Systemic Financial Risk Spillover Between Traditional-Energy and New-Energy Markets: A Quantile Time–Frequency Network with Link Prediction

Abstract

Energy transition is central to both economic development and climate-change mitigation and has become a shared global challenge. Given the close relationship between conventional energy prices and the development of the new-energy industry, this study investigates systemic risk spillovers among three crude-oil futures, two natural-gas futures, and five Chinese new-energy sector indices. We employ a quantile time-frequency-connectedness framework and an out-of-sample-validated link-prediction model to assess both realized spillovers and potential changes in the network structure. The results reveal that network connectedness is time-varying and asymmetric across quantiles, with short-horizon connectedness accounting for the majority of average system-wide connectedness. Overall connectedness also increases markedly during major crisis episodes. INE crude-oil futures and both natural-gas futures are net receivers of shocks, whereas WTI and Brent crude-oil futures consistently act as net transmitters, with Brent playing the dominant role under extreme market conditions. As the investment horizon lengthens, the solar sector shifts from a net risk receiver to a net risk transmitter. In the predicted network, the solar sector emerges as the market most likely to initiate new short-term spillover links. This finding reflects a prospective, model-implied tendency rather than a causal relationship. These findings offer useful implications for energy market policy, portfolio risk management, and investment decisions involving new-energy companies.

1. Introduction

Global economic expansion continues to drive energy demand, making the transformation of the energy system a prerequisite for sustainable growth. Since the Industrial Revolution, anthropogenic greenhouse gas emissions have risen sharply, intensifying the greenhouse effect and contributing to sea-level rise and other threats to human well-being (IPCC, 2023).

Conventional energy sources remain crucial to economic development and exert a significant influence on financial markets. However, their extensive use has caused substantial environmental pollution, prompting countries worldwide to develop alternative energy sources and reduce their dependence on fossil fuels. The combustion of fossil fuels, including oil, releases substantial quantities of carbon dioxide, thereby contributing heavily to global carbon emissions (IEA, 2024). The adoption of the Glasgow Climate Pact (UNFCCC, 2021) marked a new direction for global carbon reduction and energy transition, with major oil- and gas-consuming countries establishing climate targets to guide their green and low-carbon transitions.

Against this background, recent studies have paid increasing attention to how the relationship between conventional energy markets and clean-energy equities varies over time and across market segments. Qi et al. (2022) found that Chinese clean-energy stocks and energy commodities played different roles as spillover transmitters and receivers over short- and long-term horizons during the COVID-19 pandemic. Y. Chen and Qi (2024) examined time–frequency connectedness and causality across quantiles among Chinese energy stocks, renewable energy stocks, and commodity markets, whereas Fu et al. (2024) applied a time–frequency–quantile framework to energy commodities, clean-energy equities, and ESG assets in China. Deng and Xu (2024) further documented substantial differences in time–frequency connectedness between international crude-oil markets and individual segments of China’s new-energy industry chain. Similarly, Shi et al. (2025) found that spillovers between China’s new-energy market and other financial markets intensified under extreme shocks.

Collectively, these studies demonstrate that spillovers between conventional energy markets and new-energy equities vary over time, across market conditions and investment horizons, and among industry segments. However, because these studies examine different combinations of energy commodities, clean-energy indices, and financial assets, their findings are difficult to compare directly. In particular, relatively little evidence has been obtained from a single integrated system that jointly incorporates China’s domestic crude-oil benchmark, major international crude-oil benchmarks, regional natural-gas benchmarks, and multiple Chinese new-energy sector indices. Two empirical questions therefore remain. First, how are spillovers distributed within this broader market system? Second, do the transmitter and receiver roles of individual markets vary across quantiles and investment horizons?

To address these questions, this study uses daily observations for three crude-oil futures (INE, WTI, and Brent), two natural-gas futures (NG and IPE), and five Chinese new-energy sector indices over the period from 26 March 2018 to 28 December 2023. The starting date coincides with the launch of INE crude-oil futures, allowing China’s domestic benchmark to be examined alongside established international energy benchmarks. We employ a quantile vector autoregression (QVAR) framework to estimate the direction and magnitude of spillovers across the conditional return distribution. We then decompose the resulting connectedness measures into short- and long-term frequency components to determine whether the transmitter and receiver roles of individual markets vary across investment horizons. Finally, we employ link-prediction analysis to assess potential changes in the network structure. This empirical design allows for heterogeneity across market states and investment horizons to be examined within a consistent sample and an integrated market system.

This paper makes three contributions. First, it extends the conventional time-domain spillover framework by integrating quantile regression with frequency decomposition, thereby enabling a nuanced characterization of risk transmission across market conditions and investment horizons. This two-dimensional approach reveals asymmetric spillover patterns that would be obscured by conventional mean-based or purely time-domain specifications. Second, the proposed framework allows us to document how the transmitter–receiver roles of individual markets evolve across investment horizons and market states. In particular, it enables us to examine whether specific new-energy sectors and crude-oil benchmarks switch between transmitting and receiving risk as the investment horizon or market condition changes. Third, we combine the connectedness estimates with an out-of-sample link-prediction framework, thus extending the analysis from the measurement of realized spillovers to the prediction of newly emerging directed links over a five-day horizon. Specifically, the framework assesses whether currently weak or absent links develop into strong connections five trading days later. This forward-looking analysis provides a complementary tool for identifying markets that may assume increasingly important roles as risk transmitters or receivers as the network evolves—an issue that remains relatively underexplored in the connectedness literature.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature. Section 3 describes the empirical methodology and data. Section 4 reports the connectedness results, and Section 5 presents the link-prediction analysis. Section 6 concludes the paper and discusses the policy implications.

2. Literature Review

2.1. Linkages Between Traditional-Energy Futures Markets and Stock Markets

Existing research on the relationship between traditional-energy futures and the stock market is extensive. Researchers primarily examine the relationship between the energy market and the stock market, the underlying mechanisms, and the effects of volatility spillover (Moore & Harrison, 2009; Diaz et al., 2016; Lee et al., 2017; Jiang & Marsh, 2020). Hammoudeh and Aleisa (2004) examine the interaction between the WTI international crude-oil futures market and the stock markets of the UAE, Kuwait, Oman, and Saudi Arabia, focusing on the volatility spillover effect to elucidate the relationship between these markets. Malik and Hammoudeh (2007) highlight the notable volatility spillover effect between the five largest crude-oil companies and their respective stock markets in the United States. Arouri (2011) finds that volatility spillovers from crude-oil futures to equity markets are statistically significant in both Europe and the United States, whereas spillovers running in the opposite direction are not. Ågren (2006) reports considerable volatility spillovers from oil prices to the Japanese, Norwegian, U.K. and U.S. stock markets but not to the Swedish market. Ferrer et al. (2018) highlight that the direction of volatility spillovers can vary over time, making it essential to conduct timely analyses of volatility spillover dynamics and persistence.

2.2. Linkages Between Traditional-Energy Futures Markets and Energy Stock Markets

The emergence of the new-energy industry has led to heightened academic interest in the influence of traditional-energy markets on the new-energy stock market. Kumar et al. (2012) employ a VAR model to analyze the interrelationships between the stock prices of clean-energy firms, crude-oil prices, technology firms, and carbon trading prices. The findings indicate a significant influence of crude-oil prices and technology firms’ stock prices on the stock prices of clean-energy firms, whereas the effect of carbon trading prices on clean-energy firms is not significant. Sadorsky (2012) employs a univariate beta model to analyze the factors influencing the risk of renewable energy companies, finding that an increase in crude-oil prices positively affects the company’s risk. In a companion study, Sadorsky (2012) employs four MV-GARCH models to examine the correlation and volatility spillovers among the share prices of clean-energy firms, technology firms, and crude-oil prices. Results indicate bidirectional volatility spillovers between clean-energy and technology stocks, whereas no volatility spillovers were observed between technology stocks and crude-oil prices. Crude-oil prices demonstrated a unidirectional volatility spillover effect on clean-energy stocks. Li et al. (2022) conducted an empirical analysis of the average spillover effect and volatility spillover effect among INE crude-oil futures, Brent crude-oil futures, and the Omani crude-oil spot market utilizing a ternary VAR-BEKK-GARCH model. The findings indicate that the INE crude-oil futures price effectively aligns with the Brent crude-oil benchmark price, exhibiting a bidirectional average spillover characteristic.

The development of the new-energy industry is influenced by increasing environmental pollution and the gradual depletion of traditional-energy resources. Additionally, the characteristics of INE as an emerging crude-oil futures market indicate that relevant research has commenced relatively recently. However, the influence of traditional-energy markets on the stock market represents a complex network of interactions, yet there is a paucity of studies in the current literature addressing this perspective. Recent studies reveal complex interactions between traditional-energy markets and new-energy stock markets, encompassing price linkages, risk transmission, volatility spillovers, and inter-market anchoring. Preliminary research results provide a robust foundation for developing a comprehensive linkage model between traditional- and new-energy stock markets (Corbet et al., 2020; Nasreen et al., 2020; Qiang et al., 2023; Cevik et al., 2024). More recent studies have increasingly adopted quantile- and frequency-based frameworks to characterize risk transmission between traditional-energy and China’s new-energy markets under different market conditions. Qi et al. (2022) show that Chinese clean-energy stocks and energy commodities alternate as spillover transmitters and receivers across short- and long-term horizons during the COVID-19 period, while Y. Chen and Qi (2024) document time–frequency connectedness and causality across quantiles among Chinese energy stocks, renewable-energy stocks, and commodity markets. Fu et al. (2024) extend this to a time–frequency–quantile analysis of energy commodities, clean-energy equities, and ESG assets, and Deng and Xu (2024) report pronounced differences in the connectedness between international oil and individual segments of China’s new-energy industry chain. Focusing specifically on China’s domestic benchmark, Liu et al. (2025) apply a quantile-on-quantile connectedness approach to Shanghai (INE) crude-oil futures and China’s green markets. These studies, however, each examine only a particular combination of energy commodities, clean-energy indices, or financial assets, which motivates the unified system adopted in this paper.

