Energy commodities spillover analysis for assessing the functioning of the European Union Emissions Trading System trade network of carbon allowances

Abstract

The European Union Emissions Trading System (EU ETS) is designed to promote carbon reduction in a cost-effective and economically efficient manner. To meet their compliance requirements, participants within the EU ETS can either invest in carbon abatement, thereby improving their environmental performances, or purchase carbon allowances from the market. Our analysis aims to investigate the structural features of the EU ETS trade network of allowances and assess its robustness to spillovers propagated from energy commodities. We build the EU ETS trade network of allowances on a daily basis by leveraging a granular dataset comprising over 32k accounts that performed approximately 720k trades, involving more than 127 billion of allowances from 2005 to 2020. Importantly, our analysis is able to cover entirely the first three phases of the EU ETS, thus providing an updated view on the functioning of the EU ETS. We find that the EU ETS trade network of allowances is disassortative and characterized by very active nodes belonging to the energy sector. Moreover, we detect that energy commodity transmission channels can be substantial and may shape the way nodes transfer allowances, with the configuration of the EU ETS often being a net receiver of spillovers. These findings are robust even when we analyze extreme quantiles of the distribution to account for distressed periods. Our study shows how the functioning of the EU ETS can be influenced by spillovers imported from energy commodity markets. We discuss how investors can build portfolios that either hedge or amplify the impact of such spillovers, depending on their risk appetite. Given that energy commodities are significant inputs in the production processes of energy firms—who are highly active participants in the trade network of allowances—analyzing market responses to shocks in energy commodities is crucial for the effective implementation of the policy.

Introduction

The regulatory push toward decarbonization and a greener economy requires optimally balancing between carbon abatement investments and the preservation of economic competitiveness. The uncertainty related to the transition process toward a low-carbon economy can also trigger additional negative indirect consequences for firms, especially in carbon-intensive sectors. Various types of policy instruments have thus been proposed to encourage carbon abatement and reduce the uncertainty surrounding the adoption of cleaner and innovative technologies^1,2, such as feed-in tariffs, tax incentives, green procurement and emission trading schemes. Among the latter, the EU ETS stands as the cornerstone of climate policy in Europe and serves as a prototype for similar emissions trading systems developed globally, thus constituting a prominent example of a policy package designed to promote greenhouse gases (GHG) reductions in a cost-effective manner³.

The EU ETS was launched in 2005 and it was originally divided into three different phases with increasing environmental goals: Phase I (2005–2007), Phase II (2008–2012), and Phase III (2013–2020). To reduce aggregate emissions, participating entities are allowed to emit a total amount of emissions that is capped and decreases over time. This ensures that overall emissions progressively decline across the sectors covered by the regulation. Currently, Phase IV (2021–2030) enables carbon abatement through a mix of measures. For example, it includes a faster annual reduction rate of the total emissions, increasing from 1.74 to 2.2%. Additionally, it implements targeted carbon leakage rules for entities operating in sectors at risk of reallocating their activities to other countries, outside the European perimeter, with laxer emission constraints. Furthermore, the Green Deal and the Fit for 55 packages set the ambitious targets to cut at least 55% of GHG emissions in Europe by 2030 and to become climate neutral by 2050^4,5.

From a firm’s perspective, transforming a production process requires adopting a forward-looking perspective and evaluating the inherent long-term risks that may arise. At the same time, a sudden shock can threaten the balance between short-term performances and planned investments. Overall, such uncertainty may hinder environmental regulations from achieving cost-effective carbon abatement. For instance, uncertainty about the long-term viability of innovative cleaner technologies might lead investors to favor firms with more traditional (though less environmentally friendly) features, as these could be perceived as less risky. This preference can result in significant delays in the implementation of cleaner investments and reduce the overall effectiveness of the environmental policy^6,7.

While a firm or sector may be equipped to handle these risks and mitigate their direct losses, it may still face significant indirect effects, such as those related to the supply chain. For example, this has become particularly evident recently, as unpredictable events have disrupted or created uncertainty in global supply chains^8,9. In this context, energy commodities play a critical role in various supply chains. For instance, in the EU, the Gross Domestic Product (GDP) produced per unit of gross available energy increased from 6.25 to 8.59 € per kilogram of oil equivalent (Kgoe) in the time interval from 2000 to 2020. Due to their strong connections with capital markets, energy commodities have been studied as potential sources of disruption that could impact financial stability and the broader economy^10,11. Importantly, shocks in the energy commodities markets influence the production activities and performances of firms with energy-related businesses. These firms represent a significant group within the EU ETS, both in terms of emissions and participation in the trading of allowances^12,13,14,15. For these reasons, this paper aims to examine the impact of such energy-related shocks on the functioning of the EU ETS trade network of allowances, an area that has not yet been thoroughly analyzed (see the “Literature review” section).

The contribution of our study is manifold. First, we provide a detailed investigation of the EU ETS trade network of allowances. We apply a network theory perspective to describe how entities operating in the EU ETS acquire and transfer allowances, thereby creating a network of trades. By utilizing the most granular level of data, represented by the accounts involved in the trades, we investigate the trade network using daily transactions from 2005 to 2020 (see “Methods” section for details). This means that our sample comprises more than 32k accounts that perform about 720k trades corresponding to more than 127 billion of allowances. Some studies have already adopted a network theory perspective to analyze the functioning of the EU ETS^12,16,17,18. As far as we know, this is the first paper exploiting such detailed data on a daily basis to study the trade network of the EU ETS for the whole first three Phases of the program, covering the recent period characterized by significant events that are not well addressed in the existing empirical literature. This approach allows us to study the dynamics of the EU ETS trade network with a high level of temporal granularity, which is crucial for identifying imbalances originating from financial markets. Moreover, by constructing the trade network at the account level, we can accurately represent the actual operators involved in the trading of allowances.

Second, we highlight the crucial role of entities involved in energy-related business activities in shaping the configuration of the trade network. This is a relevant aspect to consider in designing policy interventions, as we observe that a key player in the trade network is a group typically short in allowances and at a low risk of carbon leakage. The energy sector is important for many economic activities due to the low elasticity of substitution of energy inputs, which means these inputs cannot be easily replaced by alternative production factors, such as labor or capital. As a result, energy inputs significantly affect the supply chains of various productions and their carbon footprint. Our results corroborate previous findings^12,13,14,15 highlighting the active role of energy participants within the EU ETS. In addition, we demonstrate that the distribution of the trades of the energy sector differs from that of the carbon leakage sectors. Specifically, the energy sector is more active as a purchaser of allowances, while carbon leakage sectors are more active as sellers. Interestingly, these opposite roles persist even during Phase III, when significant events and policy amendments affected the functioning of the EU ETS.

Third, due to the centrality of the energy sector, we scrutinize the stability of the trade network of allowances with respect to spillovers from economic dimensions relevant for the functioning of the EU ETS, particularly those related to energy commodities. Specifically, we rely on the spillover tests^19,20 applied on a system including energy commodities (namely, oil, natural gas, and coal) and the EU ETS configuration. This analysis complements the study of Wang et al.²¹, who investigate bilateral interactions between trading behaviors and carbon prices to disentangle the different roles of compliance versus non-compliance trading, and extends previous studies that were confined to examining the relationships between carbon prices and related energy commodities.

Participating entities under the EU ETS regulation are required to surrender an amount of allowances covering their emissions produced during the compliance year, with one tonne equivalent of carbon dioxide corresponding to one European Union Allowance (EUA). The cap-and-trade structure of the EU ETS allows firms to trade these allowances. For instance, firms that manage to reduce their emissions can either hold their allowances in excess to comply with future needs (i.e., banking of allowances), or sell them to other firms that have a shortage of allowances. In order to be complaint with the EU ETS regulation, firms therefore can either invest in innovative low-carbon technologies that favor carbon abatement or purchase additional allowances from the market. The evolution of the European carbon market has been significant since its inception^22,23. For example, at global level²⁴ the value of carbon market transactions in 2020 was around 238 € billion, with a traded volume of 10.7 billion tonnes of allowances. The EU ETS alone accounted for nearly 90% of this global carbon market trading. Hence, by analyzing the EU ETS trade network of allowances, we provide a novel and comprehensive representation of the main carbon market and its interactions with other financial markets. Our analysis reveals that the EU ETS configuration is likely to absorb spillovers from energy commodity markets. This is a significant aspect when evaluating the policy effectiveness of the EU ETS, as spillovers from energy commodity markets can influence the way EU ETS entities exchange allowances, potentially more than the carbon market itself. When assessing the functioning of the EU ETS as a cap-and-trade system designed to stimulate better environmental performances, we find that the trade network of allowances usually acts as a net receiver of spillovers from commodity-related markets. For instance, supply and demand shocks impacting energy-related raw commodities can prompt the anticipated obsolescence of carbon-intensive technologies and foster the development of viable green alternatives and more sustainable productions^25,26.

To further investigate the behaviour of spillovers during rare events, we also apply a recent quantile-based connectedness approach²⁷. Estimation by quantile regression may be more appropriate than simple conditional mean estimators when dealing with extreme events, such as distressed market periods, thus offering an informative and novel view for the study of the resilience of the trade network during turbulent market phases. The evidence of heavy tails in the data suggests that the behaviour at the mean of the conditional distribution may differ significantly from that at the tails. Our analysis reveals that the spillover intensity is stronger in the tails than at the conditional mean. Importantly, we confirm that the EU ETS remains a net receiver of spillovers even in the tails of the conditional distribution, but with greater intensity. This indicates that during distressed periods, the EU ETS configuration is likely to absorb more substantial spillovers from energy commodity markets, thereby modifying the way allowances are exchanged even more intensively.

