## 3 Materials and Methods
the normality test, ARCH test, Breusch-Godfrey serial correlation test [33], LM test [34], and Ramsey RESET test [35]. Additionally, the cumulative sum of squares (CUSUMSQ) test [36; 37] was used to confirm the stability of the model. The EKC, developed by [PERSON] and [PERSON] [38], is modeled as a functional relationship between environmental degradation (\(E\)) and income per capita (\(\gamma\)), with quadratic and cubic formulations capturing various trajectories. Following [PERSON] et al. [39] and [PERSON] et al. [22], we briefly present the main cases of the EKC and their implications. (a) Inverted U Shape (Quadratic EKC)  
The EKC equation is as follows:  
\[E=\beta_{0}+\beta_{1}Y+\beta_{2}Y^{2}+\varepsilon \tag{4}\]  
where \(\beta_{1}>0\), \(\beta_{2}<0\), and \(\varepsilon\) is the error term. At low-income levels, industrialization and resource exploitation lead to increased \(E\). After the turning point denoted by \(Y^{*}\), economic growth fosters cleaner technology, regulatory reforms, and structural transitions to service-based economies, reducing \(E\). Policies supporting renewable energy and industrial efficiency are critical for reaching \(Y^{*}\) quickly. Developing economies should focus on sustainable practices to avoid the early steep rise in degradation. (b) U Shape (Quadratic EKC)  
The EKC equation is expressed in Equation (5):  
\[E=\beta_{0}+\beta_{1}Y+\beta_{2}Y^{2}+\varepsilon \tag{5}\]  
where \(\beta_{1}<0\), \(\beta_{2}>0\), and \(\varepsilon\) is the error term. Early stages feature low environmental pressure due to subsistence-level activities. Industrialization at higher \(Y\) levels leads to increased emissions and resource use. Economies must balance growth with sustainability to prevent long-term environmental degradation. Rapid adoption of green technologies can counteract the upward trend in \(E\). The constant term \(\beta_{0}\) does not influence the turning point because it does not depend on \(Y\). It only shifts the curve vertically, leaving the location of the turning point unaffected. The turning point \(Y^{*}\) of the quadratic EKC is expressed in Equation (6):  
\[Y=-\frac{\beta_{1}}{2\beta_{2}} \tag{6}\]  
For the inverted U shape (quadratic EKC), the turning point represents a maximum. For the U shape (quadratic EKC), the turning point represents a minimum. (c) N-Shaped Curve (Cubic EKC)  
The EKC equation is as follows:  
\[E=\beta_{0}+\beta_{1}Y+\beta_{2}Y^{2}+\beta_{3}Y^{3}+\varepsilon \tag{7}\]  
In Equation (7), \(\beta_{1}>0\), \(\beta_{2}<0\), \(\beta_{3}>0\), and \(\varepsilon\) is the error term. The second phase sees technological improvements and regulatory reforms lowering \(E\). The third phase may occur due to unsustainable consumption, diminishing returns from technology, or shifting environmental externalities. Advanced economies face the risk of a rebound in environmental degradation. Long-term sustainability requires strong policies promoting circular economies and innovation. (d) Inverted N Shape (Cubic EKC)  
The EKC equation is as follows:  
\[E=\beta_{0}+\beta_{1}Y+\beta_{2}Y^{2}+\beta_{3}Y^{3}+\varepsilon \tag{8}\]In Equation (7), \(\beta_{1}<0\), \(\beta_{2}>0\), \(\beta_{3}<0\), and \(\varepsilon\) is the error term. Degradation is minimal at low-income levels. Industrialization increases \(E\), but at higher income levels sustainability initiatives stabilize or reduce \(E\). Early-stage economies must adopt green growth strategies. Advanced economies must avoid stagnation in environmental policies to maintain \(E\) at a lower level. For cubic EKC, turning points are derived by solving the quadratic Equation (9):  
\[\frac{\partial E}{\partial Y}=\beta_{1}+2\beta_{2}+3\beta_{3}Y^{2}=0 \tag{9}\]  
The N-shaped EKC illustrates how environmental pressures, such as emissions or resource depletion, initially rise with economic development and later decline due to reforms and technological advancements, but eventually resurge as economies continue to grow.