2.3. Review of Complex Connected Network Methods

Within the literature on volatility spillovers across financial markets, the connectedness framework introduced by Diebold and Yilmaz has become a widely used approach. Utilizing forecast-error variance decomposition from a vector autoregression (VAR) model, they developed a volatility spillover index (DY index) to measure the degree of volatility transmission between markets. Diebold and Yilmaz (2012) refine the method by adopting a generalized VAR framework, which makes the forecast-error variance decomposition invariant to the ordering of the variables. The refined DY index effectively measures both overall and directional volatility spillovers.

The Diebold–Yilmaz model primarily emphasizes time-domain analysis; however, Baruník and Křehlík (2018) expand this framework into the frequency domain by developing a spillover index method derived from the spectral representation of the generalized prediction error variance decomposition. This method encompasses the Diebold–Yilmaz model by deconstructing intra-system linkages into various frequency components and effectively resolves market relationships within the frequency domain. The Baruník–Křehlík spillover index method has gained significant traction in the international academic community since its introduction. Ferrer et al. (2018) and McCarthy and Orlov (2012) have examined the stock prices of U.S. clean-energy companies, crude-oil prices, and various financial variables. They employed methods to investigate the relationships among these variables across time and frequency dimensions, revealing that the linkages predominantly manifest in short-term dynamics. A further extension is the TVP-VAR, which allows the coefficient and covariance matrices to vary over time and therefore captures changes in shock intensity and in transmission paths more flexibly. Building on these developments, quantile-based connectedness has recently been introduced to capture how transmission patterns differ across market states and has been increasingly applied to energy–finance networks, including quantile-on-quantile analyses of China’s crude-oil and green markets (Liu et al., 2025; Shi et al., 2025).

This study employs a quantile time–frequency joint network approach, building on the work of Diebold and Yilmaz (2012, 2014) and Baruník and Křehlík (2018). This approach characterizes dependence across the entire conditional return distribution rather than at the conditional mean alone and therefore describes the link between traditional-energy futures and equity markets under different market states. This method examines the relationship between the two time-series from a joint time–frequency perspective, elucidating their evolutionary characteristics across various time scales and offering a robust framework for comprehending the interaction mechanism between them.

2.4. Review of Link-Prediction Methods

Link prediction uses the observed nodes and structure of a network to estimate the likelihood that a link will form between two nodes that are not yet connected (Lv et al., 2009; Gong et al., 2014). With the rapid development of computer network technology, more opportunities and demands have been provided for the applied research of link prediction, and link-prediction methods have developed rapidly. Broadly, existing approaches fall into several families: similarity-based methods, maximum-likelihood methods, probabilistic and Markov-chain models, and, more recently, machine-learning-based methods. Among similarity-based approaches, Liben-Nowell and Kleinberg (2003) propose a link-prediction framework built on common neighbors, showing that node pairs sharing more common neighbors are more likely to form new connections. Building on such structural indices, Wu et al. (2022) analyze thirty-six datasets across seven network types and identify the most suitable predictor for each type.

More recently, link prediction has been increasingly formulated as a supervised binary-classification task, in which structural and node-level features are used to train a classifier—such as logistic regression, random forests, gradient-boosted decision trees (e.g., XGBoost; T. Chen & Guestrin, 2016), or graph-representation-learning models—to distinguish node pairs that will form a link from those that will not. In parallel, research has advanced toward dynamic and temporal networks and graph-neural-network-based approaches (Xiong et al., 2026), where model performance is assessed by temporal, out-of-sample evaluation rather than by fitting the full network in sample. These methods have also been increasingly applied in financial settings, such as financial knowledge graphs and market-structure networks. Nonetheless, at present few scholars have studied link prediction between energy futures and new-energy stocks, which is the gap this study addresses.

3. Data and Methodology

3.1. Data

3.1.1. Data Processing

In this study, we examine the quantile–frequency return connectedness between traditional-energy futures prices and share prices of new-energy firms. Specifically, crude-oil futures listed on the Shanghai International Energy Exchange (INE) are selected as a typical representative of the domestic crude-oil market; meanwhile, West Texas Intermediate (WTI) and London Brent (Brent) crude-oil futures are selected as references for international crude-oil futures due to their extensive influence in the international market. For natural-gas futures, the study focuses on the New York Mercantile Exchange (NYMEX) listed contracts based on the Henry Hub spot price of natural-gas in the US(NG) and the Intercontinental Exchange (ICE) listed contracts based on the NBP spot price of natural gas in the UK (IPE), which represent the two major natural-gas markets in North America and Europe respectively.

The new-energy companies’ share price indicators used in the study include the CSI Wind Power Industry Index, the CSI Solar 50 Index, the CSI Water Resources Index, the CSI nuclear energy and nuclear power index, and the CSI 300 Green Leading Stocks Index, aiming to comprehensively reflect the market performance of different segments of China’s new-energy industry. The data involved is sourced from the Wind database (daily closing prices of INE, WTI, Brent crude-oil futures, and UK NBP natural-gas futures), U.S. Energy Information Administration (daily closing prices of Henry Hub natural-gas futures) and CSI website (new-energy indices).

The sample runs from 26 March 2018 to 28 December 2023. The start date coincides with the launch of INE crude-oil futures and is therefore the earliest date on which all ten markets can be observed jointly; the end date is the last common trading day available when the dataset was assembled. After matching trading dates across markets, we obtain a balanced panel of 1317 daily observations per series, or 13,170 observations in total.

The sample period is also economically informative because it spans several distinct market and policy regimes, each accompanied by substantial policy responses that shaped the interaction between traditional-energy and new-energy markets.

First, and most directly relevant to our study, the sample covers the emergence of China’s low-carbon policy agenda. In September 2020 China announced its “dual-carbon” goals—reaching a carbon peak before 2030 and carbon neutrality before 2060—which were subsequently embedded in the 14th Five-Year Plan through explicit targets for the large-scale development of renewable energy. This policy shift substantially raised the strategic and financial importance of the new-energy sector, helping to explain why wind, solar, nuclear, hydro and green-power equities became increasingly systemically relevant during this period and why their risk linkages with traditional-energy futures warrant close examination. Second, the sample encompasses the COVID-19 pandemic. Our period captures the sharp collapse in global energy demand in early 2020, which was met with large-scale monetary and fiscal stimulus across major economies and a coordinated OPEC+ production adjustment. The extreme dislocation of this episode—epitomized by the negative WTI price in April 2020—generated pronounced volatility in energy markets and intensified the co-movement between traditional-energy and new-energy assets. Third, the sample includes the 2022 Russia–Ukraine conflict and its policy aftermath. The imposition of energy sanctions on Russia and the ensuing reconfiguration of global energy supply drove fossil-fuel prices sharply higher and accelerated the transition toward new energy, thereby altering the structure of risk spillovers between the two market groups. Taken together, the resulting variation in energy demand, supply conditions, policy expectations and investor risk perceptions makes this period particularly well suited to examining the time-varying and frequency-dependent connectedness between traditional-energy futures and Chinese new-energy equity markets.

We choose WTI as one of the representatives of the international crude-oil market because WTI is not only the first crude-oil futures in the U.S. but also the first crude-oil futures in the world. The price of WTI crude-oil futures has also been one of the most important reference benchmarks for international crude oil. In recent years, Brent crude-oil futures have gradually replaced WTI crude-oil futures as the international pricing oil and have become the most important international crude-oil price benchmark. Therefore, we choose Brent as the representative of the international crude-oil market. For the representative of Chinese crude-oil futures, we choose Shanghai International Energy Exchange (INE) to launch the crude-oil futures contract denominated in RMB. Within a month of listing, it had overtaken Oman crude-oil futures to become the world’s third-largest crude-oil futures contract by trading volume, after WTI and Brent crude-oil futures.

Global liquefied natural-gas (LNG) trade is mainly concentrated in the three regional markets of North America, Europe, and Asia-Pacific, forming two sets of pricing systems. The activity and financial infrastructure of the North American and UK markets have made them natural-gas trading centers. Correspondingly, the Henry Hub natural-gas futures contract on NYMEX and the NBP natural-gas futures contract on ICE have become important tools for risk management globally and are pegged to the spot prices in their respective regions.

In order to comprehensively assess the overall market performance of listed companies in the new-energy industry, this study utilizes a number of specific indices. Among them, the CSI Wind Power Industry Index includes more than 50 securities engaged in the upstream and downstream businesses of the wind power industry chain in Shanghai and Shenzhen, aiming to reflect the market conditions of the entire wind power sector, which is based on the base period of 31 December 2014, with the base point set at 1000 points. The Solar 50 Index focuses on the photovoltaic power generation and solar thermal power generation industries and selects 50 relevant listed securities as samples. The CSI Water Resources Index selects companies in the fields of water conservancy project construction, hydropower production, and water resource management as constituents to show the overall performance of the water-related industry. The index is based on the base period of 29 June 2012, and the base point is also set at 1000 points. The nuclear energy and nuclear power index is based on a sample of listed companies involved in nuclear materials, equipment manufacturing, and power plant operations and aims to portray the overall performance of the nuclear energy and power industry, with a base period of 31 December 2004 and a base point of 1000 points. In addition, the study also incorporates the CSI 300 Green Leading Stock Index, which is designed to reflect the overall performance of green leading companies in the Shanghai and Shenzhen markets by eliminating the highly polluting, high-energy-consuming, and overcapacity companies in the CSI 300 index, calculating the Green Leading Score based on the listed companies’ indicators of green production level, percentage of green revenues, negative environmental news and record of environmental penalties, etc., and selecting the top 100 listed securities with the highest scores as the samples. It aims to reflect the comprehensive performance of green leading companies in the Shanghai and Shenzhen markets.

