*2.1. Methodological Framework of ARDL*

The model's technical specification is that the economic upswing and ecological disorder are positively associated in initial stages, whereas the square of GDP helps to reduce environmental depletion. The linear-quadratic equation confirms the presence of an inverted U-shaped EKC [34] and [35]. It can be written as:

$$\text{END}\_{\text{it}} = \beta\_0 + \beta\_1 \text{EN}\_{\text{it}} + \beta\_2 \text{EGW}\_{\text{it}} + \beta\_3 \text{EGW}^2{}\_{\text{it}} + \beta\_4 \text{FCF}\_{\text{it}} + \beta\_5 \text{PG}\_{\text{it}} + \mu\_{\text{i}} \tag{1}$$

Equation (1) illustrates the linear quadratic equation to run the ARDL for the confirmation of EKC. Following the footprints of [36], Equation (2) establishes to assess the short-run ARDL results.

$$\begin{array}{llll} \Delta END\_{it} = \beta\_0 & + & \sum\_{i=1}^{k} \gamma\_1 \Delta END\_{i\ t-1} + \sum\_{i=0}^{k} \alpha\_1 \Delta END\_{i\ t-1} + \sum\_{i=0}^{k} \alpha\_2 \Delta GW\_{i\ t-1} \\ & + & \sum\_{i=0}^{k} \alpha\_3 \Delta GW^2\_{i\ t-1} + \sum\_{i=0}^{k} \alpha\_4 \Delta FCF\_{i\ t-1} + \sum\_{i=0}^{k} \alpha\_5 \Delta PG\_{i\ t-1} \\ & + & \beta\_1 ENC\_{i\ t-1} + \beta\_2 GWW\_{i\ t-1} + \beta\_3 GW^2\_{i\ t-1} + \beta\_4 FCF\_{i\ t-1} \\ & + & \beta\_5 PG\_{i\ t-1} + \mu\_{it} \end{array} \tag{2}$$

In the above equation, Δ represents the difference, whereas *t* − 1 used for cross-section shows the model's previous years. The α and β are the coefficients of underline indicators. In the next step, the Error Correction Model (ECM) develops by formulating the following equation.

$$\begin{array}{ll} \Delta END\_{it} = & \beta\_0 + \sum\_{i=1}^k \gamma\_1 \Delta END\_{i\ t-1} + \sum\_{i=1}^k a\_1 \Delta ENC\_{i\ t-1} + \sum\_{i=1}^k a\_2 \Delta GW\_{i\ t-1} + \\ & \sum\_{i=1}^k a\_3 \Delta GW^2\_{i\ t-1} + \sum\_{i=1}^k a\_4 \Delta FC\_{i\ t-1} + \sum\_{i=1}^k a\_5 \Delta PG\_{i\ t-1} + \\ & \beta\_1 ENC\_{i\ t} + \beta\_2 GW\_{i\ t} + \beta\_3 GW^2\_{i\ t} + \beta\_4 FCF\_{i\ t} + \beta\_5 PG\_{i\ t} + \delta ECM\_{i\ t} + \\ & u\_{it} \end{array} \tag{3}$$

The coefficient of ECM' δ demonstrates the speed of adjustment, and it should be with a negative sign to show the convergence towards the long run from the short run to achieve the equilibrium condition.
