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The time-frequency connectedness among metal, energy and carbon markets pre and during COVID-19 outbreak

The study investigates static and dynamic returns spillover effects between metal (gold, silver, copper and aluminum), energy (oil, natural gas and coal) and carbon markets in different frequency domains using the Diebold Yilmaz (2012) and the Baruník and Křehlík (2018) method. The results show that total connectedness in the post-COVID world is significantly higher compared to pre-COVID-19 outbreak period. The total spillover is contributed mainly by short-term spillover effects. Moreover, metal markets especially copper and silver have higher explanatory power. Spillover within markets is stronger than across these markets. In addition, the carbon market is more heavily interactive with other markets, and the metal market especially copper has relatively high explanatory power for the carbon price fluctuations in post-COVID-19outbreak periods. According to the net spillover, copper and gold has a hedge function in the short- and long-term, respectively. Furthermore, the relationship among these markets is time-varying, affected by market uncertainty such as the outbreak or major events.


Introduction
The COVID-19 outbreak, the worst public health event in nearly a century, has taken a toll on financial markets worldwide and continues to circulate (Baker et al., 2020;Le et al., 2020). Simultaneously, crude oil prices witnessed a sharp fall caused by the failed negotiations between OPEC countries and non-OPEC countries. Both severe shocks would not only result in a long-run economic crash, but also change the relationship and connectedness among markets. A growing number of studies have examined the nexus between different financial and energy markets during the ongoing COVID-19 outbreak period Albulescu, 2020;Dutta, 2017;Yousaf, 2021). However, few studies lay emphasis on the relationship between metal, energy and carbon markets before and during the COVID-19 outbreak period. Our study would fill the gap and provide more valuable information for investors and policymakers.
As global warming worsens, more and more countries have established carbon emission trading markets to reduce air pollution. Trading carbon emission rights is increasingly important and prevalent (Khanna et al., 2014). Fossil fuels for industrial production are the main source of carbon emissions. Fluctuations in prices of energy can affect its consumption and thus influence the demand for carbon emission rights. In turn, changes of carbon prices have an impact on the production and consumption of energy. A bidirectional relationship exists between the carbon market and energy market. In addition, the nexus between the carbon and three kinds of main fossil energy (oil, coal and natural) are complex. Differences in the demand and efficiency of fossil energy lead to differences in carbon emissions. Therefore, it is necessary to study the within and cross-market returns spillover of the carbon and energy market including oil, coal and natural gas.
In recent years, there has been growing interest in the energy-carbon nexus (Hanif et al., 2021;Naeem et al., 2020;Wang and Guo, 2018;Zhu et al., 2017). Some scholars found that the energy price is the determinant role on carbon market and has a unilateral impact on carbon prices (Dutta, 2017). Other studies further focused on the spillovers and co-movements between energy and carbon markets from different perspectives (Chevallier et al., 2019;Uddin Salah et al., 2018). For example, the results of Lin and Chen (2019) consistent with those of Soliman and Nasir (2018) indicated the dependence between the energy and carbon market is asymmetric and is higher in the lower tail. Yin et al. (2021) found that the carbon and coal price exist stronger synchronization in the long-run (longer than one month) using sample entropy from multi-scale and cross-over aspects. However, there is a relative paucity of dynamic analysis including these three main fossil energy and carbon market in different frequency domains.
Energy markets inevitably affect metal markets through various transmission mechanisms. On the one hand, the increase in oil prices caused by the demand for energy commodities leads to inflation. The gold market is affected due to its role to hedge inflation and other similar types of risks (Yaya et al., 2016). On the other hand, crude oil price fluctuations can affect the metal industry by increasing its production cost (Kaushik, 2018). In addition, changes in oil prices may be transferred to commodity futures markets, further affecting metal markets (Bai and Koong, 2018). In turn, the changes of metal prices have an impact on the whole commodity markets including oil, coal and natural gas (Gokmenoglu and Fazlollahi, 2015). Industrial metals also play a major part in mining industries and clean energy, the price of which is closely linked with traditional energy prices (Chen et al., 2022;Rehman and Vo, 2021;Yahya et al., 2020). Therefore, the role of industrial metals needs to be further study (Yousaf, 2021).
