## 4 Data and Estimation Results
### Bivariate SWARCH Model  
Table 5 presents the estimated results of the bivariate SWARCH model with state-varying correlations. This model employs a regime-switching approach to address the non-uniform volatility-correlation relationships that may help us re-examine the intensity and direction of the volatility spillover effect in carbon-energy markets (see Section 2.2). First, the estimated volatility scale parameters (i.e., \({g_{2}}^{EULA}\) and \({g_{2}}^{WTI}\)) are significantly larger than the unity. In particular, \({g_{2}}^{EULA}\) is 6.7405, and its standard deviation is 0.4385. Moreover,  
Figure 2: Volatility of return rates on EUA carbon price and WTI crude oil price.  
\(g^{2 WTI}\) is 8.1736, and its standard deviation is 0.8642. Since their 99% confidence intervals do not cover the value of one (\(g_{1}\)\({}^{ELA}\) = \(g_{1}\)\({}^{WTI}\) = 1), we recognize Regime II as a high-volatility (HV) regime and Regime I as a low-volatility (LV) regime.  
Second, as shown in Table 5, all the estimated correlations (i.e., \(\rho_{1,1}\), \(\rho_{2,1}\), \(\rho_{1,2}\), and \(\rho_{2,2}\)) are positive and significant (\(p\)-value \(<\) 1%) but display considerable divergence. Moreover, the assumption of an equal correlation (i.e., \(\rho_{1,1}\) = \(\rho_{2,1}\) = \(\rho_{1,2}\) = \(\rho_{2,2}\) = \(\rho\)) is rejected at a 1% significance level. These results confirm the non-uniform correlations across various volatility regimes in carbon-energy markets.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Coefficient** & **S.D.** & **t-Statistic** & \(p\)**-Value** \\ \hline \multicolumn{5}{c}{_ELA Equation_} \\ \(\mu^{ELA}\) & 0.1122 & 0.0383 & 2.9295 & 0.0017 \\ \(\varphi^{ELA}\) & \(-\)0.0159 & 0.0085 & \(-\)1.8706 & 0.0307 \\ \(\omega^{ELA}\) & 0.0840 & 0.0221 & 3.8009 & 0.0001 \\ \(\alpha^{ELA}\) & 0.1037 & 0.0107 & 9.6916 & 0.0000 \\ \(\beta^{ELA}\) & 0.8949 & 0.0103 & 86.8835 & 0.0000 \\ \hline \multicolumn{5}{c}{_WTI Equation_} \\ \(\mu^{WTI}\) & 0.0607 & 0.0348 & 1.7443 & 0.0406 \\ \(\varphi^{WTI}\) & \(-\)0.0196 & 0.0105 & \(-\)1.8667 & 0.0310 \\ \(\omega^{WTI}\) & 0.1033 & 0.0232 & 4.4526 & 0.0000 \\ \(\alpha^{WTI}\) & 0.0991 & 0.0100 & 9.9100 & 0.0000 \\ \(\beta^{WTI}\) & 0.8838 & 0.0118 & 74.8983 & 0.0000 \\ \hline \multicolumn{5}{c}{_Time-varying correlation_} \\ \(\tau\) & 0.0026 & 0.0016 & 1.6250 & 0.0521 \\ \(\pi\) & 0.9751 & 0.0102 & 95.5980 & 0.0000 \\ \(\lambda\) & 0.0110 & 0.0035 & 3.1429 & 0.0008 \\ \multicolumn{5}{c}{_Log-likelihood function_} \\ \multicolumn{5}{c}{Notes: This table presents the estimation results of the bivariate GARCH-CCC model for EUA and WTI returns. See Table 1 for sample descriptions and data sources.} \\ \end{tabular}
\end{table}
Table 4: Estimated results of the bivariate GARCH-DCC model.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Coefficient** & **S.D.** & **t-Statistic** & \(p\)**-Value** \\ \hline \multicolumn{5}{c}{_ELA Equation_} \\ \(\mu^{ELA}\) & 0.1290 & 0.0358 & 3.6034 & 0.0002 \\ \(\varphi^{ELA}\) & \(-\)0.0037 & 0.0032 & \(-\)1.1563 & 0.1238 \\ \(\omega^{ELA}\) & 0.1047 & 0.0241 & 4.3444 & 0.0000 \\ \(\alpha^{ELA}\) & 0.1226 & 0.0110 & 11.1455 & 0.0000 \\ \(\beta^{ELA}\) & 0.8777 & 0.0102 & 86.0490 & 0.0000 \\ \hline \multicolumn{5}{c}{_WTI Equation_} \\ \(\mu^{WTI}\) & 0.0937 & 0.0268 & 3.4963 & 0.0002 \\ \(\varphi^{WTI}\) & 0.0036 & 0.0131 & 0.2748 & 0.3917 \\ \(\omega^{WTI}\) & 0.1687 & 0.0327 & 5.1590 & 0.0000 \\ \(\alpha^{WTI}\) & 0.1370 & 0.0120 & 11.4167 & 0.0000 \\ \(\beta^{WTI}\) & 0.8393 & 0.0143 & 58.6923 & 0.0000 \\ \hline \multicolumn{5}{c}{_Correlation_} \\ \(\rho\) & 0.1716 & 0.0167 & 10.2754 & 0.0000 \\ \multicolumn{5}{c}{_Log-likelihood function_} \\ \hline \hline \multicolumn{5}{c}{Notes: This table presents the estimation results of the bivariate GARCH-CCC model for EUA and WTI returns. See Table 1 for sample descriptions and data sources.} \\ \end{tabular}
\end{table}
Table 3: Estimated results of the bivariate GARCH-CCC model.