Fourth, the dynamic pattern of the spillover index reflects the evolving connectedness of the system, with increasing values indicating that spillovers are more likely to propagate through the dimensions of the system. This representation of spillovers is relevant for EU ETS entities that trade allowances to meet their surrendering requirements, as well as for market players trading allowances for investment purposes. We propose leveraging the behavior of the system’s connectedness to build portfolios with directional exposure to the dynamics of the spillover index, potentially neutralizing such exposure by appropriately selecting the weights in the portfolio allocation. We illustrate this approach using a five factor model²⁸ enriched by the inclusion of the spillover index. To do so, we construct a representative sample of European firms involved in energy-related business activities and then we estimate the daily exposure of each stock to the system’s connectedness. Firms are increasingly recognizing the importance of signaling their commitment to achieving enhanced environmental outcomes to attract investors and improve market performance. This trend is evidenced by the recent proliferation of investment funds that focus on environmentally friendly assets^29,30. Given the impact of climate change risks on financial asset prices, investors are faced with the challenge of identifying investment opportunities that align with environmental goals²⁹. We demonstrate how our approach can be utilized to build portfolios with varying elasticity to the dynamics of the spillover index, interpreted as an additional risk factor that contributes to explaining the cross-sectional heterogeneity of stock market performances. From a financial perspective, we provide an illustrative example of how exposures to the system’s connectedness, and consequently to the propagation of spillovers, can be managed through appropriate portfolio allocation.

In the Supplementary Information (SI) we conduct several robustness analyses. All results support our main findings. Spillovers from the EU ETS are unlikely to heavily impact on the other dimensions of the system and, if they did, the effects are generally short-lived. In contrast, we confirm that transmission channels from energy commodities can be substantial, particularly during turbulent market phases.

The paper is structured as follows. The “Literature review” section outlines how our study contributes to existing research on the EU ETS and spillovers within the energy commodity system. Next, the “Results” section describes the configuration of the EU ETS trade network of allowances, highlighting its key features. It then discusses the role of the energy sector as a pivotal actor in this topological configuration. Finally, we present the results of the spillover tests conducted in both time and frequency domains, as well as within a quantile framework. These sections are intended to introduce portfolio analysis that incorporates spillover dynamics to manage the exposure to the connectedness of the system of EU ETS-related energy commodities.

Literature review

Literature on emissions schemes^31,32 suggests that carbon abatement is performed by those entities having lower abatement costs, with the overall environment capacity to absorb emissions without harm that is a finite and exhaustible resource. The EU ETS mimics this framework since there is a finite amount of allowances declining in time, with such a scarcity inducing a value to the EUAs. More generally, the EUAs price represents a reference value to be compared to its theoretical optimum, i.e. the marginal abatement cost, with larger differences in marginal costs across entities implying greater opportunities for gain from trade, contributing to lower total welfare costs.

Interestingly, existing literature has highlighted varying effects of EU ETS participation across different sectors. For instance, firms in energy-intensive sectors react differently to the EU ETS regulations, while sectors more exposed to carbon leakage tend to benefit from more favorable allocations of allowances^33,34. Moreover, financial intermediaries, which voluntarily participate in the EU ETS marketplace of allowances, are often more active than regulated entities^12,35. Our network representation allows us to uncover the presence of a subset of entities, predominantly in the energy sector, that are highly active in the trading of allowances. Their behavior can therefore significantly impact the overall functioning of the trade network and the economic efficiency of the EU ETS.

More generally, entities participating in the EU ETS can adapt cost-effective strategies to their needs, protecting both their competitiveness and shareholder value by considering a large variety of factors affecting the functioning of the system^36,37,38. These factors include not only carbon prices, but also the presence of internationally coordinated policies, technological changes, mitigation options, financing opportunities, taxation issues, and expectations on energy commodity markets. Our analysis focuses on the latter aspect due to the importance of the energy sector within the EU ETS. Specifically, we investigate whether shocks within the energy commodity system could propagate and impact the overall functioning of the EU ETS trade network of allowances. For example, if a shock in the energy commodity market significantly alters how firms trade allowances, it could disrupt the normal functioning of the EU ETS, potentially changing the interactions between entities with surplus and deficit allowances or their propensity to bank allowances. This, in turn, may affect how EUAs prices reflect the cost-effectiveness of carbon abatement opportunities.

We measure the interdependencies between the EU ETS and the energy commodities by relying on the spillover tests based on the methodologies proposed by Refs.^19,20,27. We study such spillovers to understand potential channels of instability affecting the trade network of allowances. For example, Ref.³⁹ discusses how the strengthening of feedback effects between the carbon market and other capital markets would generate a system’s connectedness more vulnerable, which may lead to the failure of the EU ETS. They present the main channels of such connections distinguishing between the correlated-information channel, also known as “return spillover” that is based on the price discovery process⁴⁰, and the risk premium channel, also known as “volatility spillover”, that occurs when a shock in one market may adversely influence the risk appetite of the participants in any markets⁴¹. Our study contributes to this stream of research by analyzing spillovers within a framework that integrates both financial time series and a topological indicator, namely the Assortativity coefficient (see “Methods” section for details). This coefficient serves as a proxy for the configuration of the EU ETS trade network, providing a novel perspective to study how firms’ decisions to trade allowances reflect external shocks such as those originated from the energy commodity markets. By incorporating this network indicator, we also link our findings to the literature on systemic risk and financial stability, where the resilience of a system to contagion and shock propagation is often associated with its tendency to vary the assortativity mixing^42,43.

In addition, we investigate the connectedness in the tails of the distribution to assess the role of the EU ETS during distressed periods, when market uncertainty is more likely to propagate. While several works have analyzed the EUAs price dynamics and determinants (see Ref.⁴⁴ for an overview), only a few studies have focused on how market conditions affect the way EU ETS entities manage their allowances. For example, trading behavior in the EU ETS has been linked to various features^14,18,45, including the size of the firm, the difference between free allocations and verified emissions, sector affiliation, the number of installations, productivity, and location. Additionally, transaction costs play a significant role in limiting the trading activity of smaller entities, which often have less trading experience and typically conduct trades indirectly through third parties. Importantly, the EU ETS may struggle to achieve efficiency if many participants are inactive or hesitant to trade allowances, like during market phases of intense information arrival, as shown in Ref.⁴⁶ who finds that firms usually tend to trade more when price volatility is lower. Our analysis contributes to the understanding of the heterogeneous levels of EUAs trading, highlighting how spillovers, particularly from energy commodity markets, can significantly impact the configuration of the trade network of allowances. This influence is especially pronounced during periods of market turbulence, when shocks are more likely to disrupt trading dynamics.

Climate-related policies may significantly raise costs of carbon-intensive firms and limit their business activities. Consequently, investing in stocks of firms with better environmental performances might mitigate investors’ uncertainty associated with the transition to a low-carbon economy, with investors facing the challenge of picking assets that align with both their environmental and financial goals^29,47,48. Several empirical works have demonstrated how hedging portfolio strategies can be implemented using the financial time series considered in the spillover analysis. This approach typically involves calculating either the optimal portfolio weights or the hedge ratios, and then evaluating the hedging effectiveness of the corresponding diversification strategies. Few examples exist for the EU ETS context and refer to the EUAs market dynamics in relation to energy commodities or traditional asset classes (see, e.g., Refs.^49,50,51, among others). However, the relationship between spillover propagation and portfolio diversification is not straightforward because higher connectedness does not necessarily lead to a higher average volatility, especially when assets are connected in a complementary way⁵². Since the spillover index reflects the strength of the system’s connectedness, investors may opt to adjust their portfolio exposures accordingly. Our study suggests leveraging the dynamics of the spillover index to manage such exposure to the connectedness of the energy commodity system. By constructing portfolios based on a representative sample of energy firms and weighting them according to their specific exposures to spillovers, we illustrate how such portfolios can generate better risk-adjusted performances than simpler strategies. This approach can be particularly useful for participants seeking to hedge against spillover risks in a system of EU ETS-related commodities.

Results

The EU ETS

The effectiveness of the EU ETS mechanism is based on energy-intensive industrial installations mandated to participate in the system. Currently, the EU ETS covers around 40–45% of the European GHG emissions from about 11,000 installations in the power sector and manufacturing industry, from all 27 EU member states plus EEA-EFTA states. Until 2020, the United Kingdom was also part of the EU ETS.

The cap-and-trade structure of the EU ETS allows entities to trade allowances. Trades of allowances can occur between two accounts that are not necessarily both liable entities under the EU ETS regulation. Spot, futures, options, and forward contracts are traded on the secondary markets, typically in organized exchanges (e.g., EEX in Leipzig, ICE in London, and in the first years also BlueNext in Paris). Fees and admission procedures make stock exchanges attractive primarily for firms that trade significant amounts of permits, while smaller entities, which trade fewer permits, are more likely to rely on financial intermediaries in over-the-counter trades. Furthermore, in accordance with Annex XIV (4) of Regulation 389/2013, there is an embargo period of three years in the transaction data publicly available in the EU ETS portal (see “Methods” section for details).

EUAs prices experienced very volatile market patterns, especially during the financial crisis of 2007–08 and the switch from the first pilot phase to the second phase of the EU ETS program^22,53. Due to the absence of enough reliable information on future emissions, the cap of Phase I was set based on estimates, which determined an excess supply of allowances relative to verified emissions. Since these allowances could not be banked for use in the following phase, the price of EUAs practically declined to zero by the end of 2007. At the beginning of Phase II, the outbreak of the financial crisis led to a drastic reduction in emissions. This resulted in a large surplus of allowances, which heavily influenced the EUAs market prices that remained very low throughout Phase II. For example, one $\hbox {tCO}_{{2}}$ was priced a few euro cents at the end of Phase I, rising to about 20–25 € at the beginning of Phase II before dropping in 2009 to about 10 €. The EUA was priced 12–15€ until 2011, when it started to sharply decrease to about 3–5 € around the beginning of 2013. Then, it ranged at about 4–8 € during the first years of Phase III. It is worth mentioning that entities with Phase II allowances were allowed to bank them and carry-over use in Phase III, when the cap requirements would be lowered at a faster pace and would likely result in a stronger market pressure on prices. However, at the end of Phase III in December 2020, the price of EUA was still low (about 30€), but sharply increasing in subsequent months. For instance, in 2021 its average price was more than 50 €, increasing in 2022-2023 to more than 80 € (e.g., at the onset of the Russia–Ukraine war, the EUA was traded at more than 90 €).