Next, we log-differentiate the original series to obtain the log returns of crude-oil futures and new-energy stocks, respectively, with the following equations:

$${\mathrm{r}}_{\mathrm{t}}=\mathrm{l}\mathrm{n}\left({\displaystyle \frac{{\mathrm{p}}_{\mathrm{t}}}{{\mathrm{p}}_{\mathrm{t}-1}}}\right)$$

where ${\mathrm{p}}_{\mathrm{t}}$ is the closing price of crude-oil futures and new-energy stocks, and ${\mathrm{r}}_{\mathrm{t}}$ is the return of crude-oil futures and new-energy stocks.

3.1.2. Descriptive Statistics

Figure 1 displays the return series and the results of the ADF tests show that all return series are stationary. Across the sample, every series displays marked volatility clustering; notably, the return dynamics of WTI, the CSI Wind Power Industry Index, and the CSI Solar 50 Index resemble one another more closely than any of them resembles the INE.

Summary statistics are given in

Table 1. Summary statistics.

	INE	WTI	Brent	IPE	NG	Wind	Solar	Water	Nuclear	Green
Mean	0.00027	0.00032	0.00029	0.00118	−0.00035	0.00011	−0.00014	−0.00008	−0.00023	−0.00015
Standard Deviation	0.02682	0.19375	0.02843	0.06713	0.04244	0.01966	0.02068	0.01211	0.01384	0.01209
Skewness	−0.402	−1.212	−1.227	0.443	0.159	0.074	−0.056	−0.293	−0.602	−0.020
Kurtosis	2.859	646.320	18.112	5.228	8.373	2.260	1.424	4.697	4.613	4.456
JB	36.563	22,710,924.1	12,862.41	315.475	1589.742	31.252	136.986	176.874	222.320	116.419
ADF	<0.01	<0.01	<0.01	<0.01	<0.01	<0.01	<0.01	<0.01	<0.01	<0.01
Kendall’s τ	INE	WTI	Brent	IPE	NG	Wind	Solar	Water	Nuclear	Green
INE	1.000 **	0.127 **	0.143 **	0.023	0.029	0.056 **	0.063 **	0.076 **	0.086 **	0.109 **
WTI	0.127 **	1.000 **	0.782 **	0.062 **	0.062 **	0.028	0.046 *	0.01924	0.030	0.037 *
Brent	0.143 **	0.782 **	1.000 **	0.072 **	0.057 **	0.045 *	0.065 **	0.026	0.045 *	0.060 *
IPE	0.023	0.062 **	0.057 **	1.000 **	0.117 **	−0.017	0.000	−0.022	−0.028	−0.006
NG	0.029	0.062 **	0.057 **	0.117 **	1.000 **	0.016	0.026	0.011	−0.005	0.030
Wind	0.056 **	0.028	0.045 *	−0.017	0.016	1.000 **	0.578 **	0.430 **	0.571 **	0.425 **
Solar	0.063 **	0.046 *	0.065 **	0.000	0.026	0.578 **	1.000 **	0.315 **	0.453 **	0.442 **
Water	0.076 **	0.019	0.026	−0.022	−0.011	0.430 **	0.315 **	1.000 **	0.593 **	0.460 **
Nuclear	0.086 **	0.030	0.045 *	−0.028	−0.005	0.571 *	0.453 **	0.593 **	1.000 **	0.451 **
Green	0.109 **	0.037 *	0.060 **	−0.006	0.030	0.425 **	0.442 **	0.460 **	0.451 **	1.000 **

Notes: ** indicates significant correlation at the 0.01 level (two-tailed); * indicates significant correlation at the 0.05 level (two-tailed).

. The mean value of all the series is positive except for NG natural-gas futures, solar index, water index, nuclear energy, nuclear power index, and green leader index, which indicates that the average price of NG natural-gas futures, solar index, water index, nuclear energy and nuclear power index, and green leader index are decreasing, while all other indices are increasing in their average price. On the contrary, the study finds that the green leader index has the lowest variance, while WTI crude-oil futures have the highest variance, followed by IPE natural-gas futures and NG natural-gas futures. In addition, all series are significantly skewed to the left with the exception of natural-gas futures.

In addition, the empirical results show that all the series are significantly leptokurtic and non-normally distributed. Finally, we find that all return series are stationary and display significant autocorrelation. According to the nonparametric Kendall rank correlation coefficient, the returns are positively correlated. Kendall rank correlations are positive for most pairs. The strongest association is between WTI and Brent (0.782), followed by Water and Nuclear (0.593) and by Wind and Solar (0.578). Correlations between the natural-gas futures and the Chinese new-energy indices are close to zero and in a few cases slightly negative; the weakest is between IPE and Solar (0.000).

3.2. Methodology

We use the quantile time–frequency-connectedness network proposed by Chatziantoniou et al. (2021) to study the quantile as well as the time–frequency propagation mechanism between crude-oil and new-energy stock markets. The quantile–frequency-connectedness approach used in this study combines the seminal studies of Diebold and Yilmaz (2012, 2014) and Baruník and Křehlík (2018) to investigate the risk spillover effects of Chinese and international energy price volatility and share prices of new-energy firms from both time- and frequency-domain perspectives.

To calculate all the connectedness metrics, we first estimate a quantile vector autoregression, QVAR(p), which can be summarized as follows:

$$x_t={\sigma }_t(\tau )+{\varphi }_1(\tau )x_{t-1}+{\varphi }_2(\tau )x_{t-2}+\cdots +{\varphi }_p(\tau )x_{t-p}+u_t(\tau )$$

(1)

where $x_t \mathrm{a}\mathrm{n}\mathrm{d}{ x}_{t-i}, i=1,\cdots ,p$ are the $N\times 1$ dimensional endogenous variable vectors. $\tau \in \left[0,1\right]$ fixes the quantile at which the dynamics are evaluated, $p$ sets the QVAR lag order, $\sigma (\tau )$ is a dimensional conditional mean vector, ${\varphi }_j(\tau )$ is an $N\times N$ dimensional QVAR coefficient matrix, and $u(\tau )$ is an $N\times 1$ dimensional error vector with an $N\times N$ dimensional error variance–covariance matrix, $\sum (\tau )$. Invoking Wold’s decomposition, the QVAR(p) admits an equivalent quantile vector moving-average QVAR(∞) form, which recasts each variable as an infinite weighted sum of past innovations: $x_t=\sigma (\tau )+{\sum }_{j=1}^p{\varphi }_j(\tau )x_{t-j}+u_t(\tau )=\sigma (\tau )+{\sum }_{i=0}^{\infty }{\phi }_i(\tau )u_{t-i}$.

The lag order, p, of the QVAR is determined by information criteria.

Table 2. Lag-order selection for the QVAR model.

Lag (p)	AIC	HQIC	BIC
1	−75.8698	−75.7066	−75.4347
2	−75.9682	−75.6567	−75.1376
3	−75.9563	−75.4964	−74.7302
4	−75.9253	−75.3171	−74.3038
5	−75.9072	−75.1507	−73.8902
6	−75.8793	−74.9744	−73.4668
7	−75.8825	−74.8293	−73.0745
8	−75.8867	−74.6852	−72.6832

Notes: Entries are information-criterion values for VAR lag orders p = 1 to 8. Bold denotes the minimum value of each criterion. The BIC and HQIC are both minimized at p = 1; the AIC is minimized at p = 2. The QVAR lag order is selected by the BIC.

reports the AIC, HQIC, and BIC values for lag orders ranging from p = 1 to 8. Both the Bayesian information criterion (BIC) and the Hannan–Quinn criterion (HQIC) are minimized at p = 1, while the Akaike information criterion (AIC) is minimized at p = 2. Following common practice in the connectedness literature (Chatziantoniou et al., 2021; Baruník & Křehlík, 2018), we select the lag order according to the BIC and therefore set p = 1.

Subsequently, the core of the connectedness approach is the generalized prediction error variance decomposition (GFEVD) (see Pesaran & Shin, 1998).

When the random error terms of each equation in a VAR model are contemporaneously correlated, the traditional orthogonal variance decomposition is influenced by the ranking of variables, and different rankings will produce different decomposition results, so the decomposition results are not robust. The orthogonalized variance decomposition depends on the ordering of the variables, so different orderings yield different results. The GFEVD is invariant to this ordering and is therefore preferred here.

The GFEVD can be interpreted as the effect of shocks in the series $j$ on the series $i$ in terms of its share of predicted variance and can be written in the following form:

$${\theta }_{ij}(H)={\displaystyle \frac{{\left({\sum (\tau )}_{jj}^{-1}{\sum }_{h=0}^H\left({\left({\phi }_h(\tau )\sum (\tau )\right)}_{ij}\right)\right)}^2}{{\sum }_{h=0}^H{\left({\phi }_h(\tau )\sum (\tau ){\phi }_h^{\prime }(\tau )\right)}_{ii}}}$$

(2)

$${\tilde{\theta }}_{ij}(H)={\displaystyle \frac{{\theta }_{ij}(H)}{{\sum }_{k=1}^N{\theta }_{ij}(H)}}$$

(3)

As the rows of ${\tilde{\theta }}_{ij}(H)$ do not sum up to one, we need to normalize them by the row sum, which results in ~${\tilde{\theta }}_{ij}$. Through normalization, we get the following identities: ${\sum }_{i=1}^N{\tilde{\theta }}_{ij}(H)=1$ and ${\sum }_{j=1}^N{\sum }_{i=1}^N{\tilde{\theta }}_{ij}(H)=N$.

