## 1 Introduction  
Carbon markets have attracted attention following concerns around climate change [1]. Thanks to the European Union Emissions Trading System, established in 2005, carbon assets have become an available investment option for investors. Firms, particularly firms with high energy consumption, are also concerned about carbon markets since their costs rely on carbon and energy prices. For policymakers, carbon markets provide them with information to form appropriate policies to manage carbon emissions and energy consumption. The literature has examined the connectedness between carbon and energy markets and has offered evidence of their correlations (see [2] for a summary). This study contributes to the carbon-energy literature in three ways. First, we develop a theoretical perspective on the volatility-correlation relationships in carbon-energy markets to break down the fundamental channel and the volatility spillover effect, two significant factors causing the link between the two markets. The fundamental channel is based on real and economic ties between the two markets. Economic growth stimulates energy consumption and, thus, produces carbon emissions. Given this common factor (i.e., economic growth), the correlation between carbon and energy markets is straightforward. Beyond the fundamental channel, this study highlights the volatility spillover effect and presumes that this effect increases the carbon-energy correlation. Moreover, we re-examine the direction of the volatility spillover effect in carbon-energy markets using a regime-switching approach. In brief, if the carbon-energy correlation grows due to a high volatility condition of the energy market; volatility spillover is from the energy market to the carbon market. By contrast, if a high carbon-energy correlation is observed when the carbon market faces a high-volatility condition, the volatility spillover direction is from the carbon market to the energy market. To date, the existing carbon-energy market literature has not considered these dynamic volatility-correlation relationships. In this study, we address these relationships and use them to re-examine the intensity and direction of the volatility spillover effect in carbon-energy markets. Second, to examine the dynamic volatilities and correlations in carbon-energy markets, most studies [3; 4; 5] adopt the conventional time-dependent approaches, including the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) and DCC (Dynamic Conditional Correlations) models. While the GARCH and DCC models are the most popular methodologies used for market volatilities and correlations, they suffer from some limitations. First, the GARCH-based volatilities cannot control discrete volatility jumps in markets [6; 7]. Second, prior studies use a two-step estimation method for the DCC models, which implicitly assumes independence between volatilities and correlations (see Section 2.2). This study employs a regime-switching approach to identify volatility state combinations in carbon-energy markets and jointly analyzes their correlation dynamics. Our one-step estimation method mitigates the sample selection bias due to a two-step estimation process [8]. Third, we conduct two practical tests: volatility forecasting and portfolio construction, to offer evidence to support the use of our regime-switching approach against the conventional time-varying approaches, including the GARCH and DCC models. Our empirical results show that carbon-energy correlations significantly diverge across various volatility state groupings. Accordingly, this study conducts two practical tests: portfolio risk forecasting and portfolio construction. We argue that volatilities and correlations are the two critical elements for risk management; thus, the consideration of volatility-correlation relationships may enhance the performance of the tasks. To the best of our knowledge, prior studies have not conducted these two practical tests on carbon-energy market data.