The EU ETS trade network of allowances

Since the EU ETS is the reference marketplace in Europe for entities transferring allowances to meet their compliance needs, with our analysis we aim to first characterize the configuration of this trade network of allowances and its time evolution. The daily configuration (Fig. 1) of the EU ETS in our sample is sparse, with on average about 115 links per day involving only about 129 accounts, and very volatile in time. The network is even sparser during Phase I, becoming more erratic in Phase II while exhibiting more recurrent patterns in Phase III. Interestingly, we observe some regular spikes in both active edges and nodes mainly at the end of April of each year, when accounts have to surrender to government accounts an amount of allowances corresponding to their carbon emissions during the year. It seems therefore that accounts are more actively involved in the trade of allowances when they need to meet compliance requirements. In addition, a few other relevant spikes refer to the months of December, when futures contracts on carbon markets usually expire, and to October 2008 after the collapse of financial markets due to the subprime mortgage crisis. Hence, while the trading of allowances is primarily driven by compliance needs, it is also influenced by instability propagated through capital markets. It is worth reminding that Directive 2014/65/EU of the European Parliament and of the Council of 15 May 2014 classified EUAs as financial instruments.

The fact that the network is sparse is also confirmed by the transitivity coefficient. The probability that the adjacent nodes of a node are connected is very low and is on average about 1%, indicating a low propensity of clustering in the trade network. This quantity appears higher in the early years when fewer accounts actively engaged in transactions. However, as the network expanded and more accounts became active, the transitivity coefficient substantially decreased, reaching more stable values in the latter half of the sample period. This is not surprising given the low attitude of EU ETS liable entities to trade allowances especially in the early stages of the program. For example, most EU ETS liable entities were not or hardly participating in the system during Phase I³⁵ and Phase II¹², with non-regulated entities instead very active.

Interestingly, we also observe that the EU ETS network is typically disassortative in the whole period (mean: − 0.199, st. dev.: 0.114), similarly to several infrastructure networks. During the pilot Phase, the assortativity is very erratic and in some days even positive (mean: − 0.254, st. dev.: 0.132), in Phase II it is much more stable (mean: − 0.235, st. dev.: 0.087), while in Phase III it shows a volatile pattern with several days assuming a positive value although on average the coefficient is still negative (mean: − 0.158, st. dev.: 0.107). The disassortative behavior is often associated with core-periphery structures. Figure 2A shows the cumulative distribution function of the degree for each year separately. A bunch of nodes is involved in hundreds of bilateral transactions with different counterparts. This suggests the presence of a core of very active nodes surrendered by many other nodes that instead are marginally participating in the trade network. Note how on average almost 95% of the nodes have degree values lower than 10, meaning that these entities perform trades with less than 10 counterparts in a year. Also, at least half of the annual observations correspond to nodes with less than 3 counterparts, with more recent years indicating that such percentage increases to almost 90%. Furthermore, Fig. 2B confirms that peripheral nodes rely on transactions with very active counterparts. Very central nodes tend to be connected with nodes that are poorly active in the system, hence on average with nodes with a low degree. Conversely, more peripheral nodes, having few edges, are more likely to trade with counterparts that are instead very active, hence on average with a higher degree. Such negative relationship appears steeper in more recent periods.

Fig. 1

Network statistics. We report the number of nodes (Panel A) and edges (Panel B), the transitivity (Panel C) and the assortativity (Panel D) coefficients. We consider only active accounts performing at least one transaction. Transitivity is computed as the number of triangles and connected triples in the network. Assortativity measures the level of homophily of the network based on the degree of the nodes. The red dashed line signals the end of Phase I (2005–2007), while the blue dashed line signals the end of Phase II (2008–2012).

A disassortative core-periphery network might be more robust to shocks during business-as-usual periods, but more fragile under exceptional scenarios when key nodes are under stress or withdraw from the system⁵⁴, thus affecting the overall stability and functioning of the network. As shown in Figures 1 and 2, the EU ETS network tends to be disassortative but with a high variability in the way these links are formed. Given a certain disassortativity level characterizing the configuration of the network, its variation in time can then be interpreted as a disturbance impacting the usual set of bilateral trades, especially if the coefficient switches its sign. We opt to use the assortativity coefficient as a synthetic indicator to proxy the configuration of the EU ETS on a daily basis (see “Methods” section for details). For instance, the assortativity indicator has been deeply investigated in the literature on systemic risk and financial stability^42,43, where the resilience of a system to contagion and shocks propagation has been related to its tendency to vary the assortativity mixing, and to assess the robustness of interdependent networks^55,56. Additionally, the use of the assortativity coefficient to describe the EU ETS network of trades was introduced in Ref.¹² to study the role of country registries and the influence of the different types of account. Their findings indicate that in the initial stages of the EU ETS, the network exhibited high variability in assortativity. Over time, however, the network evolved into a disassortative configuration, highlighting the increasing influence of voluntarily opened accounts. Our findings complement those in Ref.¹², as we confirm the assortativity configuration in a much more granular specification of the network. Moreover, we extend the assessment to Phase III, showing how the tendency for a disassortative configuration is confirmed even in a period characterized by relevant changes in the policy framework.

Fig. 2

Degree Distribution. Panel A reports the empirical cumulative distribution function (CDF) of the degree. Panel B reports the relationship between the degree and the average nearest neighbor degree (knn). In both cases the degree is expressed in logarithm. Colors refer to different years: recent observations are in yellow, while more distant years are in dark violet.

The role of the energy sector

The EU ETS regulation is applied to firms that significantly contribute to GHG emissions. Power generators contribute for almost half of the emissions¹⁵. Projections for 2030 based on “with existing measures” (WEM) or “with additional measures” (WAM) scenarios point to emissions that are 55% or 59% below the level of 2005 when the EU ETS was launched⁵⁷. The power sector is expected to account for most of the projected reductions through the switch from coal to gas and renewables in the combustion of fuels. Moreover, the energy sector is usually considered at lower risk of carbon leakage and cheaper abatement options exist in this sector than in other industrial sectors^15,58.

Firms in the energy sector receive free allowances conditional on investments for the modernisation, diversification and sustainable transformation of their productions. However, in the first phases of the program the free allowances allocated in aggregate to these firms were typically lower than their verified emissions¹³. In addition, since Phase III power generators have to purchase all their allowances, with exceptions for firms established in three lower-income Member States (namely, Bulgaria, Hungary, and Romania). Firms in the energy sector thus tend to engage in more intensive trading to meet their compliance requirements.^14,18,45.

We define the set of nodes in our sample operating in the power generation and energy related activities by considering the NACE code “D—Electricity, gas, steam and air conditioning supply” of the installations that are linked to the accounts utilized to build the EU ETS network. For simplicity, we refer to them as the energy nodes. Approximately 26% of our accounts refer to the energy subsample. This proportion is comparable to the set of nodes related to carbon leakage sectors (28%), while the percentage of nodes in the other industrial sectors is slightly lower (about 19%). In addition, about 27% of the nodes in our sample are associated with accounts not directly linked to installations. These accounts are managed by individuals, financial intermediaries, or brokerage firms that voluntarily trade allowances but are not subject to EU ETS environmental compliance regulations.

Extant literature has recognized the emergence of different effects of the EU ETS participation across sectors. More in general, the EU ETS is a network typically dominated in terms of transactions by firms belonging to a few sectors, like financial and energy^12,14,35,46. We provide a visual representation of the EU ETS network in Fig. 3. Note how several energy nodes (highlighted in red) are very central within the network. In addition, the network is characterized by a dense subgraph of highly interconnected nodes, predominantly energy-related, surrounded by a cloud of peripheral nodes with fewer connections.

Fig. 3

EU ETS network. The network is built on the entire sample period constructed by aggregating daily transactions performed by the same pair of nodes. Red nodes stand for the Energy sector. Edges have source node color.

Figure 4 shows the average values over time of some network centrality indicators for the energy, the carbon leakage and the other industrial nodes in our sample, hence to liable entities under the EU ETS. Ref.¹⁸ analyzes the nexus between the financial (FP) and environmental (EP) performances of EU ETS firms, demonstrating that the centrality in the network of trades (expressed in terms of In-Degree and Out-Degree) helps explain the nature of the EP-FP relationship. However, they find that for energy firms, these network indicators are not statistically significant, while the impact of environmental results is substantial. They argue that energy participants in the EU ETS may have a distinctive role, dissimilar from that of firms in many other sectors. We find that the energy subsample tends to have more counterparts and to trade larger volumes compared to the other groups, especially in more recent periods. Importantly, energy nodes are more active as purchaser of permits, confirming the fact that they are typically short in allowances⁵⁹. In general, firms whose verified emissions exceed their free allocations in a given year are more likely to participate in the EU ETS market and, conditional on participation, they trade higher volumes of allowances than firms with an opposite balance of allowances¹⁴.

Fig. 4

Centrality indicators. We show the evolution in time of the average Degree (Panel A), In-Degree (Panel B), Out-Degree (Panel C), Strength (Panel D), In-Strength (Panel E) and Out-Strength (Panel F) for the energy (in red), carbon leakage (in orange) and the other (in gray) nodes in our sample (excluding non-liable entities). Values are averaged by year.

Our energy subsample refers to around 2.9k firms managing on average about 3 nodes/accounts each. Among them, the most active entities are RWE, Endesa, CEZ Group, PGE, ENEL, and Uniper that are involved in the transactions of around 3% and 36% of the allowances in the whole and energy samples, respectively. Figure 5 shows the distributions of the average values of purchasing and selling transactions per counterpart in each phase for the three groups. Note how on average the energy subsample purchases volumes higher than the other groups from their counterparts in each phase and especially in more recent periods when stricter constraints on free allocated allowances apply to this sector. For example, the median value of purchased allowances from counterparts of the energy subsample increased from about 10k in Phase I to 13.5k in Phase III, while the mean value doubled from about 163k in Phase I to more than 327k in Phase III. Coherently, in Phase III they reduce their outgoing centralities.

Our results suggest the Coase theorem might not hold, as the trading behaviour of market participants is not independent from the initial allocation of allowances. This has been previously demonstrated for Phase I⁴⁵, for the period 2005–2014¹⁴, and for small power producers¹⁵. This finding is particularly relevant for policymakers in light of significant changes in the EU ETS regulatory framework, specifically the shift from free allocation based on past emission levels or benchmarks to a greater reliance on auctioning during Phase III.

Fig. 5

Mean transaction value per counterpart. We report the distributions of the average amount of purchased (Panel A) and sold (Panel B) allowances from and to counterparts, respectively. Energy, carbon leakage and the other nodes in our sample (excluding non-liable entities) are in color red, orange and gray, respectively. Values are grouped by EU ETS phases and refer to the ratio of In-Strength over In-Degree (Panel A) and Out-Strength over Out-Degree (Panel B). Kruskal–Wallis tests and pairwise comparisons with Wilcoxon rank sum tests and Bonferroni correction indicate that the distributions of the two indicators for the three groups differ in the whole sample and in most of the single phases.