Hence, each row sum is equal to unity, representing how a shock in series $i$ has influenced the series itself and all other series, $j$.

In the next step, we can calculate all connectedness measures. The (overall) net pairwise connectedness (NPDC) can be calculated as follows:

$${NPDC}_{ij}(H)={\tilde{\theta }}_{ij}(H)-{\tilde{\theta }}_{ji}(H)$$

(4)

If ${NPDC}_{ij}\left(H\right)<0 \left({NPDC}_{ij}\left(H\right)>0\right)$, series $i$ is a net transmitter of shocks from (a net receiver of shocks to) series $j$.

To obtain information about the overall effect of variable $i$ on all other variables, $j$, we calculate the total directional connectedness to the other variables:

$${TO}_i(H)={\displaystyle \sum _{i=1,i\ne j}^N}{\tilde{\theta }}_{ji}(H)$$

(5)

Similarly, we evaluate the effect of shocking all other variables, $j$, on variable $i$ by the total directional connectedness of FROM others:

$${FROM}_i(H)={\displaystyle \sum _{i=1,i\ne j}^N}{\tilde{\theta }}_{ij}(H)$$

(6)

The (overall) NET total directional connectedness represents the difference between the (overall) total directional connectedness TO others and the (overall) total directional connectedness FROM others, which can be interpreted as the net influence series $i$ has on the predetermined network.

$${NET}_i(H)={TO}_i(H)-{FROM}_i(H)$$

(7)

The final metric is the (overall) Total Connectedness Index (TCI), which measures the degree of network interconnection and is calculated as follows:

$$TCI(H)=N^{-1}{\displaystyle \sum _{i=1}^N}{TO}_i(H)=N^{-1}{\displaystyle \sum _{i=1}^N}{FROM}_i(H)$$

(8)

In other words, this approach accounts for the average impact of a shock in one series on all other series. A higher TCI indicates stronger system-wide interconnection and therefore greater systemic risk.

Having characterized connectedness in the time domain, we now extend the analysis to the frequency domain, which disentangles how spillovers accumulate across distinct cyclical horizons. To this end, we build on the spectral decomposition of Stiassny (1996) to express the system’s interlinkages as functions of frequency. First, we consider the frequency response function, $\phi \left(e^{-if}\right)={\sum }_{h=0}^{\infty }{e}^{-i\omega h}{\phi }_h$, where $i=\sqrt{-1}$, and $f$ denotes the frequency to continue with the spectral density of $x_t$ at frequency $f$, which can be defined as a Fourier transformation of the QVMA(∞) representation:

$$S_x(\omega )={\displaystyle \sum _{h=-\infty }^{\infty }}E\left(x_t{x}_{t-h}^{\prime }\right)e^{-i\omega h}=\phi \left(e^{-i\omega h}\right){\sum }_t{\phi }^{\prime }\left(e^{+i\omega h}\right)$$

(9)

It is worth noting that the frequency GFEVD is a combination of the spectral density and the GFEVD. As in the time domain, we need to normalize the frequency GFEVD, which can be expressed as follows:

$${\theta }_{ij}(\omega )={\displaystyle \frac{{\left(\sum (\tau )\right)}_{jj}^{-1}{\left|{\sum }_{h=0}^{\infty }{\left({\phi }_h(\tau )\left(e^{-i\omega h}\right)\sum (\tau )\right)}_{ij}\right|}^2}{{\sum }_{h=0}^{\infty }{\left(\phi (e^{-i\omega h})\sum (\tau )\phi (\tau )\left(e^{i\omega h}\right)\right)}_{ii}}}$$

(10)

$${\tilde{\theta }}_{ij}(\omega )={\displaystyle \frac{{\theta }_{ij}(\omega )}{{\sum }_{k=1}^N{\theta }_{ij}(\omega )}}$$

(11)

where ${\tilde{\theta }}_{ij}(f)$ denotes the part of the first sequence spectrum at a given frequency, $f$, that can be attributed to a surge in the first series. It can be interpreted as an intra-frequency indicator.

In order to assess the short-term and long-term connectedness, instead of the connectedness of a single frequency, we aggregate all frequencies within a specific range, $D=\left(a,b\right):a, b\in \left(-\pi , \pi \right), a<b$:

$${\tilde{\theta }}_{ij}(D)={\int }_a^b{\tilde{\theta }}_{ij}(f)d\omega$$

(12)

From here, we can calculate exactly the same connectedness metrics as Diebold and Yilmaz (2012, 2014), which can be interpreted identically; however, in this case they refer to frequency-connectedness metrics that provide spillover information in a specific frequency range:

$${NPDC}_{ij}(D)={\tilde{\theta }}_{ij}(D)-{\tilde{\theta }}_{ji}(D)$$

(13)

$${TO}_i(D)={\displaystyle \sum _{i=1,i\ne j}^N}{\tilde{\theta }}_{ji}(D)$$

(14)

$${FROM}_i(D)={\displaystyle \sum _{i=1,i\ne j}^N}{\tilde{\theta }}_{ij}(D)$$

(15)

$${NET}_i(D)={TO}_i(D)-{FROM}_i(D)$$

(16)

$$TCI(D)=N^{-1}{\displaystyle \sum _{i=1}^N}{TO}_i(D)=N^{-1}{\displaystyle \sum _{i=1}^N}{FROM}_i(D)$$

(17)

In our case, we have two bands to illustrate short-term and long-term dynamics, ranging from 1 to 5 days, $D_1=\left(\pi /5, \pi \right)$, and from 6 to infinite days, $D_2=\left(\left.0, \pi /5\right]\right.$. Thus ${NPDC}_{ij}(D_1),{TO}_i(D_1),{FROM}_i(D_1),{NET}_i(D_1),TCI(D_1)$ illustrates short-term net pair connectedness, short-term total directional connectedness to others, short-term total directional connectedness from others, short-term net total directional connectedness, and short-term total directional connectedness index, while ${NPDC}_{ij}(D_2),{TO}_i(D_2),{FROM}_i(D_2),{NET}_i(D_2),TCI(D_2)$ illustrates long-term network pair connectedness, long-term total directional connectedness with others, long-term total directional connectedness with others, long-term network total directional connectedness, and long-term total connectedness index. Finally, we show the relationship between the frequency-domain measure of Baruník and Křehlík (2018) and the time-domain measure of Diebold and Yilmaz (2012, 2014):

$${NPDC}_{ij}(H)=\sum _d{NPDC}_{ij}(D)$$

(18)

$${TO}_i(H)=\sum _d{TO}_i(D)$$

(19)

$${FROM}_i(H)=\sum _d{FROM}_i(D)$$

(20)

$${NET}_i(H)=\sum _d{NET}_i(D)$$

(21)

$$TCI(H)=\sum _{d}TCI(D)$$

(22)

Intuitively, the overall connectedness measures are equal to the sum of the corresponding frequency-connectedness measures. It is important to note that all these connectedness measurements are based on a specific quantile, $\tau$. We estimate all measures over a grid of ten quantiles, τ ∈ {0.05, 0.15, …, 0.95}, spanning bearish (lower-tail) and bullish (upper-tail) market states. This grid captures connectedness dynamics across bearish (lower-tail) and bullish (upper-tail) market conditions. To formally assess whether the connectedness measures differ across horizons and quantiles, rather than relying on visual inspection, we conduct paired tests on the matched rolling-window series. For each market and quantile, τ, we test the null of equal short- and long-term net directional connectedness, NET_short(τ) = NET_long(τ), and analogously test TCI_short(τ) = TCI_long(τ). We report the paired Wilcoxon signed-rank test as our primary (nonparametric) statistic, and because the rolling-window estimates are serially dependent, we base statistical significance on a stationary block bootstrap (block length, 20; 2000 replications): a difference is deemed significant only when its bootstrap 95% confidence interval excludes zero.

4. Empirical Results

In this section, we present the findings and discuss related issues in terms of two main aspects: frequency-generated connections and quantile connectedness. We focus on the dynamic results for frequencies and quantiles, and our research framework brings together relevant results from Diebold and Yilmaz (2012, 2014), Baruník and Křehlík (2018), and Chatziantoniou et al. (2021). By decomposing connectedness across frequencies, this approach extends the conventional time-domain measure and, through the quantile dimension, further uncovers how linkages behave in the tails of the return distribution. It thus enables us to assess whether short- and long-term spillovers differ across quantiles.

4.1. Connectedness in the Frequency Domain

4.1.1. Averaged Median Dynamic Connectedness Measures

We begin with the full-sample averages at the median quantile, which abstract from time variation. These results are shown in

Table 3. Averaged dynamic connectedness.

