## 3 Research Models
### Bivariate SWARCH Model  
The GARCH (see Equations (5) and (6)) and DCC models (see Equations (9) and (10)) accommodate the variance- and correlation-clustering behaviors, respectively (the persistence of variance/correlation over time), which are commonly observed in the financial markets. This study proposes a regime-switching approach to investigate our hypothesis of intensity and direction of volatility spillover effect in carbon-energy markets (see Section 2.2). In particular, we extend [PERSON]'s (1994) Markov-switching Autoregressive Conditional Heteroskedasticity (SWARCH) model to investigate the regime-switching pattern for market volatility (i.e., switching between a low- and high-volatility regime) and the regime-switching volatility-correlation relations. Considering the two asset positions in the carbon-energy portfolio, we develop a bivariate SWARCH model as follows:  
\[h_{t}^{EUA}=g_{s_{t}^{EUA}}^{EUA}\times\left[\omega^{EUA}+\alpha^{EUA}.{(e_{t-1 }^{EUA})}^{2}/g_{s_{t-1}^{EUA}}^{EUA}\right] \tag{11}\]  
\[h_{t}^{WTI}=g_{s_{t}^{WTI}}^{WTI}\times\left[\omega^{WTI}+\alpha^{WTI}.{(e_{t- 1}^{WTI})}^{2}/g_{s_{t-1}^{WTI}}^{WTI}\right] \tag{12}\]  
We explain the state-dependent nature involved in Equations (11) and (12) as follows: First, the state variables, \(s_{t}^{EUA}\) and \(s_{t}^{WTI}\), have two possible outcomes: \(1\) or \(2\). Second, under Regime I (i.e., the state variable \(=1\)), the conditional variance of EUA is \(g_{1}^{EUA}\) times ARCH (1) process, and the conditional variance of WTI is \(g_{1}^{WTI}\) times ARCH (1) process. Under Regime II, (i.e., the state variable \(=2\)), the conditional variance of EUA is \(g_{2}^{EUA}\) times ARCH (1) process, and the conditional variance of WTI is \(g_{2}^{WTI}\) times ARCH (1) process (some researchers [35; 36; 37; 38] propose the use of the Markov-switching GARCH (MS-GARCH) model to handle regime-switching conditional variance. However, the MS-GARCH model suffers from an ad hoc assumption in merging the state-dependent conditional variances into a single value that does not depend on the regime path. [PERSON] [39] employs certain simulation methods to show that the estimates obtained from this collapsing procedure are biased and inconsistent. In the present study, we attempt to extend our empirical analysis by using a bivariate MS-GARCH model to capture the dynamic variance/correlation structure of stock and cryptocurrency returns. However, we cannot obtain valid estimation results since the Hessian matrix is singular and, thus, cannot be inverted. While the literature suggests that the MS-GARCH model offers an efficient way to model regime-switching conditional variance, it is not practical to apply a bivariate version of it to model conditional correlation given the technical problems as mentioned above. As such, in our empirical analysis, we adopt the bivariate SWARCH model, which is found to be robust in capturing regime-switching conditional correlations through their relationships with conditional variances). Third, Without loss of generality, we follow [PERSON] and [PERSON] [40] to normalize \(g_{1}{}^{FUA}=g_{1}{}^{WTI}=1\). Therefore, the conditional variances under Regime II are \(g_{2}{}^{EUA}\) and \(g_{2}{}^{WTI}\) times the variances under Regime I for EUA and WTI returns. Last, based on our subsequent empirical analysis (presented in Section 4), the estimated \(g_{2}{}^{EUA}\) and \(g_{2}{}^{WTI}\) coefficients are significantly larger than unity. Accordingly, we define Regime I as the low-volatility (LV) regime and Regime II as the high-volatility (HV) regime. Given the two separate volatility regimes for EUA and WTI, we generate four different possible states of the EUA-WTI market: (1) \(s_{t}{}^{EUA}=1\) and \(s_{t}{}^{WTI}=1\), (2) \(s_{t}{}^{EUA}=1\) and \(s_{t}{}^{WTI}=2\), (3) \(s_{t}{}^{EUA}=2\) and \(s_{t}{}^{WTI}=1\), and (4) \(s_{t}{}^{EUA}=2\) and \(s_{t}{}^{WTI}=2\).