## 3 Variables and Model
### Model
(2018) introduced the autoregressive distributed lag (ARDL) model, which offers greater flexibility by allowing variables of mixed orders of integration (i.e., \(\{1(0)\) or \(1(1)\)). The ARDL model also addresses challenges such as serial correlation and endogeneity, providing more reliable and robust estimates. Moreover, it distinguishes between short- and long-run dynamics, enabling the formulation of policies that are tailored to different time horizons. The ARDL framework, therefore, offers a comprehensive approach to understanding both short- and long-term relationships between variables, as demonstrated in the following formal model specification. \[ \begin{array}{c}\Delta\text{CO}_{2,t}=\text{b}_{0}+\sum_{i=0}^{n1}\text{b}_{ 1i}\Delta\text{CO}_{2,t-i}+\sum_{i=0}^{n2}\text{b}_{2i}\Delta\text{pt}_{t-i}+ \sum_{i=0}^{n3}\text{b}_{3i}\Delta\text{ee}_{t-i}+\\ \sum_{i=0}^{n4}\text{b}_{4i}\Delta\text{eg}_{t-i}+\sum_{i=0}^{n5}\text{b}_{5i} \Delta\text{wt}_{t-i}+\sum_{i=0}^{n6}\text{b}_{6i}\Delta\text{int}_{i-i}+ \sum_{i=0}^{n7}\text{b}_{7i}\Delta\text{fd}\text{i}_{t-i}+\\ \sum_{i=0}^{n8}\text{b}_{8i}\Delta\text{gt}_{t-i}+\text{b}_{5}\text{CO}_{2,t-1} +\text{b}_{10}\text{pt}_{t-1}+\text{b}_{11}\text{ee}_{t-1}+\text{b}_{12}\text{ ge}_{t-1}+\text{b}_{13}\text{wt}_{t-1}+\\ \text{b}_{14i}\text{in}_{t-1}+\text{b}_{15}\text{fd}\text{i}_{t-1}+\text{b}_{16 }\text{gt}_{t-1}+\text{ect}_{t-1}+\text{e}_{t}.\end{array}  \tag{2}\]  
Equation (2) illustrates the ARDL model with an unrestricted intercept and no trend. In this formulation, \(\text{b}_{0}\) represents the intercept, while the coefficients \([\text{b}_{1i},\text{b}_{8i}]\) correspond to short-run estimates and \([\text{b}_{9},\text{b}_{16}]\) reflect the long-run estimates. The terms \([\text{n}1,\text{n}8]\) denote the lag orders, and \(\text{e}_{t}\) represents the disturbance term. To assess cointegration, two standard tests are applied within the ARDL framework: firstly, an F-test (or bounds test) on the long-run coefficients, testing the null hypothesis that \(\text{b}_{9}=\text{b}_{10}=\text{b}_{11}=\text{b}_{12}=\text{b}_{13}=\text{ b}_{14}=\text{b}_{15}=\text{b}_{16}=0\), and secondly, a \(t\)-test on the lagged dependent variable, specifically testing \(\text{b}_{9}=0\). The null hypothesis \(\text{H}_{0}\) in both cases posits no cointegrating relationship, whereas the alternative hypothesis \(\text{H}_{1}\) asserts the presence of cointegration. Cointegration is confirmed if the F- and \(t\)-test statistics exceed the critical values for the upper bound, corresponding to an \(\text{I}(1)\) series. However, one limitation of the ARDL approach is its inability to account for structural breaks, which can lead to biased results ([PERSON] and [PERSON] (1979); [PERSON] and [PERSON] (1997); [PERSON] and [PERSON] (2018)). To address this issue, the Fourier transformation is incorporated into the ARDL model, enhancing its ability to handle structural breaks without requiring prior knowledge of the timing, frequency, or form of the breaks. Unlike structural break dummies, the Fourier transformation efficiently captures these breaks while maintaining strong power and size properties due to its reduced number of parameters. The model incorporating the Fourier transformation, known as FARDL, is formulated as follows:  
\[