In the past two decades, the relationship between the metal and energy market has become one of the key issues for many scholars (Cagli et al., 2019;Hammoudeh and Yuan, 2008;Shahbaz et al., 2017). However, there is still no consensus. Some scholars have failed to consider the impact of energy price fluctuations on metal prices (Chang et al., 2013;Soliman and Nasir, 2018). Others found that there exists co-movement or the bilateral relationship between metal and energy markets (Wang and Chueh, 2013;Zhang and Wei, 2010). For instance, Zhang and Tu (2016) provided supports to the result that the crude oil price has a symmetric impact on metal prices. Rehman and Vo(2021) and Sari et al. (2010) found that precious metals prices follow oil prices and the co-movement varies in the short-and long-run. Eryigit (2017) proved that gold, silver and palladium prices have different correlations with energy prices. In addition, some studies have found the cointegration relationship between precious and industrial metals (Ciner, 2001;Sari et al., 2010). Most studies only considered the precious metal markets especially the gold market, without further studying the relationship between energy, industrial and previous metal markets.
When examining the relationship among markets, various models and methods are used. On the one hand, some scholars studied the causality and spillover through structural vector autoregressive approach, Johansen co-integration test, Granger causality test, and some employed the NARDL method to examine the asymmetric nexus (Jain and Ghosh, 2013;Le et al., 2020;Popp et al., 2018). On the other hand, GARCH family models and copulas are widely adopted in investigating the volatility (Gronwald, 2012;Kirkulak-Uludag and Safarzadeh, 2021). In order to study the directional spillover and the influence size between variables, the DY method of Diebold and Yilmaz (2012) is proposed. Baruník and Křehlík (2018) further proposed the frequency model (BK) to study the connectedness in different frequency domains. A growing number of studies have adopted the DY and the BK method to investigate the relationship between financial and energy markets (Li et al., 2021;Plakandaras et al., 2018;Sang et al., 2019;Tiwari et al., 2020;Xia et al., 2019;Zhang and Yan, 2020).     Our study makes several contributions to current literature. First, unlike previous studies on the nexus between energy, metal and carbon markets that only focus on a whole period, we compared the returns spillover effects before and during the COVID-19 pandemic crisis period . The unprecedented outbreak, the implementation of low-carbon economy and increasing political uncertainties have changed the relationship within and across the metal, energy and carbon markets. These changes motivate us to examine the risk transmission in the pre-and post-COVID-19 periods, which is valuable for investors as well as policymakers to make accurate plans. It is necessary to consider the dynamic connectedness and master the law of spillover sizes and directions among the three markets.
Second, few studies examine the directional connectedness between the metal (precious and industrial) and carbon market using network diagram across different time frequency. This study fills the gap. The metal mining plays an important role in economic growth, which is the foundation of basic industries such as transportation industries and buildings. Metals extraction with high energy consumption and pollution is the main source of CO2 emission, such as electrolytic aluminum (Qi et al., 2017;Wang and Feng, 2021). Policies implemented to reduce air pollution increase the production cost, thus causing a slowdown in metal industry. Therefore, it is necessary for policymakers to find reasonable cap-and-trade regulation in the carbon market, and promote economic stability through balancing the two markets.
Third, this study provides extensive examinations on relationship within the specific market. Previous literature mainly investigated the spillover effects across these three markets  or only included gold and oil markets. However, coal and natural gas are also two kinds of major energy to economic development (Lovcha and Perez-Laborda, 2020). Similarly, industrial metals such as copper and aluminum are considered as widely traded commodities (Evrim Mandacı et al., 2020). Therefore, we consider not only oil and precious metals but also natural gas, coal, copper and aluminum. We find that the connectedness between oil and gold is smaller compared to previous studies due to greater spillover effects within markets. Our results can further provide more generalized results and give investors more opportunities to diversify and shift risks.
Finally, we adopt the DY method (Diebold and Yilmaz, 2012) based on the GFEVD framework and the BK method (Baruník and Křehlík, 2018), which enable us to track the directional and long-, medium-and short-term returns connectedness between the three markets. On the one hand, the connectedness is generally determined by different information assimilation and diverse investment horizons due to different goals, demands and risk tolerance of investors, polluters and policymakers (Jiang et al., 2020). On the other hand, the time-frequency connectedness is driven by shocks at various frequencies with various strengths. A shock with long-term effects usually have stronger impacts at low frequency. The price fluctuations, unexpected outbreak, and market fundamentals significantly affect the connectedness at different time-frequency (Alberola et al., 2008). These outcomes provide information for policymakers and investors to craft appropriate strategies according to different time-frequency spillover effects.
The paper is organized in the following way. Section 2 introduces the data and the methodology. The main findings and discussions are presented in Section 3. Section 4 is the robustness test, and Section 5 presents conclusions.