	INE	WTI	Brent	IPE	NG	Wind	Solar	Water	Nuclear	Green	FROM
INE	42.67	20.48	23.09	1.94	1.50	1.83	1.80	1.71	2.45	2.55	57.33
	(33.96, 8.71)	(15.53, 4.95)	(17.27, 5.82)	(1.54, 0.40)	(1.28, 0.21)	(1.40, 0.43)	(1.43, 0.36)	(1.35, 0.35)	(1.97, 0.48)	(2.00, 0.55)	(43.77, 13.56)
WTI	3.88	51.35	36.21	1.32	1.25	0.88	1.21	1.23	1.35	1.68	48.65
	(3.26, 0.62)	(42.49, 8.87)	(30.09, 6.02)	(1.13, 0.19)	(1.02, 0.23)	(0.69, 0.17)	(0.93, 0.21)	(1.01, 0.22)	(1.19, 0.23)	(1.27, 0.36)	(38.52, 7.87)
Brent	4.68	36.87	49.01	1.59	1.01	1.50	1.88	1.57	1.98	2.10	50.99
	(4.13, 1.06)	(30.53, 6.24)	(40.37, 8.65)	(1.25, 0.34)	(0.79, 0.23)	(1.17, 0.32)	(1.43, 0.44)	(1.31, 0.26)	(1.62, 0.36)	(1.61, 0.49)	(41.82, 9.17)
IPE	1.64	2.41	2.77	82.18	3.13	1.52	1.55	1.31	2.11	1.39	17.82
	(1.41, 0.23)	(2.22, 0.19)	(2.49, 0.28)	(68.13, 14.05)	(2.50, 0.62)	(1.18, 0.34)	(1.30, 0.25)	(1.04, 0.27)	(1.67, 0.44)	(1.18, 0.20)	(14.99, 2.82)
NG	1.35	2.49	1.95	3.09	82.74	1.74	1.82	1.61	1.59	1.61	17.26
	(1.16, 0.19)	(2.16, 0.34)	(1.68, 0.27)	(2.52, 0.57)	(69.41, 13.33)	(1.48, 0.26)	(1.49, 0.33)	(1.38, 0.23)	(1.35, 0.24)	(1.36, 0.25)	(14.57, 2.69)
Wind	1.43	0.98	1.41	0.69	0.56	33.17	17.53	14.13	19.94	11.42	66.83
	(1.14, 0.29)	(0.75, 0.23)	(1.11, 0.29)	(0.52, 0.17)	(0.44, 0.12)	(27.26, 5.91)	(14.43, 3.10)	(11.97, 2.16)	(17.08, 2.86)	(9.37, 2.04)	(55.87, 10.97)
Solar	1.45	1.32	1.78	0.53	1.09	19.67	37.59	9.93	14.90	13.37	62.41
	(1.23, 0.23)	(1.13, 0.18)	(1.56, 0.21)	(0.42, 0.11)	(0.88, 0.21)	(16.45, 3.22)	(30.81, 6.78)	(8.47, 1.46)	(12.77, 2.13)	(11.11, 2.26)	(52.72, 9.69)
Water	0.98	0.83	1.04	0.61	0.58	14.42	8.68	36.59	22.84	14.62	63.41
	(0.80, 0.18)	(0.67, 0.15)	(0.87, 0.17)	(0.47, 0.14)	(0.45, 0.13)	(11.73, 2.69)	(7.02, 1.66)	(29.60, 7.00)	(18.53, 4.31)	(11.90, 2.72)	(51.52, 11.88)
Nuclear	1.65	0.92	1.32	0.79	0.56	18.36	12.28	20.30	31.60	13.58	68.40
	(1.31, 0.34)	(0.75, 0.16)	(1.05, 0.27)	(0.68, 0.11)	(0.47, 0.09)	(14.97, 3.39)	(9.67, 2.62)	(16.69, 3.61)	(26.15, 5.44)	(11.15, 2.43)	(55.58, 12.82)
Green	1.88	1.19	1.50	0.62	0.92	12.70	13.17	15.31	15.78	38.47	61.53
	(1.60, 0.28)	(1.01, 0.18)	(1.30, 0.20)	(0.52, 0.10)	(0.72, 0.20)	(10.79, 1.90)	(11.21, 1.96)	(13.00, 2.31)	(13.58, 2.20)	(32.60, 5.88)	(52.50, 9.03)
TO	18.05	61.99	67.44	11.19	10.60	69.56	56.63	64.31	79.70	59.42
	(15.02, 3.03)	(49.90, 12.09)	(53.95, 13.49)	(9.05, 1.14)	(8.56, 2.04)	(57.37, 12.20)	(46.17, 10.46)	(53.91, 10.40)	(67.06, 12.64)	(48.49, 10.93)	TCI
NET	−39.29	15.60	16.45	−6.33	−6.66	2.73	−5.79	0.90	11.30	−2.11	59.43
	(−28.75, −10.53)	(11.38, 4.22)	(12.14, 4.32)	(−5.94, −0.39)	(−6.02, −0.64)	(1.50, 1.23)	(−6.55, 0.77)	(2.38, −1.48)	(11.48, −0.18)	(−4.01, 1.91)	(48.78, 1.65)

. More specifically, Table 3 contains, in parentheses, the time-domain values as well as the short-term and long-term connectedness values. For instance, INE has an own-variance share of 42.67%, so that shocks from the other nine markets account for 57.33% of its forecast-error variance. Of the 42.67%, 33.96% is considered to be short-term own-variance spillover, and 8.71% is considered to be long-term own-variance spillover. This means that all other factors account for 57.33% of the forecast-error variance of the INE. Specifically, WTI, Brent, IPE natural-gas futures, NG natural-gas futures, CSI Wind Power Industry Index, CSI Solar 50 Index, CSI Water Resources Index, CSI nuclear energy and power index, and CSI 300 Green Leading Stocks Index had impacts of 20.48%, 23.09%, 1.94%, 1.50%, 1.83%, 1.80%, 1.71%, 2.45%, and 2.55%, respectively. Each of these shocks can be decomposed into short-term and long-term spillover effects. To sum up, this study finds that the impact of INE crude-oil futures on the market is 18.05% and affected by 57.33% of the market, indicating that it is a net shock recipient (−39.29%). China’s contract has a much shorter trading history than its international counterparts, and its futures market is still developing relative to those in the United Kingdom and the United States. It is therefore unsurprising that INE acts as a net receiver of system-wide shocks. Looking at the mean TCI, we find that the short-term dynamic effect (48.78%) is more than four times larger than the long-term spillover effect (10.65%). Because these figures capture only period-average linkages and can therefore conceal effects that are specific to particular dates or that evolve over the sample, we now turn to the time-varying connectedness plots.

4.1.2. Median Dynamic Total Connectedness

Figure 2 shows the results of the short-term, long-term and total dynamic connectedness of the median. The red areas in Figure 2 represent short-term dynamic connectedness, the green areas represent long-term dynamic connectedness, and the black areas represent overall dynamic connectedness. A high value indicates that the markets in the network are tightly linked, so that a shock to one market is transmitted more readily to the others. It can be seen that short-term dynamic connectedness accounts for a large share of total connectedness, while long-term dynamic connectedness accounts for a small share, which indicates that risk is transmitted rapidly and dissipates quickly.

The total TCI fluctuated at the 55% level between 2018 and the end of 2019. From the end of 2019, the total TCI showed a large fluctuation and reached a peak in March 2020. During this period, long-term dynamic correlations rose sharply, and all correlations have been on the rise since, coinciding with the onset of the COVID-19 pandemic in 2020. The index then declines gradually and only begins to rise again in 2022.

4.1.3. Median Net Total Directional Connectedness Measures

This section focuses on the net directional volatility spillover effects of each series. A net positive spillover indicates that the market is a net sender of shocks to other markets in the network system, while a net negative spillover indicates that the market is a net recipient of shocks in the system. In this section, the total spillovers are broken down into short- and long-term spillovers, denoted by black, red, and green areas, respectively, on the image. As can be seen from the image above, the long-term effect of each series is relatively stable, while the short-term response to market shocks is sensitive. Furthermore, a variable may alternate its function as either a transmitter or receiver depending on its market performance over time. Figure 3 presents the results. Positive values indicate net contributions to shocks within the system, while negative values reflect net absorption.

Focusing on the crude-oil futures market, both WTI and Brent are strong net transmitters of shocks and reach high spillover levels after 2020. Specifically, the price of WTI, both in the short and long term, is a net source of shocks. The situation is similar for Brent crude-oil futures, which remained a net source of long-term shocks throughout the study period but experienced a brief period of net acceptance of short-term shocks between April and June 2020. In contrast, INE is a net receiver throughout the sample. Turning to the natural-gas futures market, both IPE and NG natural-gas futures show the characteristics of being net shock recipients. Looking further at the new-energy indices, the Solar 50 Index and the CSI 300 Green Leading Index generally act as short-term net receivers but long-term net transmitters. By contrast, the Nuclear Index is a strong net transmitter in the short-term band, whereas its average long-term NET is close to zero and slightly negative. Finally, the Wind index transmits shocks at the long horizon and receives them at the short horizon between March 2018 and August 2020; after August 2020 the two roles reverse.

It is worth noting that all series experienced a sharp swing in April 2020. WTI crude-oil futures acted as a net transmitter of shocks, whereas the remaining series acted as net recipients. This episode coincided with the sharp contraction in oil demand during the COVID-19 pandemic and severe constraints on available storage capacity at Cushing, Oklahoma. On 20 April 2020, the May 2020 WTI futures contract fell below zero and settled at −US$37.63 per barrel. This unprecedented negative settlement provides important market context for the pronounced short-term spillover observed during this period.

4.1.4. Median Net Pairwise Directional Connectedness Measures

Finally, we turn to the pairwise dimension to examine the bilateral linkages among the crude-oil futures in greater detail. Consistent with the net directional results, a positive value identifies a market as a net transmitter of shocks, whereas a negative value identifies it as a net receiver. Taking Figure 4 for illustration, the red shading captures the short-term component of the bilateral connectedness and the green shading, the long-term component. Analyzing connectedness at the pairwise level allows for interrelationships between individual market pairs, and their evolution over the sample period, to be traced explicitly.

The relationship between INE crude-oil futures and other crude-oil futures indicates that the latter consistently occupies a dominant position within the propagation mechanism. At the pairwise level, INE is predominantly affected by shocks from the other two crude-oil benchmarks in the short-term band: a disturbance to WTI or Brent produces a net change in INE, whereas the reverse effect is much weaker. Furthermore, the stability of the movement strength of WTI–Brent surpasses that of INE-WTI and INE–Brent. The intensity of WTI–Brent movements appears to be less influenced by various factors in both the short and long term. In contrast, the intensity of China and INE crude-oil futures may fluctuate due to different influences, suggesting that WTI and Brent crude-oil futures prices are nearly synchronized.

