## 3 Materials and Methods
This pattern has been observed across advanced economies, emerging economies, and specific sectors, reflecting a complex interplay of industrial growth, policy interventions, and consumption behaviors. 1. Monotonic Increase (Linear EKC) The EKC equation is expressed in Equation (10): \[E=\beta_{0}+\beta_{1}Y+\varepsilon\] (10)  
In Equation (10), \(\beta_{1}>0\). This scenario often occurs in economies heavily reliant on resource-intensive industries and weak environmental governance. Continuous growth without sustainability considerations leads to escalating environmental challenges. Urgent policy interventions are required to transition to green technologies and practices. 1. Monotonic Decrease (Linear EKC) The EKC equation is expressed in Equation (11): \[E=\beta_{0}+\beta_{1}Y+\varepsilon\] (11)  
In Equation (10), \(\beta_{1}<0\). Strong environmental standards, service-oriented economies, and early adoption of green technologies drive consistent reductions in \(E\). This case underscores the potential for leapfrogging directly into sustainable development pathways. High-income countries can serve as models for green economic policies. The form of the EKC (quadratic or cubic) determines the policy focus. For inverted U shapes, policies should aim to expedite the transition beyond \(y^{*}\). For N shapes, sustainable consumption and circular economy practices are essential to mitigate the upward trend at high income levels. \(\beta_{0}\) represents the intercept of the curve. It indicates the level of environmental degradation E when income \(Y\) is zero, assuming no other factors are influencing environmental degradation. In practice, \(\beta_{0}\) may not have a meaningful interpretation if income levels of zero are not realistic in the studied economy, but it still provides the baseline value for the environmental degradation when the income effect is not yet present. \(\beta_{1}\) measures the immediate impact of income on environmental degradation. Specifically, it shows how much environmental degradation E changes for a one-unit increase in income (\(Y\)), assuming the quadratic effect of \(Y^{2}\) is held constant. \(\beta_{1}>0\) indicates that increasing income leads to higher environmental degradation in the initial stages of economic development (typically associated with industrialization, resource extraction, and higher energy consumption). \(\beta_{1}<0\) suggests that increasing income leads to lower environmental degradation in the early stages, possibly due to improvements in technology, cleaner energy sources, or efficiency gains. \(\beta_{2}\) captures the non-linear effect of income on environmental degradation, i.e., how the effect of income on environmental degradation changes as income increases. \(\beta_{2}>0\) suggests a U-shaped relationship between income and environmental degradation. \(\beta_{2}<0\) suggests an inverted U-shape (the classic EKC). \(\beta_{3}\) represents the cubic effect of income on environmental degradation. If \(\beta_{3}>0\), the relationship between income and environmental degradation can initially rise, reach a peak, decline, and then increase again at higher income levels. This suggests that, after reaching a point where degradation starts to decline, further income growth may lead to a resurgence of environmental degradation, possibly due to increased consumption, higher demand for resources, or unsustainable growth practices. If \(\beta_{3}<0\), the relationship may show an initial increase in degradation, followed by a decrease, and possibly stabilize or continue decreasing as income grows. This could indicate that environmental degradation begins to reverse more sharply as economies mature, with improvements in technology, policies, or economic structure helping mitigate environmental impacts. If \(\beta_{3}=0\), the model behaves like a quadratic EKC. A cubic EKC with \(\beta_{3}\) allows for more nuanced patterns in the data.