Data
This article aims to analyze the time-frequency connectedness among metal, energy and carbon markets before and after the COVID-19 outbreak on 1 January 2020. We utilize daily data spanning the period from January 2014 to March 2022, which is divided into two periods: Pre-COVID-19 period: 1 January 2014 to 31 December 2019; the post-COVID-19 period: 1 January 2020 to 31 March 2022. We take the price of WTI crude oil (WTI), NYMEX natural gas futures (Gas), DCE coal futures (Coal), COMEX gold (Gold), COMEX silver (Silver), LME copper futures (Copper), LME aluminum futures (Al) and the ICE European emission allowance futures (EUA) from the Wind database. Moreover, we compute the returns using first log difference on these indices. Table 1 summarizes the statistical description of all the variables. According to the ADF tests, variables are stationary, satisfying the conditions for both DY and BK models. The standard deviations of all variables become larger during the COVID-19 pandemic period, indicating that risks increase after the outbreak.

Methodology
The models and methods widely used to study the nexus among markets, such as VAR model, GARCH family models and the waveletbased technique mentioned in the introductory section, have the inability to examine the directional and net spillover effects. The DY (Diebold and Yilmaz, 2012) and the BK (Baruník and Křehlík, 2018) method are not only used to investigate the dynamic and directional connectedness but also to examine the spillover at different time-frequency domains. Variance share consists of own variance share and cross variance share. Cross variance share shows H-period ahead uncertainty of y i that is explained by the changes of another variable y j . Proposed by Diebold and Yilmaz (2012), the spillover index model (DY) is based on a vector generalized autoregressive model (VAR) that measures the forecast error variance decomposition. The DY model can calculate the cross variance share in the following way. The n-dimen- where t = 1,...,T, ε t is white noise. We can rewrite model (1) as follows: Therefore, Y t can be written as the following recursive form: where ψ(L) = [Φ(L)] − 1 . The GFEVD of H-step forecast can be used to calculate cross variance share, written as follows: where H is the forecast horizon. The off-diagonal elements represent the  W. Jiang and Y. Chen forecast error variance decomposition and reflect the connectedness between k and j. The volatility spillover is denoted as the ratio of the sum of the off-diagonal elements to the sum of the whole matrix: (θθ H ) k,j is the standardized spillover effects defined as: where ∑ n j=1 (θ H ) k,j = 1. S ⋅k (H) measures the spillover effects from the variable k to other variables in our system, and S i⋅ (H) measures the spillover from the other variables to the variable k. The net spillover of the variable k can be calculated as follows: Therefore, we get the net spillover of the variable k to j: Furthermore, considering that spillover effects among metal, energy and carbon markets may be different at different frequencies, we adopt the BK model. The generalized causation spectrum under the frequency ω is defined ω =∈ (− π, π) as follows: where ψ is the Fourier transform, thus ψ(e − iω ) = ∑ h e − iωh ψ h h = 1,...,H, The forecast horizon H is not useful due to the unconditional error variance decomposition. Therefore, Barunik and Křehlik (2018) define the variance decompositions on frequency band d, d = (a, b), a, b ∈ ( − π, π). (f(ω)) k,j is weighted by the frequency shares of the variance of the j − th volatility. The generalized variance decompositions on frequency band d are presented as where Γ k (ω) is a weighting function defined by Baruník and Křehlík (2018). The scaled generalized variance decompositions can be calculated: where θ ∞ ) dω. Now, the total spillover effects under the frequency band d can be specified as The frequency connectedness under the frequency band d is To sum up, the within connectedness calculates the spillover effects within a specific band. The frequency connectedness decomposes the overall connectedness into three independent parts in our paper.