4.2. Connectedness Across Quantiles

Section 4.1 examines the frequency decomposition of connectedness at the median quantile, which represents relatively normal market conditions. This section extends the analysis across the conditional return distribution. Specifically, we examine whether overall, short-term, and long-term connectedness differ between lower-tail, median, and upper-tail market states. Therefore, the results in this section complement rather than replace the median-quantile findings. Differences between Section 4.1 and Section 4.2 indicate that the magnitude and direction of spillovers depend jointly on the investment horizon and the prevailing market state.

In this section, we shift the focus to quantile market risk. This perspective is thus more general than the previous one, since we previously fixed the quantile median. To provide an overview of the quantile–frequency-connectedness benefits, we focus on the time-varying market connectedness conditional on the survey quantile, as shown in Figure 5. The heat map can be read as a three-dimensional representation of dynamic total connectedness, with time on the vertical axis and the quantile on the horizontal axis. In addition, based on the median dynamic total connectedness information, we can further extract information about the tail-end connectedness behavior. Darker shading denotes higher connectedness. Connectedness is strongest at both ends of the distribution for returns below the 20th percentile and above the 80th percentile. At this aggregate level, the overall dynamic total connectedness therefore appears broadly symmetric across the two tails. This observation is consistent with the results of Chatziantoniou et al. (2021). The shading on the vertical axis reflects periods of higher uncertainty across quantiles, which may signal widespread economic and financial crises.

The COVID-19 pandemic commenced in 2020, marked by a notable increase in market interconnectedness across all quantiles. Between 2021 and early 2022, connectedness declined across all quantiles. From March to September 2022, connectedness rose, coinciding with the Russia–Ukraine conflict. At the aggregate level, total connectedness appears visually high in both tails, suggesting approximate symmetry between extreme negative- and positive-return states. However, the generally darker shading in the lower quantiles indicates that downside connectedness may be stronger. This possibility is examined more formally using the frequency-specific results below.

Figure 5b presents short-term dynamic total connectedness across time and quantiles. Connectedness generally increases as returns become more extreme. However, the frequency-specific results reveal that this apparent symmetry does not hold for short-term connectedness. The formal symmetry tests show that short-term connectedness is significantly stronger in the lower quantiles, particularly during crisis periods. A formal test of TCI(τ) = TCI(1 − τ) (

Table A6. Short-term band symmetry test: TCI(τ) = TCI(1 − τ).

Quantile Pair (τ/1 − τ)	Full Sample		COVID-19 (January 2020–2021)		Russia–Ukraine (February 2022–2023)
	Diff	p	Diff	p	Diff	p
0.05/0.95	−1.97	0.609	0.63	0.935	15.20	0.033
0.15/0.85	2.27	0.159	−1.87	0.320	13.18	0.000
0.25/0.75	2.63	0.002	3.80	0.000	5.92	0.000
0.35/0.65	1.64	0.000	2.62	0.000	1.70	0.006
0.45/0.55	0.44	0.002	0.98	0.000	0.22	0.401

Notes: Entries are the sample-average difference D = TCI(τ) − TCI(1 − τ) of the short-term (1–5 day) connectedness index, together with the p-value of an intercept-only regression estimated with Newey–West HAC standard errors. A positive value of D indicates that downside (lower-quantile) spillovers exceed upside (upper-quantile) spillovers. Figures in bold are significant at the 5% level.

, Newey–West HAC) confirms that short-term connectedness is asymmetric, with significantly stronger spillovers at the lower (negative-return) quantiles. Over the full sample this downside dominance is significant across the intermediate quantiles (0.25/0.75, 0.35/0.65, 0.45/0.55; D > 0, all p < 0.01). Moreover, the asymmetry is markedly amplified during crisis periods: during the Russia–Ukraine conflict the downside dominance extends to the extreme left tail (0.05/0.95: D = 15.20, p = 0.03; 0.15/0.85: D = 13.18, p < 0.01), whereas during the COVID-19 period it remains concentrated in the intermediate quantiles. These results indicate that short-term risk transmission is systematically stronger during periods of negative returns (crises) than during periods of positive returns, and that this downside asymmetry is an event-driven feature that intensifies under extreme market conditions. At the lower end, there is a strong correlation between the early-2020 and mid-2022 markets. The 2020 peak coincides with the COVID-19 pandemic. The occurrence of large public health emergencies has hindered the normal operation of energy markets and increased investor anxiety. Negative shocks and heightened uncertainty accompanied sharp price swings in energy markets and a sharp rise in market correlations. In 2022, price volatility in the energy market due to the Russia–Ukraine conflict increased market correlation.

Figure 5c shows a different pattern for the long-term band. Connectedness is again elevated in both tails, but during early 2020 and mid-2022 the lower-quantile spillovers are comparatively weak, indicating that the two crises operated mainly through the short-term component.

We examine the net directional connectedness results next. Warmer shading indicates net transmission and cooler shading, net reception. There are two important findings in this paper. First, in the short run, INE crude-oil futures are completely net recipients of shocks, but in the long run, the net acceptance effect weakens, and under extreme market conditions, INE crude-oil futures turn into shock producers. Over time, the net acceptance of IPE gas futures and NG gas futures strengthened. Second, unlike the median-quantile results reported in Section 4.1.3, the quantile-dependent results reveal that the roles of the Solar 50 Index and the CSI 300 Green Leading Index vary across market states. Although both indices tend to be short-term net receivers and long-term net transmitters at the median quantile, their roles may reverse at some tail quantiles, where they become short-term net transmitters and long-term net receivers. By contrast, the Nuclear Index generally remains a net transmitter at both horizons, with substantially stronger net transmission at the long horizon, particularly over the central quantiles.

4.3. Result Comparison with Previous and Existing Studies

The majority of the visual analyses and estimated findings presented in the text (Table 1 and Table 2; Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6) have some degree of validation in the literature already in existence. In the results of our study, the connectedness on realized volatility is asymmetric, with negative information having a stronger influence than positive information. Some literature results show that extreme events will lead to increased risk spillovers of oil price fluctuations (Lee et al., 2017; Lei et al., 2023). This study reached a similar conclusion. Our work revealed that this is also true of natural-gas futures prices. Specifically, INE crude-oil futures and natural-gas futures are identified as net risk receivers. According to their network study, the other asset classes function as net recipients of spillovers, whereas WTI, heating oil, and green bonds behave as net transmitters of risk. On the other hand, the traditional-energy futures and new-energy stock markets are the focus of our research. The results shown in Figure 5 are consistent with the conclusions of some of the literature, such as work by Kamel et al. (2023), which indicates that the COVID-19 pandemic and Russian–Ukrainian conflict have greatly increased economic uncertainty worldwide and significantly impacted the markets for energy futures. Kamel et al. (2023) likewise conclude that the disruption to energy markets caused by the Russia–Ukraine conflict was more severe than that caused by the COVID-19 pandemic.

Different from other articles that use the time-varying frequency-connectedness approach, building on this network, we conduct an out-of-sample link-prediction analysis and extract a rooted directed spanning arborescence to identify the most likely initiator of new spillover links.

4.4. Robustness Checks

We assess robustness in two ways; detailed results are reported in Appendix A. First, we re-estimate total connectedness using the frequency-connectedness approach of Baruník and Křehlík (2018). Second, we apply the quantile-connectedness approach of Chatziantoniou et al. (2021) to the net directional and total connectedness measures. In both cases only the magnitudes change: the time profile and the sign pattern are unchanged, indicating that our findings are robust.

A potential concern is that our results may depend on the specific choice of the rolling-window length and the GFEVD forecast horizon. To address this, we re-estimate the time-frequency connectedness model under a range of alternative specifications—rolling windows of 100, 150 and 200 days, combined with forecast horizons of 15, 20 and 100 steps—and compare the outcomes with our baseline (a 100-day window and a 15-step-ahead horizon). The detailed results are reported in Appendix B;

Table A2. Sensitivity of the total connectedness index (TCI) across frequency bands.

Window	Horizon	TCI (Total)	TCI (Short Term)	TCI (Long Term)
100	15	52.65	43.17	9.49
150	15	50.24	41.07	9.17
200	15	48.84	39.74	9.10
100	20	52.67	45.44	7.23
100	100	52.74	43.25	9.50

reports the average total connectedness index (TCI) decomposed into the total, long-term and short-term frequency bands; while

Table A3. Sensitivity of average net directional connectedness (NET)—total band.

Window	Horizon	INE	WTI	Brent	IPE	NG	Wind	Water	Nuclear	Green	Solar
100	15	−38.89	19.63	19.03	−7.51	−8.42	6.32	1.01	12.54	−0.79	−2.92
150	15	−39.47	18.55	18.67	−5.26	−5.21	5.35	−0.31	12.23	−1.38	−3.18
200	15	−39.56	18.05	18.39	−4.15	−3.54	4.62	−0.90	12.27	−1.58	−3.59
100	20	−38.90	19.72	18.99	−7.50	−8.41	6.29	1.03	12.55	−0.83	−2.94
100	100	−38.89	20.02	18.96	−7.49	−8.46	6.14	1.15	12.61	−0.98	−3.06

,

Table A4. Sensitivity of average net directional connectedness (NET)—long-term band.

Window	Horizon	INE	WTI	Brent	IPE	NG	Wind	Water	Nuclear	Green	Solar
100	15	−10.63	4.66	4.08	−0.77	−0.72	1.83	−0.17	0.97	0.52	0.22
150	15	−10.63	4.29	4.39	−0.65	−0.63	1.76	−0.74	1.22	0.79	0.20
200	15	−10.91	4.15	4.65	−0.51	−0.50	1.56	−0.88	1.38	0.79	0.26
100	20	−8.19	3.59	3.12	−0.57	−0.54	1.43	−0.13	0.72	0.38	0.19
100	100	−10.59	4.65	4.08	−0.77	−0.73	1.83	−0.17	0.98	0.52	0.21

and

Table A5. Sensitivity of average net directional connectedness (NET)—short-term band.

