variability not explained by the model, ensuring that the residuals are normally distributed with a mean of zero. This formulation allows for the precise estimation of the impact of key variables on the outcome while maintaining statistical rigor. In addition, this study employs the innovative Fourier autoregressive distributed lag methodology to achieve its research objectives. The following section provides a detailed discussion of this approach. Traditional methods for testing cointegration, such as the [PERSON] and [PERSON] (1966) two-step procedure and the [PERSON] (1977) test, are widely recognized in the literature. However, these techniques are constrained by certain limitations, particularly the requirement for all variables to be integrated at the same order. To address these issues, [PERSON] et al.