Static connectedness using DY model
The DY and BK method are on the basis of a vector generalized autoregressive (VAR) model. We choose the optimal lag length for the multivariate VAR according to the Hannan-Quinn criterion (HQ), Final Prediction Error (FPE), Akaike information criterion (AIC), Schwartz criterion (SC).
Tables 2 and 3 demonstrate the spillover effects of metal, energy and carbon markets based on DY method. It can be observed that the total connectedness is 19.76% and 26.88% in the pre-COVID-19 and post-COVID-19 period, respectively. The total spillover effects increase significantly after the COVID-19 outbreak, approximately 26.88% of connectedness in all the markets from inter-connectedness. According to the previous studies, unexpected crisis (SARS outbreak, financial crisis and oil price crash) causes an increase in the connectedness among markets (Zhang and Broadstock, 2020). Different than any of previous shocks, the COVID-19, regarded as a global pandemic, spread throughout the world in a relatively short time and has been spreading. World Health Organization (WHO) declared that over 496 million confirmed cases and over 6 million deaths have been reported globally by 10 April 2022. The global public health risk remains very high. In addition, threats and shocks to the energy, metal and carbon markets are more pronounced due to the quarantine measures, restriction on transaction, lockdown orders at the domestic and global level (Farid et al., 2022). The outbreak has influenced policy-making and expectations of investors, altering and increasing the connectedness among markets.
As shown in Table 3, metal market is the largest spillover transmitters within the market all the time such as the "to" spillover of copper  (5.76%), silver (6.09%), aluminum (5.45%). The results are similar to Naeem et al. (2020) andUddin Salah et al. (2018). Silver is both the largest contributor and receiver. On the one hand, metal production leads to high carbon emissions and energy consumption, implying the mental market connects the carbon market and energy market. On the other hand, metal is a key raw material to clean energy, which becomes the focus in the post-COVID-19 period and is developing rapidly to reduce carbon emissions (Yahya et al., 2020). Meanwhile, clean energy has become a substitute for traditional energy (Chen et al., 2022). Therefore, metal prices prove of utmost importance for prices of energy and carbon trading.
Moreover, the result further illustrates that the spillover effect of EUA has become stronger from 0.45% to 1.08%. The closer nexus between EUA and other markets indicates the vulnerability of carbon markets to shocks from other markets. Energy use and production of metals, adversely affected by the COVID-19 outbreak, are main source of carbon emissions. Carbon trading as a financial tool plays a more important role in energy and metal markets (Yin et al., 2021). Meanwhile, cap-and-trade system in European carbon market as a policy instrument is more widely used in post-COVID-19 outbreak period. Moreover, an increasing number of institutional investors and portfolio managers use carbon markets to obtain portfolio diversification benefits. Based on all these factors, carbon market is more integrated with the other markets. Therefore, the carbon market may be not appropriate for hedging duo to its rising participation in the post-COVID-19 period, which is consistent with the result of .
Spillovers within markets are greater than that across markets all the time. The connectedness within metal market is more integrated than energy market such as gold-silver (33.08%-35.40%), copper-aluminum (20.99%-20.40%). From the cross-market perspective, the largest spillovers among pairs occur changing from oil-copper (4.48%, 5.45%) to coal-aluminum (6.90%, 11.05%). In addition, gold amongst precious and industrial metals is least impacted by the spillover from energy markets, especially the shocks from oil and natural gas. Gold price fluctuations are mainly affected by other metal prices, which is ignored by previous studies. Therefore, gold has the stable nature, speculators can use it to diversify risks from energy markets (Batten et al., 2015). During the COVID-19 pandemic period, coal is relatively more integrated with copper and aluminum. Natural gas acts neither as a main transmitter nor a recipient among these markets, which is similar to the result by Rehman and Vo (2021). In summary, the energy market is relatively less connected with precious metal market, while it is more affected by industrial metal and carbon markets during the COVID-19 outbreak.

Static connectedness using BK model
Furthermore, Table 4 shows the results using BK approach by decomposing total connectedness into long-(more than 30 days), medium-(5-30 days), and short-term connectedness (1-5 days). The frequency bands are d1 ∈ [3.14, 0.63], d2∈ [0.63, 0.10] and d3 ∈ [0.10, 0.00]. In the post COVID-19 outbreak period, overall connectedness is clearly higher in all frequency domains. Both before and after the outbreak, total spillover effects mainly come from short-term spillover (16.23%-21.83%), followed by medium-(2.98%-4.27%) and long-term frequency (0.56%-0.79%). The total connectedness declines over time, and about half of the spillover effects are decreased to almost zero in the long-term. The above results prove the importance of dividing the overall connectedness into different frequency domains. The reason may be that the returns spillover effects are mainly driven by market participants with short investment horizons. The risks of these markets are transferred quickly to each other within a week.
As shown in Table 5, regarding the short-term horizons, the across market spillover effects mainly come from the connectedness among oil, coal, copper, aluminum and EUA. Short-term spillover effects are more linked with the speculators and day traders who do not have longer investment horizons. During the COVID-19 outbreak period, EUA exhibits more spillover effects, and still receives risk spillovers from energy and metal markets in the long-term. Since natural gas has lower connectedness with EUA, investors could benefit if they add natural gas in their portfolio to be a hedging need. In addition, gold has lower connectedness with oil before and after the pandemic crisis, compared to previous results of Li et al. (2021) and Shah et al. (2021). The reason is that the spillover between gold and oil in previous studies is overestimated due to neglect of the effects from other energy and mental markets. Connectedness within markets contributes more to the total connectedness.

