## 4 Results
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Coefficient** & **T-Statistics** & **Prob.** \\ \hline D(\(CO_{2}\) (\(-1\))) & \(-0.08\) & \(-2.25\) & \(0.087\) * \\ D(\(CO_{2}\) (\(-2\))) & \(0.52\) & \(11.64\) & \(0.000\) *** \\ D(GDP) & \(186.36\) & \(14.22\) & \(0.000\) *** \\ D(GDP(\(-1\))) & \(112.23\) & \(10.19\) & \(0.000\) *** \\ D(GDP2) & \(-26.45\) & \(-14.27\) & \(0.000\) *** \\ D(GDP2(\(-1\))) & \(-15.52\) & \(-9.90\) & \(0.000\) *** \\ D(GDP2(\(-2\))) & \(0.06\) & \(2.43\) & \(0.071\) * \\ D(GDP3) & \(1.25\) & \(14.35\) & \(0.000\) *** \\ D(GDP3(\(-1\))) & \(0.70\) & \(9.55\) & \(0.000\) *** \\ D(GDP3(\(-2\))) & \(-0.014\) & \(-5.99\) & \(0.003\) *** \\ D(RENC) & \(-0.86\) & \(-17.50\) & \(0.000\) *** \\ D(RENC(\(-1\))) & \(-0.33\) & \(-7.25\) & \(0.001\) *** \\ D(RENC(\(-2\))) & \(0.12\) & \(2.45\) & \(0.069\) * \\ D(ACEL) & \(0.09\) & \(2.61\) & \(0.059\) * \\ D(ACEL(\(-1\))) & \(1.33\) & \(33.80\) & \(0.000\) *** \\ D(ACEL(\(-2\))) & \(0.92\) & \(16.71\) & \(0.000\) *** \\ D(FDI) & \(-0.006\) & \(-2.55\) & \(0.063\) * \\ D(FDI(\(-1\))) & \(0.06\) & \(30.07\) & \(0\) *** \\ D(FDI(\(-2\))) & \(0.03\) & \(19.47\) & \(0\) *** \\ CointEq(\(-1\)) & \(-1.42\) & \(-31.64\) & \(0\) *** \\ R-squared & & & \(0.98\) \\ Adjusted R-squared & & & \(0.96\) \\ \hline \hline \end{tabular}  
* and *** indicate significance of the variables at 10% and 1% levels, respectively.  
\end{table} 
Table 8: ECM-ARDL (3, 2, 3, 3, 3, 3, 3) model for short-run estimated results. The coefficient of D(FDI) is negative and significant at the 10% level, suggesting that FDI has a modest effect in reducing \(CO_{2}\) emissions in the short run. The positive and highly significant coefficients of D(FDI(\(-1\))) and D(FDI(\(-2\))) indicate that, over time, FDI might lead to increased \(CO_{2}\) emissions, possibly due to investments in energy-intensive industries. The cubic EKC Equation (12) is as follows:  
\[CO_{2}=53.64 GDP-7.31 GDP2+0.33 GDP3-1.04 RENC-0.76 ACEL-0.05 FDI-125.5 \tag{12}\]  
Deriving the right-hand side of the EKC equation with respect to GDP and setting it to 0, we obtain Equation (13):  
\[53.64-14.62 GDP+0.99 GDP2=0 \tag{13}\]  
The turning points for the cubic EKC occur at the following GDP values: \(GDP_{1}=7.97\), \(GDP_{2}=6.80\), as observed in Figure 2. These GDP values represent the points where the slope of \(CO_{2}\) emissions with respect to GDP changes, indicating the transition from one phase of the EKC to another. \(GDP_{2}=6.80\) is the point where \(CO_{2}\) emissions peak and begin to decline as GDP increases. The first turning point (\(GDP_{2}=6.80\)) marks the peak in \(CO_{2}\) emissions during the early stages of economic development. This phase reflects the effects of industrialization, urbanization, and significant reliance on fossil fuels for energy production. At this stage, environmental policies and clean technologies have a minimal impact. In contrast, the second turning point (\(GDP_{1}=7.97\)) highlights a resurgence of CO\({}_{2}\) emissions after an initial decline. This increase can be attributed to the rebound effect, where higher income levels and increased consumption demand offset earlier environmental gains. To address this, targeted policies must focus on maintaining the downward trajectory of emissions through investments in renewable energy, circular economic practices, and stringent environmental regulations. Comparative analysis with emerging economies such as Brazil and China underscore the need for proactive measures to decouple economic growth from \(CO_{2}\) emissions at advanced stages of development. The null hypotheses of diagnostic and stability tests are shown in Table 9. The results prove that the ARDL model is stable. Figure 2: Cubic EKC for India. The test results confirm that the ARDL model is stable, with no autocorrelation, heteroscedasticity, or specification errors. Stability is further visually reconfirmed through the CUSUMSQ test, strengthening confidence in the robustness of the estimated model.