Spillover analysis within the energy-commodity system

The spillover analysis within the energy-commodity system is significant because substantial price variations in energy commodities can influence the optimal energy mix, leading to shifts towards alternative fuels and contributing to the anticipated obsolescence of carbon-intensive production technologies^25,26,60. Exploring interdependencies across energy commodities is crucial for understanding how disturbances propagate through energy supply chains. Despite this relevance, few studies address how such disturbances propagate within energy supply chains comprehensively. Ref.⁵² is a notable exception, as they outline how upstream shocks dominate the system’s connectedness, although downstream shocks are increasingly influential in creating short-term spillovers. However, their analysis is limited to the oil supply chain and does not include other related energy commodities such as gas and coal.

Carbon prices can be influenced by the market dynamics of energy-commodities, generating relevant transmission channels of spillovers. For instance, it has been found that EUA forward prices depend on the price of electricity as well as on the gas-coal difference⁶¹, and that the electricity demand plays a crucial role in the information spillover channel with the carbon market playing as the net information receiver^62,63. Similarly, Refs. ^64,65 find the net receiver behavior of the carbon market against oil and the stock market. In addition, bidirectional non-linear mean spillover effects might emerge between EUA spot and futures prices, while their volatility spillovers result to be vulnerable to financial crises and extreme events⁶⁶.

For these reasons, we propose to examine spillovers within a comprehensive system including the EU ETS network configuration (in terms of Assortativity) with Oil, Natural Gas, Coal and EUA (see “Methods” section for details). Energy commodities not only influence the carbon price but refer to typical economic dimensions affecting the business activities of energy-intensive firms. As outlined in previous sections, energy firms are among the EU ETS liable entities more actively participating in the trading of allowances and more directly involved in carbon abatement targets.

Spillover analysis in the time domain

In what follows, we set $H=10$, VAR order $p=5$ and a rolling window of 250 observations. Additional parameters are considered in the SI, Figures S2–S3. With this analysis, we aim to investigate if shocks hitting energy-commodity markets might affect the overall functioning of the EU ETS.

The results of the spillover analysis are shown in Fig. 6. The substantial change in the regulatory framework occurred between Phase I and Phase II heavily affected the way entities traded allowances (see also Fig. 1). As for the spillover index (Fig. 6A), it reached a value of about 22 at end of December 2007. Then, it fluctuated around 10–20 before declining between the end of Phase II and the beginning of Phase III, thus showing a relatively modest connectedness. By contrast, a sharp increase emerges in 2016. This can be related to some relevant events occurring in the underlying commodity markets. In the period 2014–2016, the oil price plunged due to a growing supply glut, the booming of the US shale oil production and a decrease in the global economic activity, including in oil-exporting economies. For example, the daily spot price of oil reached in January 2016 its minimum in 13 years. This significant decline in oil prices had notable repercussions on the market dynamics of the natural gas, which was strongly linked to oil prices during that period. Moreover, after five years of price decline, in 2016 the coal price showed a sharp increase. Such rebound was probably due to a supply shock related to the tightening of the production in the international coal markets as global demand had decreased since 2014; for example, policy changes in China limited production capacity by reducing the number of working days in Chinese mines, causing a drop in the domestic production. In addition, the Conference of the Parties (COP) held in Paris from 30 November to 11 December 2015 unanimously adopted to keep global warming below $2^\circ {\hbox {C}}$ above pre-industrial levels and continue efforts to limit it to $1.5^\circ {\hbox {C}}$, thus pointing to more investments in the decarbonization of economic activities.

Fig. 6

Spillover Analysis. Panel (A) Shows the Spillover Index with H=10 and VAR(P=5). Panels (B, C, D, E) show the price dynamics of Brent Oil (orange), Natural Gas (blue), Coal (brown) and EUA (green) respectively. Table S1 in the SI shows how in general these market series are leptokurtic, which is typical for financial assets, and have significant serial auto-correlation.

Fig. 7

Net Spillovers. Green colors refer to net transmitter, red color to net importer of spillovers. Net spillovers are constructed subtracting the directional From spillovers from the directional To spillovers. The mean (median; st. dev.) of the net spillovers of EU ETS, Oil, Natural Gas, Coal, EUA are respectively: − 0.037 (− 0.112; 0.926), 0.022 (0.002; 0.826), 0.195 (0.169; 0.648), − 0.045 (− 0.161; 0.826), − 0.135 (− 0.233; 1.006).

Once decomposed the system’s connectedness into the net contribution of each series to spillovers, we first note (Fig. 7) that the structure of the EU ETS (Panel A) is more often that of a net receiver of spillovers, although presenting some short phases in which it plays as net transmitter. On average its net spillover is about − 0.037. The EUA tends to import spillovers from the rest of the system, with exceptions in particular at the end of Phase I, when it reaches its maximum, and between Phase II and Phase III. Changes in energy commodity prices can affect carbon emissions through both income or substitution effects, thereby influencing EUA prices and the way firms trade allowances. In particular, Natural Gas shows long phases in which it plays as net transmitter, while Oil and Coal tend to frequently alternate their net position. Figure S1 in SI shows how the EU ETS is on average a net importer of spillovers especially from Natural Gas (mean: − 0.073) and Coal (mean: − 0.016), it is able to net transmit spillovers to EUA (mean: 0.047), while the bilateral relationship with Oil is almost null (mean: 0.005).

These results are coherent with empirical works analyzing similar energy systems. For instance, it has been observed that in a system composed by EUA, WTI oil, Brent oil and natural gas markets, the WTI oil market transmits the strongest spillover effect to the system and the spillover effect of natural gas to carbon market is substantial, with some major policy changes and events generating important variations in the connectedness of the system⁶⁴. Conversely, coal typically plays as an information recipient in systems with oil, natural gas, clean energy, and EUA^67,68. Interestingly, our results highlight the relevance of the transition from Phase I to Phase II, which led to a substantial increase in the connectedness of the system, with the EUA emerging as the main net transmitter during that period. Furthermore, we offer a novel representation of the connectedness within the energy system, illustrating how the EU ETS trade network generally acts as a net receiver of spillovers, particularly from natural gas.

Spillover analysis in the frequency domain

Figure 8 shows the evolution of the spillover index decomposed by frequency bands. We note how the high-frequency connectedness is the most relevant component in all dates, with medium- and long-term spillovers which account only for a small portion of the connectedness.

Fig. 8

Dynamic analysis in the frequency domain. Decomposition of the spillover index for the returns series. We apply VAR(5) and H=100 with a rolling window of 250 days. Frequency bands refer to 1–4 days (dashed line), 5–14 days (dotted line) and >15 days (solid line).

These relationships are largely confirmed even during Phase III, when the MSR significantly reduced the surplus of allowances limiting the “waterbed” effect and contributing to the raise of the EUAs price^69,70, thus strengthening the long-run price signaling mechanism of carbon markets. Table S2 in the SI reports the static analysis. The generalized variance decomposition indicates that short-term frequency is the most contributing component of the spillovers, observing also a decreasing pattern when moving to medium- and long-term impacts. This is an interesting feature of the system since the European carbon market and the configuration of the EU ETS trade network of allowances may be affected by cross shocks within the energy-commodity system, but such cross-effects are quickly absorbed. The literature underscores that spillovers are mainly generated by short-term factors⁷¹. We support this view by demonstrating that energy commodity price dynamics react swiftly to unexpected events, confirming that available information is rapidly incorporated into these markets⁷².

Quantile analysis

Finally, we investigate the behaviour of the connectedness in the tails of the conditional distribution using the quantile approach²⁷. Figure 9 shows the results for the total dynamic connectedness. We confirm very low values of connectedness in the interval 2008–2009 (as shown also in Fig. 6A), but here we observe how also a huge portion of the conditional distribution indicates low connectedness in that period, thus even for extreme observations in the tails. Conversely, the beginning of 2016 corresponds to a higher and more uniform connectedness across quantiles, suggesting that connectedness that prevails at the conditional mean can be generalized to the entire distribution, thus supporting the fact that it refers to a very critical period in terms of spillovers propagation.

Note however that in general the connectedness is very strong for highly negative or positive changes in the series, with impacts appearing almost symmetrical. These levels of connectedness in the tails are also generally much higher than those observed at the conditional mean, thus suggesting stronger spillovers during turbulent periods. Interestingly, the time variation observed between different quantiles of the conditional distribution still confirms higher levels of connectedness during the third phase.

Fig. 9

Dynamic total connectedness. We employ a 250-days rolling-window QVAR model with lag length of order 1 and H=10. Darker shades correspond to higher levels of connectedness.

Figure 10 shows the net directional spillovers. It is noteworthy how both the magnitude and the sign of the net directional connectedness observed in the central part of the conditional distribution can differ significantly from those at the tails. This variation highlights the importance of considering different quantiles of the distribution when analyzing spillover effects, as extreme values may reveal distinct dynamics compared to the mean tendencies. However, some relevant results confirm the evidences discussed in Fig. 7. For instance, Natural Gas is often a net transmitter of shocks also when observing the tails of the conditional distribution. From 2016 on-wards this series is consistently a net transmitter of connectedness, with net effects which are similar across the quantiles of the distribution. A somehow similar role is played by Oil, but at the beginning of the sample period.

More generally, our analysis reveals that the average spillover intensity is stronger in the tails than at the conditional mean or median and, more importantly, that the role of net transmitter or net receiver of spillovers might vary depending on the selected quantile of the conditional distribution. Interestingly, we confirm that the structure of the EU ETS trade network of allowances is typically that of a net importer of spillovers even when focusing on the tails of the conditional distribution. However, in the tails the magnitude of the negative net spillovers is even higher.

Fig. 10

Net directional effects in the quantiles. Panel (A) Shows the EU ETS configuration, while Oil, Natural Gas, Coal and EUA are reported in Panels (B, C, D, E), respectively. Green colors refer to net transmitter, red color to net importer of spillovers.