Dynamic connectedness using DY model
The assumption of above analysis is that the spillover effects are constant. To capture dynamic effects of economic events and market turmoil in the entire sample period, we adopt the rolling window to study the dynamic connectedness between metal, energy and carbon markets. Fig. 1 demonstrates the dynamic connectedness in the pre-and postpandemic crisis periods. In the beginning, the spillover keeps in a low level due to recover from European debt crisis. The spillover effect dramatically increased from almost 25%-40% throughout the last second quarters of 2015. It climbed again from approximately 30%-37%, because a surge in production of OPEC caused the oil price plunge in late 2015. The Chinese stock market disaster, the Paris Climate Change Conference held in 2015 and Brexit in 2016 also led to EUA price increase. After reaching the highest points, the total connectedness dropped and fluctuated. During the last three quarters of 2017, the connectedness stayed flat almost at 25%. Afterwards, the weakening of the global economy, Sino-US trade friction and production cut agreement of OPEC strengthened the total spillover from 25% to 36%. Notable spillover effects occurred in face of the COVID-19 pandemic, and the connectedness remained high in the first and second quarters of 2020. Affected by the emergence of novel variants of the virus, the connectedness experienced an increase again. The Omicron variant appeared in the second half of 2021 and has become the dominant variant circulating globally. These dynamic results indicate that connectedness and risk transmission increase when adverse shocks occur in the market or economic conditions get worsen, confirmed by the findings of Badshah et al. (2019), Li et al. (2016) and Naeem et al. (2020).

Dynamic connectedness using BK model
Next, we compute the dynamic total connectedness in three timedomain frequencies (1-5 days, 5-30 days, more than 30 days) using a rolling window approach. From Fig. 2, we conclude that the total spillover is comparatively higher during the market uncertainty period such as economic crisis, political events or oil prices shocks. Specifically, the short-run connectedness fluctuates between 20% and 40%, and its trend is similar to the overall spillover trend. That indicates that shortrun connectedness is a top contributor of total connectedness between metal, energy and carbon markets in line with the results of Tables 4 and  5 The medium-term and long-term spillovers oscillate 3%-8% and 0.5%-2% respectively. The total spillover effect becomes smaller and relatively stable over time, implying that metal, energy and carbon markets digest information quickly.

Net spillover results
The net connectedness equals the difference between the directional "to" spillovers and the directional "from" spillovers. Fig. 3 plots timevarying net spillover effects of the eight variables use the DY method.
The value of the net connectedness alternates between positive and negative before the COVID-19 outbreak. That means the role of these eight variables is changing over time. However, during the ongoing COVID-19 pandemic, copper and aluminum are the main net transmitters of spillover. Although the value of net spillover of EUA increases rapidly in face of the COVID-19 outbreak, EUA plays the role of a net receiver from other variables in most of the sample period. The nexus between carbon market and other markets is moving closer due to the low carbon reduction policy and investors' choices to spread risks; this implies that carbon market is not the best choice as hedging assets in the ongoing COVID-19 era.
Figs. 4 and 5 plot the net spillover at different frequency bands using the BK method. Consistent with the results of Tables 4 and 5, short-term exhibits a larger connectedness. As shown in Fig. 4, in pre-COVID-19 outbreak era, the net connectedness of all markets alternates between positive and negative at all time-frequency. During the COVID-19 outbreak period, copper has positive short-term net connectedness and offers hedging properties in the short term (1-5days). The reason may be that copper is a vital industrial metal used in manufacturing and is likely to be more demanded when recovering from crisis and expanding economy (Azimli, 2022). The response of copper market to an improvement in economic policy tends to be positive (Roache and Rossi, 2009). In addition, investors with long-run goals can add gold into their portfolio, because gold is the net transmitter and less affected by energy prices in the medium-and long-term.