Window	Horizon	INE	WTI	Brent	IPE	NG	Wind	Water	Nuclear	Green	Solar
100	15	−28.26	14.97	14.95	−6.74	−7.70	4.49	1.18	11.57	−1.31	−3.15
150	15	−28.84	14.27	14.28	−4.61	−4.58	3.59	0.43	11.01	−2.17	−3.38
200	15	−28.65	13.90	13.73	−3.65	−3.04	3.06	−0.02	10.90	−2.37	−3.86
100	20	−30.72	16.13	15.86	−6.93	−7.87	4.85	1.16	11.83	−1.20	−3.13
100	100	−28.30	15.37	14.88	−6.71	−7.73	4.31	1.32	11.64	−1.50	−3.28

Notes: Entries are time-series averages of the connectedness measures over the full sample. The baseline specification is a 100-day rolling window with a 15-step-ahead GFEVD, based on a QVAR(1) model estimated across the quantile grid τ ∈ {0.05, 0.15, …, 0.95}. Positive (negative) NET values indicate net transmitters (receivers) of shocks. The short-term band corresponds to a 1–5-day horizon and the long-term band, to horizons of 6 days and beyond.

report the average net directional connectedness (NET) of each market within the total, long-term and short-term bands, respectively.

The results are highly robust across all specifications. The total connectedness index remains within a narrow range of 48.84–52.74, while the short-term band ranges from 39.74 to 45.44 and consistently dominates the long-term band, which ranges from 7.23 to 9.50. This confirms that system-wide risk transmission is driven primarily by higher-frequency, short-horizon shocks, and that its overall magnitude is insensitive to the modelling choices.

This short-horizon dominance is not merely descriptive. Paired tests confirm that short-term connectedness significantly exceeds long-term connectedness at every quantile, with mean differences ranging from approximately 37 to 52 percentage points and block-bootstrap 95% confidence intervals excluding zero throughout, including the extreme tails (Appendix C;

Table A7. Short- versus long-term connectedness: paired horizon tests by symmetric quantile pairs.

Quantile Pair	τ	TCI (Long)	TCI (Short)	Diff. (Short − Long)	Wilcoxon p	Bootstrap 95% CI	Sig.
0.05/0.95	0.05	25.06	72.01	+46.95	<0.001	[37.41, 56.19]	***
	0.95	22.85	73.99	+51.13	<0.001	[44.26, 57.77]	***
0.15/0.85	0.15	18.49	67.70	+49.21	<0.001	[45.54, 53.08]	***
	0.85	16.64	68.66	+52.03	<0.001	[49.19, 54.77]	***
0.25/0.75	0.25	14.91	59.19	+44.27	<0.001	[41.99, 46.51]	***
	0.75	13.39	60.25	+46.86	<0.001	[45.36, 48.33]	***
0.35/0.65	0.35	12.28	51.04	+38.76	<0.001	[37.59, 39.90]	***
	0.65	11.34	52.12	+40.78	<0.001	[39.60, 41.98]	***
0.45/0.55	0.45	10.75	48.08	+37.33	<0.001	[36.32, 38.32]	***
	0.55	10.49	48.66	+38.17	<0.001	[37.10, 39.25]	***

Notes: N = 1219 matched rolling-window observations per quantile. Positive differences indicate that short-term connectedness exceeds long-term connectedness. The dominance of the short-term band is statistically significant at every quantile, including the extreme tails, and is robust to the block-bootstrap correction for serial dependence. *** indicates that the stationary block-bootstrap 95% confidence interval excludes zero.

). At the market level, INE, IPE and NG generally display stronger net-receiving positions in the short-term band over the central quantiles, whereas WTI, Brent, Nuclear and Wind generally display stronger short-term net-transmission positions. These findings provide formal statistical support for the dominance of short-horizon connectedness.

More importantly, the main directional pattern remains broadly stable across the alternative parameter specifications. INE remains the dominant net receiver, with NET values of approximately −39 in the total band, −28 in the short-term band and −11 in the long-term band. WTI and Brent remain net transmitters, whereas IPE and NG remain net receivers. Nuclear is a strong short-term transmitter but has a much weaker long-term transmitting position. Solar and Green generally change from net receivers in the short-term band to net transmitters in the long-term band. Thus, the signs and relative positions of the principal markets are largely robust to changes in the rolling-window length and forecast horizon.

The most noticeable difference concerns the 20-step forecast horizon, under which the short-term TCI rises to 45.44, while the long-term TCI declines to 7.23. Nevertheless, the short-term component remains dominant under every specification, and the principal directional conclusions are unchanged.

5. Link Prediction

The connectedness analysis above characterizes the risk-spillover relationships that have already materialized between the traditional-energy futures markets and the new-energy equity markets. The historical network structure alone, however, cannot indicate which currently weakly connected market relationships are likely to emerge in the future. To make the risk-network analysis more forward-looking, we build a link-prediction task on top of the dynamic quantile time-frequency-connectedness network, using current and historical network states to predict directed risk-spillover relationships that may arise in the future.

Unlike approaches that perform static extrapolation from an averaged network, we take the daily dynamic connectedness matrices as the prediction basis and split the sample into training, validation, and test sets in strict chronological order. The objective of the link-prediction task is not to judge whether existing relationships will persist but to identify directed market pairs that are not yet strongly connected but may become strongly connected within the next five trading days.

5.1. Prediction Task and Experimental Design

Let $w_{ij,t}$ denote the short-term risk-spillover intensity from market i to market j on trading day t. At the median quantile τ = 0.50 and in the short-term frequency band, we classify the directed market relationships in the top 20% of directed connectedness intensity on each trading day as strong linkages and represent their binary connection state as

$$A_{ij,t}=I (w_{ij,t} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{t}\mathrm{o}\mathrm{p} 20\% \mathrm{o}\mathrm{f} \mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y} \mathrm{o}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \mathrm{d}\mathrm{a}\mathrm{y})$$

This paper focuses on the formation of new risk-spillover links. For a directed market pair that is not yet strongly connected on trading day t, that is $A_{ij,t}=0$, current and historical information are used to predict whether the pair forms a strong linkage five trading days later: $y_{ij,t}^{(5)}=A_{ij,t+5},A_{ij,t}=0.$

When $y_{ij,t}^{(5)}=1$, the market pair transitions from a non-strong connection to a strong connection five trading days later; when $y_{ij,t}^{(5)}=0$, it remains not strongly connected five trading days later. Thus, we predict whether a currently non-existing risk-spillover relationship will emerge in the future, rather than whether an existing linkage can persist. The five-day horizon corresponds to a one-week risk-warning window and is long enough to limit the mechanical persistence of the network state that would dominate at shorter horizons. For each fixed directed market pair, we compute the connection state five trading days ahead and retain only the currently non-connected pairs as candidates for new-link prediction.

For each candidate directed market pair (i, j, t), we construct the predictor vector

$${\mathit{x}}_{ij,t}=[{\mathit{e}}_{ij,t},{\mathit{n}}_{i,t},{\mathit{n}}_{j,t},{\mathit{s}}_t,{\mathit{m}}_{ij,t},{\mathit{c}}_{ij}]$$

(23)

where ${\mathit{e}}_{ij,t}$ captures the recent connectedness intensity of the directed relationship and its changes; ${\mathit{n}}_{i,t}$ and ${\mathit{n}}_{j,t}$ respectively describe the network positions of the source and target nodes, including in-degree, out-degree, TO, FROM, and NET; ${\mathit{s}}_t$ denotes the overall connectedness state of the system; ${\mathit{m}}_{ij,t}$ denotes the returns, volatilities, and correlation of the two markets; and ${\mathit{c}}_{ij}$ denotes the market categories to which the source and target nodes belong. These variables respectively reflect the dynamics of the directed relationship itself, the positions of the two nodes within the risk network, the aggregate risk state of the system, and the price state of the markets. The model accordingly estimates the conditional probability that a currently non-connected pair forms a strong linkage five trading days later:

$${\widehat{p}}_{ij}\left(H\right)=P\left(A_{ij,t+5}=1\right|A_{ij,t+5}=0,{\mathit{x}}_{ij,t})$$

(24)

To ensure that model evaluation respects the chronological order of genuine prediction, this paper splits the sample by the prediction target date. Observations whose target date is no later than 31 December 2021 are used for model training, and observations in 2022 are used for hyperparameter selection, early stopping, probability calibration, and classification-threshold determination. Once the model specification is fixed, the model is retrained on all observations with target dates up to the end of 2022, and the 2023 network is used as an independent out-of-sample test set. As a result, the 2023 link-formation outcomes take no part in model selection or parameter tuning.

Because new links account for only a small fraction of all candidate relationships, ordinary classification accuracy is easily affected by the large number of negative samples; we therefore evaluate model performance primarily using AUPRC, AUROC, F1, P@10, and the Brier score. AUPRC measures precision across different recall levels and is the primary ranking metric in this paper; AUROC measures the model’s overall ability to rank positive against negative samples; F1 jointly reflects precision and recall at a given classification threshold; P@10 is the fraction of actual new links among the ten candidate edges with the highest predicted probability each trading day; and the Brier score measures the mean squared error between the predicted probabilities and the realized binary outcomes, with lower values indicating more accurate probability forecasts. This paper uses the threshold that maximizes F1 on the 2022 validation set to generate the binary classifications.

Under random ranking, the expected values of both AUPRC and P@10 equal the positive-link prevalence of the test set, namely 0.01836. Accordingly, 0.01836 serves as the random-prediction baseline for both AUPRC and P@10 in this paper.