Portfolio strategy

In this work we opt for the direct use of the dynamics of the spillover index to study the market behavior of stocks and build portfolio strategies. The dynamics of the spillover index signals time variations in the connectedness of the underlying series considered in the system, with increasing values of it indicating that spillovers are more likely to propagate to the other dimensions of the system. However, how spillovers propagation relates to portfolio diversification is not obvious since an increased connectedness does not necessarily imply higher average volatility if stocks are connected in a complementary way⁵².

We propose interpreting the spillover index as a factor that can help explain the market behavior of stocks related to the underlying system. We label it as the Spillover Propagation Risk (SPR) factor. We thus analyze a reduced-form of asset pricing that is not derived from assumptions about the beliefs or preferences of the investors prescribing which dimensions should appear in the stochastic discount factor model. Given the nature of the system analyzed in previous sections, we illustrate the functioning of the proposed SPR factor analyzing a sample of stocks broadly belonging to the energy sector in Europe. The construction of our spillover index refers in fact to Oil, Natural Gas, Coal and Carbon market series, which are reasonable dimensions influencing business activities of energy-related firms. Importantly, these firms may behave differently to spillovers in the system due to aspects such as their positioning in the value chain, different adopted technological solutions or production efficiency performances. Specifically, we collect market data of 70 stocks of firms engaged in a variety of activities and services ranging from production and distribution of energy to related operations such as research, logistics, consulting, storage, and exploration (see “Methods” section for details). Among them, some firms are more focused on traditional business lines while others are more involved in research and renewable energy resources. Distress impacting the commodity dimensions considered in the spillover analysis may thus have different consequences on the business activities of these firms and, in turn, on their market performances.

We take into account that other sources of risk influence stocks’ market dynamics by relying on the five factor model²⁸ to which we add the SPR factor. Specifically, we estimate separately for each stock i a (5+1) factor model using a rolling window of 250 days. In formula: $R_{it} - RF_{t} = \alpha _{i} + \beta ^{MKT}_{i}MKT_{t} + \beta ^{SMB}_{i}SMB_{t} + \beta ^{HML}_{i}HML_{t}+ \beta ^{RMW}_{i}RMW_{t}+ \beta ^{CMA}_{i}CMA_{t} + \beta ^{SPR}_{i}SPR_{t} + \epsilon _{it}$, where $\hbox {R}_{{it}}$ is the natural logarithm return of stock i at day t, MKT, SMB, HML, RMW and CMA are the five factors retrieved along with the risk free rate RF from Kenneth R. French’s website, SPR is the dynamic spillover index, and $\epsilon _{it}$ is the idiosyncratic component of the asset’s excess return. More specifically, MKT is the excess return on the market. SMB (Small Minus Big) is the differential of the average return of a portfolio of small stocks minus the average return of a portfolio of large stocks; similarly, HML, RMW and CMA are differential portfolios returns computed for High Minus Low, Robust Minus Weak, and Conservative Minus Aggressive portfolios of stocks, respectively. These factors are widely applied in finance to explain cross-sectional heterogeneity of stocks’ market performances.

Importantly, the SPR factor is constructed using series (EUA, Oil, Natural Gas, Coal and EU ETS) that are different from those used to build the portfolios (70 stocks of energy-related firms). Two stocks with different signs of beta with respect to SPR tend to exhibit opposite market dynamics when the SPR varies. For instance, on average, a stock with a high and positive SPR beta will see an upward market dynamic when the connectedness of the system increases. Conversely, a stock with a high and negative SPR beta will experience a decline in market performance when the spillover index rises. Indeed, a stock with an SPR beta pointing to zero means that on average its excess market returns are unrelated to the dynamics of the spillover index. Factor loadings of the SPR are reported in Fig. 11. Note how over the whole period (Panel A) the estimated betas are almost evenly distributed around zero, although with high heterogeneity between and within stocks (Panel B).

Fig. 11

Distribution of the SPR beta. Panel (A) displays the density over the whole period aggregating across stocks; Panel (B) is the distribution in time for each stock separately.

We propose to exploit such factor loadings to build portfolios and map their performances in time with respect to the dynamics of the spillover index. In so doing, we can for instance build portfolios that have a directional exposure to the dynamics of the spillover index of the system or even neutralize such exposure by appropriately selecting the weights in the portfolio allocation. To show the functioning of illustrative strategies, we rank stocks in terms of the SPR betas and then we select the top and bottom stocks based on the estimated coefficients. Specifically, we consider subsamples corresponding to the first/last tertiles or deciles of the distribution of the SPR beta. We label them Top-ter, Top-dec, Bottom-ter, and Bottom-dec, where top groups refer to stocks with higher and positive SPR betas, bottom groups refer to stocks with higher and negative SPR betas, while ter and dec are abbreviations for tertiles and deciles of the distribution respectively. All remaining stocks are allocated to the Medium group. In addition, we consider a simple 1/N strategy equally investing in all 70 stocks. Portfolios rebalancing is performed on a daily basis using the updated betas available on a certain day to rank stocks and invest in the following period. Figure S4 in the SI shows the distributions of each factor withing each group. We notice that the betas with respect to the other factors are very comparable among groups, meaning that heterogeneity in terms of market performances between groups is not expected to be systematically related to different exposures to these risk factors.

We then propose to build zero-cost investment strategies which buy a certain Top or Bottom group and sell the other group. Hence, each strategy is long in a group, e.g. buying Top-ter investing a certain amount of money, and is short in the opposite group, e.g. selling an equivalent amount of money of Bottom-ter. As a result, the overall invested amount of money is zero. We construct the following four strategies: TB-ter, TB-dec, BT-ter, BT-dec, where the first letter (T or B) refers to the group that is bought and the second letter indicates the group that is sold. For instance, TB-ter buys Top-ter and sells Bottom-ter. Such strategies amplify the exposure to the SPR factor, representing therefore a way to implement an active portfolio allocation based on whether the investor feels bullish or bearish with respect to the impact of the dynamics of the connectedness on the market performances of these stocks.

Figure 12 shows the corresponding cumulative performances. Note how those zero-cost investment strategies that are long in the Bottom groups (in green colors) obtain very high performances because over the whole sample period the spillover index (in black color) shows more and longer phases of downturn. Hence, having a long position in stocks negatively correlated with the index and a short position in positively correlated stocks enhances the overall performances of these zero-cost investment strategies. By contrast, the opposite cases (in red colors) exhibit a deterioration of the portfolio performances. Additionally, the results of the deciles groups appear more volatile than those of the tertiles groups, suggesting diversification benefits at portfolio level that reduce dispersion. Finally, the Medium groups (in blue colors) and the 1/N strategy (in gray) are in a much stricter, and similar, range of performances. The 1/N strategy can also be considered as a simple benchmark for a passive allocation strategy, representing the overall market dynamics of the stocks in the sample under analysis.

Fig. 12

Zero-cost cumulative returns. TB-ter, TB-dec, BT-ter, BT-dec are shown in red, orange, green and light green, respectively. The first letter (T or B) refers to the group that is bought and the second letter indicates the group that is sold. The 1/N strategy is in gray, while Medium-ter and Medium-dec are in blue and cyan respectively. The spillover index is in black color and is scaled in terms of its first observation in the period. To ease visual comparability, the zero-cost investment strategies are summed to an initial value of 1.

These findings indicate that factor investing based on SPR can be implemented to build portfolio strategies positively or negatively related with the dynamic of the spillover index, also being potentially neutral to such factor exposure. Table 1 reports summary statistics for each strategy. Top and Bottom groups have opposite average SPR betas, confirming different exposures to the dynamic of the spillover index. From a financial perspective, Bottom strategies obtain on average positive daily returns, while the opposite is observed for Top strategies. Both groups have similar standard deviations of market returns. The corresponding Sharpe Ratios (SR) indicate that Bottom strategies generate more favourable risk-adjusted performances. Medium groups have slightly positive financial performances and SPR betas, very similar to those obtained by strategy 1/N. By contrast, zero-cost investment strategies obviously have much more pronounced (and opposite) average SPR betas and get also more extreme risk-adjusted results. Given the dynamic evolution of the spillover index, TB strategies are penalized and obtain negative results, while BT strategies result the best performing ones in the analysis. Overall, these results confirm the visual inspection of Fig. 12.

Table 1 Market performances of the portfolio strategies.

Finally, to further evaluate the proposed strategies, we calculate other measures of performance that are adjusted for risk. In particular, we consider the Modified Cornish–Fisher Value at Risk (VaR), the Expected Shortfall (ES) and the Ulcer index (Martin Ratio) as measures of risk in the modified version of the Sharpe Ratio. We also include the gain–loss ratio (Omega Ratio), the ratio of the frequency of negative over positive returns (DRatio), the ratio of the annualized return over the absolute value of the maximum drawdown of an investment (Calmar ratio), and the mean value of the drawdowns over the entire sample period (Pain Index). In general, these indicators are typically applied in portfolio management when a simple measure of risk is not considered appropriate. For instance, when returns are not symmetrically distributed, the Omega ratio can be preferred because it includes higher moments of the returns distribution, while the maximum drawdown is the maximum loss accrued in a period with respect to the peak value recorded in that period, thus representing the maximum loss suffered due to the trading activity. These measure clearly indicate that Top and TB strategies perform poorly, while Bottom and BT strategies are among the best performing ones.

Our investment strategies based on the SPR factor do not consider features key to their implementation, such as transaction costs, liquidity, investability and capacity, while the estimation of the corresponding factor loadings has limitations, such as the choice of an arbitrary time window of 250 days or factor contamination. These are all aspects that can be easily addressed by elaborating on the proposed illustrative strategies. Optimization procedures can be also applied to select portfolio weights in line with investor’s preferred exposures to SPR.