Connectedness network results
The static and dynamic spillover of the above section cannot illustrate the direction and size connectedness very accurately and clearly. Therefore, we adopt the complex network diagram to display the pairwise net spillover and core variables. The strength of the spillover effects is described by the thickness of the line, and the direction of the arrow represents the relationship between two variables in the positive net direction. Figs. 6 and 7 depict the net pairwise directional spillover in the pre-and post-COVID-19 periods.
It can be observed in Fig. 6 that metal markets have deeper colors in most time-frequency domains. That implies that metals play leading roles in the total spillover consistent with the results in Tables 2-5 During the ongoing COVID-19 period, the largest spillover effects exist within the metal market. Copper among industrial metals play the dominant role in the short-term, which is supported by the results of Rehman and Vo (2021). The spillover effect of gold, with a deeper color in the post COVID-19 period, is stronger especially in the medium-and long-term. In addition, carbon market is more vulnerable to the shocks from other markets in short-and medium-term.

Robustness test
In this section, we replace the WTI crude oil with Brent oil to be a robustness test. The results of static and dynamic connectedness are demonstrated in Tables 6-9 and Figs. 8-12. The overall spillover effects across these three markets are larger in the COVID-19 pandemic period than in the pre-COVID-19 period. Meanwhile, copper and silver exhibit largest spillover effects, implying that metal market contributes more to the total connectedness. EUA is more integrated with other variables during the COVID-19 period. According to net spillover effects, copper and gold offer safe-haven properties in short-and long-term, respectively. The total and net spillovers between metal, energy and carbon markets go to their peak during a market crisis or economic shocks. These results are in line with the above results, implying our findings are robust.
Considering the spillover may be sensitive to the size of the rolling window , we change the size of the rolling window (50 days, 100 days, 150 days) for robustness testing. Table 10 exhibits that total connectedness is larger in the short-run than in the long-run. The trend of spillover effects across different window sizes are the same, but the influence size is slightly different. Specifically, the shorter the rolling window is, the higher the spillover effect. The standard deviation of total spillover shows that the shorter the window length is, the stronger the fluctuation of the connectedness. Overall, these results prove our findings are robust.

Conclusion
In this paper, we study the static and dynamic connectedness between metal (gold, silver, copper and aluminum), energy (oil, natural gas and coal) and carbon markets with the method of DY and BK in preand post-COVID-19 outbreak periods. First, we investigate total spillover effects within and across these three markets using the DY approach. Then, the BK method is adopted to decompose the overall connectedness into high, medium and low frequency (1-5 days, 5-30 days, more than 30 days). Third, we are based on the rolling window to examine dynamic time-frequency spillover effects. Finally, we display the net spillover effects among these variables more clearly through several complex networks. In addition, our results are robust. We have some meaningful findings as follows: First, total connectedness between metal, energy and carbon markets becomes higher after the COVID-19 outbreak. The spillover effect is most significant in the short-term and sharply decreases over time. Moreover, metal markets especially copper and silver not only contribute more to total connectedness, but also are susceptible to energy and carbon markets. Third, the connectedness is larger within markets than that across markets, therefore the spillover between gold and oil is relatively smaller. Fourth, the carbon market becomes more vulnerable to the risk transmission from other markets, meanwhile it has a greater spillover effect on other markets in all time-frequency domains. The carbon market tends to play an indispensable role in the markets worldwide. According to the net spillover of BK method, copper and gold have the hedge ability in the short-and long-term, respectively. Finally, we find that the nexus among metal, energy and carbon markets is time-varying. It can be observed from the rolling window that the global crisis or political events increase the overall connectedness.
The above findings have significant implications for investors and policymakers. On the one hand, when policymakers regulate markets and make policies, they need to fully consider the larger risk transmission and spillover effects among these markets in the post-COVID-19 world. Moreover, policymakers should lay emphasis on the frequency spillover sizes and directions of carbon trading prices because of its increasing interaction with other markets. Energy plays an extremely important role in the development of economy and society. It is necessary to reduce the risk spillover of industrial metal markets to the energy markets. In addition, government should attach importance to the spillover between carbon and metal prices. The analysis of spillover effects in different frequencies enables policymakers to make more accurate policies to achieve their goals more effectively.
On the other hand, speculators had better pay special attention regarding risk management regard of the greater spillover of metal markets. Buyers and sellers of EUA must consider the spillover effects of energy and metal markets both in the short-and long-term. In addition, investors need to attach importance to the short-term spillover. It is necessary to choose the potential diversification opportunities in various assets based on the connectedness in different time-frequency domains.

Funding
This work was supported by the National Social Science Fund of China (20BJL020).

Declaration of competing interest
None.

Data availability
Data will be made available on request.