## 6 Conclusions and Future Research Directions  
The volatility spillover effect in carbon-energy markets has been documented in the literature. Our study offers contributions to the literature in three respects. First, we develop the non-uniform volatility-correlation relationships and use them to re-examine the intensity and direction of the volatility spillover effect in carbon-energy markets. Second, we develop a bivariate SWARCH model with four-state volatility combinations and correlations. Third, we employ the realized data (EUA and WTI) to run the model empirically using a one-step estimation process. Finally, we perform two practical tests, portfolio volatility forecasting and portfolio construction, to validate our proposed regime-switching approach. Using a two-volatility-state setting for each carbon and energy market (HV versus LV), we establish a four-volatility-state system for carbon-energy markets (i.e., LV-LV, HV-LV, LV-HV, and HV-HV). We then develop the non-uniform volatility-correlation relationships using these volatility state groupings. While existing studies have provided evidence of the volatility spillover effect in carbon-energy markets, to the best of our knowledge, they have neither addressed the non-uniform volatility-correlation relationships, nor used them to empirically prove the intensity and direction of the volatility spillover effect. We fill these gaps by developing a theoretical hypothesis, constructing a specific econometric method, and conducting two practical risk management tests. Our empirical findings are presented as follows: First, the carbon-energy correlation under the HV-HV state is larger than that under the LV-LV state. This result offers evidence of the volatility spillover effect in carbon-energy markets. Second, when the carbon and energy markets are experiencing an opposite volatility regime (i.e., one in an HV regime and the other in an LV regime), a higher carbon-energy correlation is observed when the energy market is in an HV regime. This result indicates that the direction of the volatility spillover effect is from the energy market to the carbon market. Third, our proposed \begin{table}
\begin{tabular}{c c} \hline _Panel A: Portfolio mean return_ & \\ \hline \hline \multicolumn{2}{c}{Mean return} \\ \hline \hline \multicolumn{2}{c}{Bivariate GARCH-CCC model} & 0.0436 \\ \multicolumn{2}{c}{Bivariate GARCH-DCC model} & 0.0427 \\ \multicolumn{2}{c}{(\(-0.3524\))} \\ \multicolumn{2}{c}{Bivariate SWARCH model with state-varying correlations} & 0.0452 \\ \multicolumn{2}{c}{(0.1611)} \\ \hline \hline \multicolumn{2}{c}{_Panel B: Portfolio return volatility_} \\ \hline \hline \multicolumn{2}{c}{Return volatility} \\ \hline \hline \multicolumn{2}{c}{Bivariate GARCH-CCC model} & 1.7981 \\ \multicolumn{2}{c}{Bivariate GARCH-DCC model} & 1.7840 \\ \multicolumn{2}{c}{(\(-1.5996\))} \\ \multicolumn{2}{c}{Bivariate SWARCH model with state-varying correlations} & 1.6392 \\ \multicolumn{2}{c}{(\(-7.2752\)) ***} \\ \hline \hline \multicolumn{2}{c}{_Panel C: Sharpe ratios_} \\ \hline \hline \multicolumn{2}{c}{Bivariate GARCH-CCC model} & 0.0242 \\ \multicolumn{2}{c}{Bivariate GARCH-DCC model} & 0.0239 \\ \multicolumn{2}{c}{Bivariate SWARCH model with state-varying correlations} & 0.0276 \\ \hline \hline \end{tabular} Notes: This table lists the performance of portfolio construction (via a minimum variance strategy) for each model. Two portfolio performance measures are adopted: mean return and return volatility. We use the bivariate GARCH-CCC model as a benchmark to calculate the t-statistic for the difference (the figure in the parenthesis). *** denotes significance at the 1% level.  
\end{table} 
Table 8: Performance of EUA–WTI portfolio construction. regime-switching approach may offer better performance on carbon-energy portfolio risk forecasting and portfolio construction than the conventional GARCH and DCC models. Lastly, we address several future research directions.