Our approach aims to intuitively demonstrate that, once controlled for common risk factors, firms within a specific system, such as the European energy system, may respond differently to the dynamics of the spillover index. This heterogeneity can be leveraged to construct portfolios that vary in their vulnerability to the system’s connectedness. By exploiting these differences, investors can build portfolios that either hedge or amplify the impact of spillovers, depending on their risk appetite. This is relevant since high connectedness does not mean that the market dynamics of the underlying series in the system are necessarily positively correlated. If the spillover index increases, then cross shocks are more substantial, but this does not imply that portfolio diversification would necessarily deteriorate⁵². Hence, the spillover analysis, although relevant to understand shocks transmission in a system, may result difficult to interpret and exploit for portfolio purposes. Since the spillover index reflects how strong is the extent of connectedness in a system, investors may seek to reduce, increase or even neutralize their portfolio exposures to such dynamics depending on how they expect this connectedness will affect their assets under management. For instance, a more connected system indicates easier shocks propagation and synchronization⁷³; during period of market distress, this might be interpreted as a source of uncertainty by long investors while being appealing for short investors. Instead, diversification with respect to the spillover index can be implemented using strategies such as long––short positions based on the tails of the SPR beta distribution. Alternatively, portfolio optimization can be employed to neutralize the exposure to the spillover index by constructing a strategy with a synthetic aggregate SPR beta of zero. Our proposed approach aims to balance the portfolio’s exposure to spillovers, thereby reducing the overall risk and uncertainty associated with fluctuations in the connectedness of a system of EU ETS-related energy commodities.

Discussion

Despite the extensive literature, there remains a lack of clear understanding regarding the impact of policy packages designed at national and international levels to foster the transition toward sustainability and achieve net-zero GHG emissions targets^38,74,75. Importantly, it is still unclear whether different regulatory frameworks create positive synergies that strengthen the credibility of single instruments, or if they instead overlap, leading to conflicts in their interaction and weakening the overall environmental impact^76,77.

Carbon pricing is a critical component of the EU ETS as it helps minimize the risk of adverse economic impacts, stimulates innovation, and generates revenues that can support participants adversely affected by the system³⁸. Initially most EUAs were allocated for free⁵³, but the proportion of auctioned allowances has significantly increased over time, reaching 57% of total EUAs between 2013 and 2020. Revenues from these auctions accrue to Member States’ national budgets, with at least 50% of these funds required to be used for climate- and energy-related actions. For example, total auctioning revenues amounted to 38.8 € billion in 2022, of which 29.7 € billion went directly to EU Member States, 5.4€ billion to the Modernisation Fund and 3.2€ billion to the Innovation Fund. In addition, due to the large surplus of allowances transferred from Phase II to Phase III, the EU decided to implement a “back-loading” process postponing the auctioning of 900 million allowances to the end of Phase III.

Robust EUAs prices provide therefore a relevant economic rationale and incentive for the promotion of investments in cleaner and low-carbon intensive technologies⁷⁸. Consequently, a well-functioning market of allowances is crucial for the EU ETS to operate effectively and achieve its environmental goals^22,79,80. However, the very low market prices observed in the early years of the program led to the proposal of a price floor to correct potential underlying distortions and design flaws. Additionally, the potential abundance of allowances supported the implementation of a market stability reserve (MSR) to adjust the number of auctioned allowances based on the surplus^81,82. In the short term, the surplus of allowances can in fact undermine the orderly functioning of the carbon market, while in the longer term it may hinder the ability of the system to meet more stringent carbon reduction targets in a cost-effective way. Low carbon prices provide therefore a weaker incentive to reduce emissions, with the creation of the MSR as an example of a policy intervention aimed at supporting carbon market dynamics and increasing the resilience of the EU ETS in the face of future imbalances. Moreover, the MSR is designed to enhance the system’s resilience to major shocks and foster synergies with other climate policies implemented by individual member states, such as renewable energy (RES) expansion measures or coal phase-outs. However, initiatives like the coal phase-out would have minimal impact on environmental targets if the freed-up emission allowances were used elsewhere (“waterbed effect”). These aspects have sparked significant debate about the functioning and evolution of the EU ETS as an efficient mechanism for reducing carbon emissions and addressing climate change issues^{53,69,82,83,84}.

The EU ETS is unlikely to achieve efficiency if many participants are inactive and reluctant to trade allowances during phases of intense information arrival⁴⁶. For example, trading behaviour has been found to relate to the size, location, sector affiliation, number of installations, productivity levels and the difference between free allocations and verified emissions^14,18,45. In addition, transaction costs have been identified as a significant driver behind the limited trading activity of smaller entities. These smaller firms often have less trading experience and typically perform trades indirectly through third parties, which can further exacerbate the impact of transaction costs. Our analysis shows that the EU ETS trade network of allowances exhibits a core-periphery structure, with a group of highly active nodes dominating the trading of allowances. In particular, we contribute to the understanding of this heterogeneous participation by demonstrating how firms involved in energy-related business activities play a substantial role in the functioning of the trade network. This insight underscores the significance of energy firms in shaping the dynamics and efficiency of the EU ETS trading system. This is an interesting result because these firms are typically short in allowances and at a low risk of carbon leakage.

A disassortative core-periphery network might be more robust to shocks during business-as-usual periods, but it becomes more fragile under exceptional scenarios when key nodes are under stress or withdraw from the system. Our study identifies a subset of nodes whose variations in the trading patterns are more likely to deeply influence the overall functioning of the system. These nodes operate in energy-related business activities and, therefore, might be affected by shocks occurring in energy commodity markets. The spillover analysis within the energy-commodity system can provide insights into how price variations in energy commodities influence the optimal energy mix of production activities and, in turn, trading patterns. For instance, Ref.⁸⁵ finds that oil supply and demand shocks have a positive effect on EUAs returns, whereas oil risk shocks have a negative effect. Additionally, oil shocks tend to be more pronounced under bearish and normal market conditions. More in general, substantial energy price variations can stimulate the diffusion of alternative fuels and contribute to the anticipated obsolescence of carbon-intensive production technologies^25,26,60. A relevant example is the outbreak of the Russia–Ukraine war, which caused wholesale electricity prices in European markets to reach several record highs in 2022 causing severe production imbalances⁸⁶. The European Power Benchmark averaged €230/MWh, a 121% increase compared to 2021, with an unprecedented peak in August. This led to significant energy price increases and uncertainty regarding supply. Consequently, the share of renewables in the energy mix rose to 39%, and the installed renewable capacity grew by 16% in 2022 compared to the previous year. Interestingly, despite EUAs prices increasing to €80/tCO2 in 2022, these levels were still insufficient to support coal-to-gas fuel switching in power generation due to exceptionally high gas prices throughout most of the year. Such market dynamics are crucial for evaluating firms’ commitment to improving their environmental performance, as very low EUAs prices can deter investment in cleaner technologies aimed at reducing carbon emissions.

The EU ETS is the cornerstone of the European policy to enhance carbon abatement. Its correct functioning should reflect the trade-off between investing in cleaner technologies and more efficient production processes versus meeting compliance targets through the trading of allowances. Besides the strategic interactions between participants with surpluses and deficits of allowances, our spillovers analysis indicates that shocks in the energy commodity markets can propagate and impact the configuration of the EU ETS trade network. This finding suggests that dimensions related to commodity markets, beyond just carbon prices, can significantly influence how EU ETS firms exchange allowances. Consequently, fluctuations in energy commodity prices and market dynamics play a crucial role in shaping trading behaviors and network configuration within the EU ETS. This is a relevant aspect because transmission channels, such as those described in this paper with energy commodities, may extend beyond the perimeter of the EU ETS policy application but affect how allowances are exchanged. If the EU ETS is an efficient solution for cost-effective carbon reduction, its trade network of EUAs should primarily reflect the differential in the environmental performances of firms and be influenced by factors such as allowances allocation or carbon price dynamics. However, there are still some problems in the trading operation process, such as traceability mechanism, low degree of information sharing, market opacity, sectoral biases, and trade distortions. Future policy amendments might consider for instance the impact of distributed ledger technologies on climate policies to provide swift and automatic settlement. This could help overcome issues that affect the fairness and security of the trading mechanism and enhance the operating efficiency of the EU ETS⁸⁷. It is worth recalling that EUAs are perfectly homogeneous, dematerialized goods that are exchanged electronically, without any transportation or delivery costs.

From the policymaker perspective, we show that sources of instability not directly related to the EU ETS policy, e.g. energy commodities market dynamics, can affect the businesses of EU ETS firms and influence how these firms trade allowances, potentially undermining the effectiveness of the EU ETS as a cap-and-trade solution to promote carbon abatement in a cost-efficient manner. Ref.¹⁶ exploits the trade network to evaluate how the network configuration affects price formation. Interestingly, they detect structural causes leading to the emergence of information asymmetries in the carbon market. Our analysis confirms that relevant events occurring within the EU ETS, such as the switch from the pilot Phase to Phase II, deeply influence the connectedness of the system, with the configuration of the EU ETS and the EUAs market dynamics becoming more central in the propagation of spillovers.

Impact investing has gained momentum in the research agenda^6,29,30,48. Our illustrative strategies offer examples of how to manage sources of distress and uncertainty in the EU ETS resulting from spillovers. The dynamic spillover index, which reflects the time evolution of the connectedness within the energy system, can be used to implement these strategies. By monitoring changes in the spillover index, investors can adapt their exposures to mitigate risks and optimize their trading strategies in response to evolving market conditions. Investors can thus construct portfolios by investing in stocks of firms operating within this system with varying degrees of exposure to the system’s connectedness, which is interpreted as the risk factor representing the likelihood of spillovers propagation in that specific system. Strong regulatory actions are essential to mitigate the potentially adverse consequences of climate change, which is primarily driven by the combustion of fossil fuels. However, uncertainty regarding the implementation of these climate policies will particularly impact firms with carbon-intensive business models⁴⁷. An unexpected increase in climate change concerns boosts the market performance of firms with better environmental outcomes, while the prices of brown firms decrease⁴⁸. In our framework, the SPR betas guide the stocks selection, enabling investors to adjust their positions based on their vulnerability to the system’s connectedness. From the perspective of the EU ETS participants, we show how our approach can be exploited to obtain better risk-adjusted performances and manage the risk of spillovers from a system of EU ETS-related energy commodities.

Conclusion

Despite the extensive research on various aspects of the EU ETS, hardly any attention has been paid to the structure governing the EU ETS trade network of allowances. Existing literature has highlighted several points: emission levels affect the trading performance of emitting firms¹⁴, some liable entities were only marginally involved in the trading system^12,35, firms with higher deficits in allowances generally make more purchases to meet compliance requirements^58,88. Additionally, while some studies have examined the interlinkages between carbon markets and other asset classes, none have specifically analyzed how market dynamics of energy commodities affect the trading behavior of EU ETS entities.

Our paper leverages a granular dataset comprising over 32k accounts that performed approximately 720k trades, involving more than 127 billion of allowances from 2005 to 2020, to build the EU ETS trade network of allowances on a daily basis. To the best of our knowledge, this represents the most granular and temporally extensive sample used to study trades of allowances within the EU ETS. We contribute to the existing literature in four main ways. First, we demonstrate that the EU ETS trade network of allowances exhibits a disassortative configuration, resembling a core-periphery structure. This feature is consistent across all the first three Phases of the EU ETS. This finding is crucial for effective policy implementation, as the presence of highly active nodes significantly influences the functioning of the network and its resilience to distress. Second, we identify that the group of highly active nodes dominating the trading of allowances largely comprises firms involved in energy-related business activities. These firms are typically short in allowances and have a low risk of carbon leakage, highlighting their significant role in the trading network, particularly in meeting compliance requirements. Third, we show that energy commodity spillovers are substantial and significantly impact how EU ETS entities transfer allowances. By employing spillover analysis in both time and frequency domains, we find that energy market episodes generate more pronounced spillovers. Our main findings remain robust across various checks and are consistent even when analyzing the tails of the distribution within a quantile framework, which we introduce to examine distressed periods. Finally, since sources of instability not directly related to the EU ETS policy, such as the dynamics in the energy commodities markets, can influence the trading behavior of EU ETS firms, potentially undermining the effectiveness of the EU ETS as a cost-efficient cap-and-trade solution for promoting carbon abatement, we propose an illustrative factor investing approach to manage these exposures. This involves managing exposures to the energy system’s connectedness and the propagation of spillovers through portfolio allocation. We show that by incorporating the spillover dynamics into portfolio decisions, investors can manage the associated risks and optimize their investment strategies in response to the connectedness of the EU ETS-related energy system.

The EUAs price serves as a reference value to be compared with its theoretical optimum, i.e. is the marginal abatement cost. Larger differences in marginal costs across firms create greater opportunities for gains from trade, thereby contributing to lower total welfare costs. However, the EUAs price might not be driven solely by variables related to marginal abatement costs. Instead, the main determinants of carbon prices can include external factors such as fuel prices and weather shocks²². The spillover analysis computed on a daily basis provides a system-wide tool to monitor the transmission channels of shocks from energy markets. From a policy-maker’s perspective, the computation of the spillover index can be employed to scrutinize the trade network of allowances and its variation from business-as-usual days. For example, the bilateral net spillovers can be exploited to assess which energy commodity dimension is more likely to transmit spillovers to the EU ETS configuration, while an increasing value of the spillover index can inform the participants that cross-shocks are more likely to propagate. Furthermore, incorporating a network-like assessment into the regulatory risk dashboard can help identify, at the micro level, the nodes/firms that are either more vulnerable to distress or have a greater influence on the configuration of the trade network.

We recognize that this study has limitations. We opt for a simple network representation to uncover the main features of the trade system. Future research could benefit from employing more sophisticated network centrality measures or a multi-layer network representation to better distinguish the roles of participating entities and the significance of their links. Furthermore, while we focus on an energy system that includes the most significant commodities affecting EUAs, other economic dimensions, such as those related to manufacturing, may also influence EU ETS firms’ business activities and therefore the way they operate in they trade network of allowances. Hence, the spillover analysis could be extended to cover additional financial time series, possibly disentangled by geographical area to capture heterogeneity across countries. Additionally, given the variability of energy markets, future studies could incorporate spillover analysis relative to market return volatility, estimated using methods such as GARCH-like models.

Methods

Data

We considered internal and external transactions occurring in the EU ETS, corresponding to transaction code types 10-0, 3-21 and 3-0. This means that flows of allowances related to compliance (e.g., allocation, surrendering, cancellation) purposes were excluded from the analysis. We also removed those transactions for which the identifier of one of the counterparts in the bilateral trade is missing. EU ETS data were retrieved from the EUTL transaction log website: https://ec.europa.eu/clima/ets/. Additional information was retrieved from the “EUETS.INFO” (https://www.euets.info/) database and the “EU ETS (JRC-EU ETS-FIRMS)” (https://data.jrc.ec.europa.eu/dataset/bdd1b71f-1bc8-4e65-8123-bbdd8981f116) database. The sample begins at date 2005-02-07 and ends at 2020-04-30, thus covering entirely the first three phases of the EU ETS. In accordance with Annex XIV (4) of Regulation 389/2013, there is an embargo period of three years in the transaction data.

For the carbon market of European allowances (EUA), we considered the secondary market price of futures contracts (contract CFI2Zc1). Similarly, we also retrieved the ICE Europe Brent Crude (LCOc1), the TRPC Natural Gas Title Transfer Facility (TRNLTTFMc1) and the ICE Europe Rotterdam Coal (ATWMc1) data. We considered futures contracts since they are less affected by short-run noise and are more actively traded than spot contracts. For comparability reasons, we restricted our sample to the interval from 2006-07-18 to 2020-04-30. These series were collected through Thompson Reuters Eikon (https://eikon.refinitiv.com/).

We also collected daily market data of 70 stocks for the portfolio exercise. This sample is composed by the following firms: A2A SPA, Acea SPA, Albioma SA, Ascopiave SPA, BKW AG, Bonheur ASA, Centrica PLC, Cez AS, CropEnergies AG, Dno ASA, Drax Group PLC, E.ON SE, EDP Energias de Portugal SA, Electricite de France SA, Elia Group SA, Enagas SA, Endesa SA, Enel SPA, Engie SA, Eni SPA, Equinor ASA, ERG SPA, Etablissements Maurel et Prom SA, Euronav NV, EVN AG, Exmar NV, Fortum OYJ, Fugro NV, Galp Energia SGPS SA, Grupa Lotos SA, Harbour Energy PLC, Hera SPA, Hunting PLC, Iberdrola SA, Iren SPA, Koninklijke Vopak NV, MOL Magyar Olaj-es Gazipari NYRT, National Grid PLC, Naturgy Energy Group SA, Nel ASA, Neste OYJ, OMV AG, Orron Energy AB, Polski Koncern Naftowy Orlen SA, Polskie Gornictwo Naftowe i Gazownictwo SA, Red Electrica Corporacion SA, Repsol SA, Rubis SCA, RWE AG, Saipem SPA, Saras SPA, SBM Offshore NV, Schoeller Bleckmann Oilfield Equipment AG, Severn Trent PLC, Shell PLC, Siemens Gamesa Renewable Energy SA, Snam SPA, SSE PLC, Tecnicas Reunidas SA, Tenaris SA, Terna Rete Elettrica Nazionale SPA, Tethys Oil AB, TGS ASA, TotalEnergies SE, Tullow Oil PLC, Vallourec SA, Veolia Environnement SA, Verbio Vereinigte Bioenergie AG, Verbund AG, Vestas Wind Systems A/S.

Finally, to estimate the factor model we used the European daily 5 factors available from Kenneth R. French’s website: https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

Network representation of the EU ETS

We represent the network of trades of allowances as a directed graph $G = (V, E)$, where V is the set of nodes representing the accounts operating in the EU ETS and E is the set of oriented edges which stand for the transferred allowances between pairs of nodes. This object can be conveniently synthesized by an adjacency matrix W whose element $w_{ij}$ stands for the flow of allowances from a source node i to a target node j, with $W_{ij} = 0$ if there is no trade from i to j. The EU ETS network is directed, meaning that the matrix W does not necessarily have symmetric edges. The number of edges of each node represents its degree, with high values of it indicating that the node is involved in transactions with many different counterparts. Nodes very central in terms of degree trade with several other nodes in the network, either in terms of purchaser or seller of allowances, and are therefore more likely to influence the functioning of this trade network than nodes that instead perform only a few trades, the latter having therefore a low value of degree.

In our analysis, we opt to synthesize such network configuration by using the Assortativity coefficient. It measures the propensity of the nodes to connect to similar counterparts, that is, to other nodes in the network having similar degree. The assortativity coefficient corresponds to the Pearson’s correlation of the degrees of the nodes in each connected pair and it ranges between $+1$ and $-1$. Assortative mixing patterns occur when the coefficient assumes positive values pointing to $+1$, while the network is said to be disassortative when the coefficient is negative and pointing to $-1$. For example, if the network has a high and positive assortativity coefficient, it means that nodes that are more active in the trading of allowances are more likely to be connected to other nodes that are also very active in the trade network. Instead, in a disassortative network low degree nodes tend to have more connections with very central nodes. We label the Assortativity series as EU ETS.

Spillovers in the time domain

The measurement of the spillover index relies on the approach proposed by Refs.¹⁹ (hereinafter DY) to assess the contributions of shocks to variables to the forecast error variances. Such connectedness is built on the variance decomposition of a P-th order VAR approximating model of a covariance stationary process $Y_{t}=(Y_{1,t},Y_{2,t},\ldots ,Y_{K,t})^{'}$ of K endogenous variables:

$$\begin{aligned} Y_{t} = \sum _{i=1}^{P} \Phi _{i}Y_{t-i} + \epsilon _{t} = \Phi (L)Y_{t} + \epsilon _{t} \end{aligned}$$

(1)

where $\Phi (L)$ is a ($K\times K$) P-th matrix of polynomials in the lag operator L and $\epsilon _{t} \sim (0,\Sigma )$ stands for a vector of disturbances whose covariance $\Sigma$ is possibly non-diagonal. The time index is $t=1,2,\ldots ,T$ and the variable index is $k=1,2,\ldots ,K$.

Each variable is thus regressed on P lags of both the variable itself and the other variables in the system. The moving average representation with infinite order, VMA($\infty$), of the VAR system is $Y_{t} = \sum _{j=0}^{\infty } \Psi _{j}\epsilon _{t-j}$, where coefficient matrices $\Psi _{j}$ are recursively defined as $\Psi _{j} = \Phi _{1}\Psi _{j-1} + \Phi _{2}\Psi _{j-2} + \ldots + \Phi _{p}\Psi _{j-p}$ with $\Psi _{0}=I$ and $\Psi _{j}=0$ for $j<0$.

Following Refs.^89,90 (hereinafter KPPS) a generalized vector autoregressive framework is employed to compute variance decompositions that are invariant to the ordering of the variables in the system. KPPS H-step-ahead forecast error variance decomposition is thus obtained as:

$$\begin{aligned} \theta _{ij}(H) = \frac{\sigma _{jj}^{-1}\sum _{h=0}^{H-1}(e_{i}^{'}\Psi _{h}\Sigma e_{j})^{2}}{\sum _{h=0}^{H-1}(e_{i}^{'}\Psi _{h}\Sigma \Psi _{h}^{'}e_{i})} \end{aligned}$$

(2)

where $\Sigma$ is the covariance matrix of $\epsilon$, $\sigma _{jj}$ is the j-th diagonal element of $\Sigma$ and $e_{j}$ is a selection vector equal to one for the j-th element and zeros otherwise. The main diagonal elements of the ($K\times K$) matrix $\theta (H)$ represent the own contribution of shocks to each variable i to its own forecast error variance, while the off-diagonal elements refer to the cross contributions (spillovers) of the other variables j to the forecast error variance of i.

The own and cross-variable variance contributions do not necessarily sum to one (in general, $\sum _{j=1}^{K}\theta _{ij}(H) \ne 1$). Based on DY, each entry of the variance decomposition matrix is thus normalized by its row sum as follows:

$$\begin{aligned} \tilde{\theta }_{ij}(H) = \frac{\theta _{ij}(H)}{\sum _{j=1}^{K}\theta _{ij}(H)} \end{aligned}$$

(3)

with $\sum _{j=1}^{K}\tilde{\theta }_{ij}(H) = 1$ and $\sum _{i,j=1}^{K}\tilde{\theta }_{ij}(H) = K$ by construction.

Total directional spillovers received by variable i from all other variables j is then defined in percentage terms as:

$$\begin{aligned} TDS_{i\leftarrow \bullet }(H) = \frac{\sum _{j=1, i\ne j}^{K}\tilde{\theta }_{ij}(H)}{\sum _{i,j=1}^{K}\tilde{\theta }_{ij}(H)} \cdot 100 = \frac{\sum _{j=1, i\ne j}^{K}\tilde{\theta }_{ij}(H)}{K} \cdot 100 \end{aligned}$$

(4)

and, similarly, total directional spillovers from variable i to all the other variables j is defined as:

$$\begin{aligned} TDS_{\bullet \leftarrow i}(H) = \frac{\sum _{j=1, i\ne j}^{K}\tilde{\theta }_{ji}(H)}{\sum _{i,j=1}^{K}\tilde{\theta }_{ji}(H)} \cdot 100 = \frac{\sum _{j=1, i\ne j}^{K}\tilde{\theta }_{ji}(H)}{K} \cdot 100 \end{aligned}$$

(5)

More in general, a system-wide measure of connectedness is the total spillover index defined as:

$$\begin{aligned} TS(H) = \frac{\sum _{i,j=1, i\ne j}^{K}\tilde{\theta }_{ij}(H)}{\sum _{i,j=1}^{K}\tilde{\theta }_{ij}(H)} \cdot 100 = \frac{\sum _{i,j=1, i\ne j}^{K}\tilde{\theta }_{ij}(H)}{K} \cdot 100 \end{aligned}$$

(6)

which provides the (cross) contribution of spillovers from shocks to all variables to the total forecast error variance.

The net directional connectedness between variable i and the other variables of the system is defined as $NS_{i}(H) = TDS_{\bullet \leftarrow i}(H) - TDS_{i \leftarrow \bullet }(H)$ and represents the difference between the gross spillovers transmitted from variable i to the system and spillovers received by i from the other variables.

Spillovers are computed using daily variations. For the market series, from the closing price $P_{t}$ at time t, we compute daily market variations as the natural logarithmic return $r_{t} = ln(P_{t}) - ln(P_{t-1})$. For the Assortativity coefficient, we construct a similar measure of daily variation, but in this case to deal with negative and zero values we utilize the simple difference $EUETS_{t} - EUETS_{t-1}$.

Spillovers in the frequency domain

Following the spectral decomposition approach proposed by Ref.²⁰ (hereinafter BK), we estimate the connectedness in short-, medium-, and long-term frequencies to assess if shocks to one variable generate a persistent (lower frequencies) or immediate (higher frequencies) connectedness.

A Fourier transform of the coefficients $\Psi _{h}$ is employed to compute the frequency response function $\Psi (e^{-i\omega })=\sum _{h}e^{-i\omega h}\Psi _{h}$. In turn, the spectrum density of series $Y_{t}$ over the specific frequency $\omega$ is given by:

$$\begin{aligned} S_{Y} (\omega ) = \sum _{h=0}^{\infty }E(Y_{t}Y_{t-h})e^{-i\omega h} = \Psi (e^{-i\omega h})\Sigma \Psi ^{'}(e^{+i\omega h}) \end{aligned}$$

(7)

where $i^{2}=-1$ is the imaginary unit. $S_{Y} (\omega )$ provides information on the distribution of the variance of $Y_{t}$ over the frequency components.

We can measure how shocks to the j-th variable impact on the portion of the spectrum of the i-th variable at a given frequency $\omega \in (-\pi , \pi )$ by computing the generalized causation spectrum:

$$\begin{aligned} \theta _{ij}(\omega ) = \frac{\sigma _{jj}^{-1}[(\Psi (e^{-i\omega })\Sigma )_{ij}]^{2}}{(\Psi (e^{-i\omega })\Sigma \Psi ^{'}(e^{+i\omega }))_{ii}} = \frac{\sigma _{jj}^{-1}\sum _{h=0}^{\infty }(\Psi (e^{-i\omega h})\Sigma )_{ij}^{2}}{\sum _{h=0}^{\infty }(\Psi (e^{-i\omega h})\Sigma \Psi ^{'}(e^{+i\omega h}))_{ii}} \end{aligned}$$

(8)

with h, $\Sigma$ and $\sigma _{jj}$ with the same meanings as in Eq. 2. Quantity $\theta _{ij}(\omega )$ can be interpreted as the within-frequency causation for a certain $\omega$ value.

As in the case of the time domain analysis, $\theta _{ij}(\omega )$ can be normalized for the K dimensions as follows:

$$\begin{aligned} \tilde{\theta }_{ij}(\omega ) = \frac{\theta _{ij}(\omega )}{\sum _{j=1}^{K}\theta _{ij}(\omega )} \end{aligned}$$

(9)

Desired frequency bands can be calculated by integration. This helps us to compute short-, medium-, and long-term connectedness rather than its value at a single given frequency. Hence, with $d = (a,b) : a,b \in (-\pi , \pi )$ and $a<b$, the generalized forecast error variance decomposition on frequency band d is:

$$\begin{aligned} \tilde{\theta }_{ij}(d) = \int _{a}^{b}\tilde{\theta }_{ij}(\omega )d\omega \end{aligned}$$

(10)

Similarly to DY for the time domain, given a certain frequency band d we can define measures of connectedness in the frequency domain. Hence, the within spillover is weighted by the power of the series on the selected frequency band, while the frequency spillover provides the decomposition of the total spillover index highlighting the most contributing frequency. The total within connectedness on band d is:

$$\begin{aligned} WC(d) = \left( 1-\frac{Tr\{\tilde{\theta }_{ij}(d)\}}{\sum _{i,j=1}^{K}\tilde{\theta }_{ij}(d)}\right) \cdot 100 \end{aligned}$$

(11)

while the contribution of a given frequency band d to the aggregate spillover is:

$$\begin{aligned} FC(d) = WC(d) \cdot \Gamma (d) = \frac{\sum _{i,j=1}^{K}\tilde{\theta }_{ij}(d) - Tr\{\tilde{\theta }_{ij}(d)\}}{\sum _{i,j=1}^{K}\tilde{\theta }_{ij}(\infty )} \cdot 100 \end{aligned}$$

(12)

with the term $Tr\{\cdot \}$ referring to the trace of the variance decomposition matrix and the weighting function ($\Gamma (d) = \sum _{i,j=1}^{K}\tilde{\theta }_{ij}(d) / \sum _{i,j=1}^{K}\tilde{\theta }_{ij}(\infty ) = (\sum _{i,j=1}^{K}\tilde{\theta }_{ij}(d))/K$) representing the contribution of frequency band d to the whole VAR system.

It is worth noting that the sum of all BK spillovers over disjointed frequency intervals is equal to the total spillover measure proposed by DY. Finally, BK directional and net measures of connectedness can be constructed in line with DY measures.

Quantile connectedness

DY and BK measures of connectedness represent the spillovers at the mean of the conditional distribution of $Y_{t}|{Y_{t-1},Y_{t-2},\ldots ,Y_{t-p}}$, thus assuming that the relationships emerging at the conditional mean can be generalized to the entire conditional distribution.

This assumption may be hard to hold during extreme market phases. Following the approach of Ref.²⁷, we propose therefore to measure a quantile connectedness to study spillovers in the extreme of the conditional distribution. This provides a more accurate measurement of the tail behaviour of spillovers, being possibly asymmetric and very different from the mean or median relationships. To do so, we first estimate a quantile VAR (QVAR) as:

$$\begin{aligned} Y_{t} = \mu (\tau ) + \sum _{i=1}^{P}\Phi _{i}(\tau )Y_{t-i} + \epsilon _{t}(\tau ) \end{aligned}$$

(13)

with $\tau$ representing the quantile, $\mu$ a $K \times 1$ conditional mean vector, and the other quantities in line with Eq. 1. Then, we transform QVAR(P) into its QVMA ($\infty$) using the Wold’s Theorem: $Y_{t} = \mu (\tau ) + \sum _{i=1}^{P}\Phi _{i}(\tau )Y_{t-i} + \epsilon _{t}(\tau ) = \mu (\tau ) + \sum _{j=o}^{\infty }\Psi _{j}(\tau )\epsilon _{t-j}$. Finally, we consider similar measures of connectedness as those introduced for DY and BK.

This approach helps us to distinguish between the common and the idiosyncratic components of the error process. The quantile framework disentangles the impacts from rare events from business as usual periods. It contributes to detect whether time-varying connectedness from large shocks might vary not only in the tails of the conditional distribution with respect to its body, but also asymmetrically. Large positive or negative idiosyncratic shocks might thus generate different levels of connectedness.

Code packages

The main statistical analysis is performed in R. In particular, for the spillovers analysis we use the packages Spillover, frequencyConnectedness, ConnectednessApproach; for the network analysis we rely on igraph; for the portfolio performance we use PerformanceAnalytics.