5.2. Link Prediction with XGBoost and Model Comparison

This paper employs XGBoost to estimate the probability that a currently non-connected pair forms a strong linkage five trading days later (T. Chen & Guestrin, 2016). XGBoost fits a sequence of shallow decision trees, each of which corrects the observations that the preceding trees fit poorly, and aggregates their outputs as a weighted sum. The predictors in this paper span several categories, including edge-weight changes, node network positions, and market states; the predictive signal provided by any single predictor may be weak, and the formation of a new link is more likely to depend on the joint effect of multiple signals. Gradient boosting, which aggregates many weak rules, is therefore well suited to this task.

For a candidate directed market pair (i, j, t), the predicted probability is expressed as

$${\widehat{p}}_{ij,t}={\displaystyle \frac{1}{1+exp{[-{\sum }_{m=1}^M{\eta f}_m({\mathit{x}}_{ij,t})]}^{\prime }}}$$

(25)

where $f_m(·)$ denotes the m-th decision tree, η is the learning rate, M is the number of trees, and ${\widehat{p}}_{ij,t}$ is the probability that the market pair forms a strong linkage five trading days later. Given the scarcity of new-link samples, the model up-weights positive samples during training and controls model complexity through shallow trees and early stopping on the validation set.

To evaluate the predictive performance of XGBoost, we compare it with logistic regression, random forest, and DirectedGAE. Logistic regression serves as a traditional statistical-learning benchmark; random forest represents an ensemble method based on averaging independent decision trees; and DirectedGAE performs information propagation and node-representation learning over the full directed network. Because the network in this paper has few nodes and the input variables already incorporate edge-weight histories, node positions, and the system connectedness state, it is not clear a priori whether complex graph-representation learning offers a distinct advantage over traditional machine-learning methods; a comparison through a unified out-of-sample experiment is therefore required.

Table 4. Out-of-sample link-prediction performance (2023 holdout).

Model	Seeds	AUPRC (Mean)	AUPRC (SD)	AUROC	F1	P@10	Brier
XGBoost	10	0.4227	0.0072	0.9645	0.4788	0.1213	0.01303
DirectedGAE	10	0.4244	0.0110	0.9585	0.4548	0.1191	0.01334
Logistic	10	0.4130	0.0000	0.9641	0.4448	0.1185	0.01347
Random Forest	10	0.3768	0.0090	0.9619	0.4692	0.1200	0.01333

Notes: All metrics are averaged over 10 random seeds, and the standard deviation (SD) is reported for AUPRC. Given the strong class imbalance in the 2023 holdout (positive-link prevalence = 0.01836), AUPRC is the primary criterion, and its random baseline equals this prevalence. Higher values are better for AUPRC, AUROC, F1, and P@10, whereas lower values are better for the Brier score. Models are ordered by mean AUPRC.

shows that the AUPRC of all four models is markedly above the random baseline of 0.01836, indicating that the network and market features constructed in this paper have good predictive power for new links. DirectedGAE attains the highest AUPRC, at 0.4244, with XGBoost close behind at 0.4227, a difference of only 0.0017. At the same time, XGBoost achieves the best results on AUROC, F1, P@10, and the Brier score, and its across-seed standard deviation of AUPRC (0.0072) is also lower than that of DirectedGAE (0.0110). Logistic regression reaches an AUPRC of 0.4130, close to the two more complex models, further indicating that the predictors constructed in this paper already contain sufficient information on link formation; random forest, with an AUPRC of 0.3768, is relatively weaker overall.

Weighing ranking ability, classification results, identification of key candidate edges, probability accuracy, and across-seed stability, we ultimately adopt XGBoost to generate the subsequent predicted network. It should be noted that the marginal advantage of DirectedGAE in AUPRC suggests that complex graph-representation learning has some potential, but under the small-scale network and the existing structured features of this paper, its overall advantage remains unclear.

5.3. The Predicted Network and the Rooted Directed Spanning Tree

Having selected XGBoost as the final prediction model, we construct the predicted network from the new-link probabilities it generates for the 2023 test sample. First, the predicted probabilities for the same directed market pair across trading days and random seeds are averaged to obtain the mean new-link formation probability from market i to market j:

$${\overline{p}}_{ij}={\displaystyle \frac{1}{S\left|{\mathcal{T}}_{ij}\right|}}{\displaystyle \sum _{s=1}^S}\sum _{t\in {\mathcal{T}}_{ij}}{\widehat{p}}_{ij.t}^{(s)}$$

(26)

where S is the number of random seeds, and ${\mathcal{T}}_{ij}$ is the set of dates on which the pair (i, j) appears as a candidate non-connection during the test period. The full set of ${\overline{p}}_{ij}$ constitutes the XGBoost predicted network, in which a larger edge weight indicates a higher average probability that the directed market pair forms a new connection five trading days later.

On this basis, the predicted out-score and predicted in-score of node i are computed as

$$S_i^{out}=\sum _{i\ne j}{\overline{p}}_{ij},S_i^{in}=\sum _{j\ne i}{\overline{p}}_{ji}$$

(27)

and the predicted net-transmission score of a node is defined as ${NET}_i^p=S_i^{out}-S_i^{in}$, where $S_i^{out}$ measures node i’s overall propensity to form new links toward other markets, and $S_i^{in}$ measures other markets’ overall propensity to form new links toward node i. When ${NET}_i^p>0$, the node acts more as a transmitter of new links in the predicted network; when ${NET}_i^p<0$, it acts more as a receiver. These metrics are node-level aggregations of the XGBoost predicted probabilities and are not equivalent to the TO, FROM, and NET computed earlier from the realized connectedness matrices.

The node scores show that Solar has the highest predicted net-transmission score, at 0.6758, and it is therefore selected as the root of the directed spanning tree. The out-, in-, and NET scores of the nodes are reported in

Table 5. Directed node scores in the predicted-link network (XGBoost).

Market	S_out	S_in	NET_P	Root
Solar	1.2677	0.5920	+0.6758	Yes
Nuclear	0.8130	0.4281	+0.3849	No
WTI	0.3882	0.0379	+0.3503	No
Water	0.6098	0.3091	+0.3007	No
Brent	0.3192	0.0470	+0.2722	No
NG	0.0805	0.0769	+0.0035	No
IPE	0.0299	0.0836	−0.0536	No
Wind	0.6063	0.8252	−0.2189	No
INE	0.0453	0.6782	−0.6329	No
Green	0.7599	1.8420	−1.0821	No

Notes: The scores are computed on the predicted-link (out-of-sample) network produced by the XGBoost model. S_out and S_in aggregate the predicted probabilities of new links originating from and pointing to each market, respectively, and NET_P = S_out − S_in measures a market’s net role as a transmitter (NET_P > 0) or receiver (NET_P < 0) of newly forming links. Solar has the highest NET_P and is therefore selected as the root of the rooted directed spanning tree. Markets are ordered by NET_P.

.

The predicted net-transmission scores reveal a prospective asymmetry that both complements and cross-validates the results for the realized network. Solar has the highest predicted net-transmission score (${NET}^p=+0.676$) and is the market most likely to actively initiate new short-term spillover links over the coming week, followed by Nuclear (+0.385), WTI (+0.350), Water (+0.301), and Brent (+0.272); natural gas (NG) is close to neutral (+0.004). At the other end are Green (−1.082) and the Shanghai INE crude-oil futures (−0.633), which are the principal receivers of future new links, with Wind (−0.219) and IPE (−0.054) next. Two points deserve emphasis. First, the receiver roles of INE and IPE in the predicted network are directionally consistent with their net-receiver roles in the realized connectedness network (Section 4), indicating that the two approaches point in the same direction. Second, and in contrast, Solar appears mainly as a net receiver in the realized network but becomes the principal initiator of new links in the predicted network. This contrast is the key prospective finding of this paper: it indicates that, conditional on the structural persistence of the sample period, the risk topology may be evolving from one in which new-energy equities mainly absorb shocks toward one in which Solar generates new transmission channels, consistent with the high policy sensitivity, rapid capacity expansion, and elevated idiosyncratic volatility of the photovoltaic market. It should be noted that these scores summarize associative tendencies at the level of predicted probabilities and should not be interpreted as causal.

The full predicted network contains a large number of directed relationships. To extract the principal predicted paths that connect all markets, we further construct a rooted directed minimum spanning tree. First, the average predicted probabilities are converted into edge costs: $c_{ij}=-log{\overline{p}}_{ij}$.

Because the negative logarithm is monotonically decreasing, a higher predicted probability corresponds to a lower edge cost. Then, taking Solar—the node with the highest predicted net-transmission score—as the root, the rooted directed minimum-cost spanning tree is solved:

$$\mathcal{T}=arg{min}_{\mathcal{T}}\sum _{(i,j)\in \mathcal{T}}{c}_{ij}$$

(28)

where the root has in-degree 0, every other node has in-degree 1, and the root can reach all remaining nodes along directed paths. The tree contains 10 nodes and 9 directed edges, and it extracts, from the full predicted network, the minimum-cost connection skeleton covering all markets. The edges of the tree and their corresponding predicted probabilities are reported in

Table 6. Edges of the rooted directed spanning tree (XGBoost; root = Solar).

Source → Target	Probability	Cost (−log p)	Level
Solar → Wind	0.4766	0.741	1
Solar → Nuclear	0.2390	1.431	1
Solar → NG	0.0236	3.748	1
Solar → Brent	0.0141	4.259	1
Solar → WTI	0.0089	4.724	1
Nuclear → Green	0.5289	0.637	2
WTI → INE	0.3479	1.056	2
Wind → Water	0.1409	1.960	2
NG → IPE	0.0358	3.329	2

Notes: The tree is the rooted directed minimum spanning arborescence (Chu–Liu–Edmonds) computed on the predicted-link network, with Solar—the market with the highest NET_P—fixed as the root. Each directed edge represents the most economical predicted transmission path; the edge cost is defined as −log(predicted probability), so that higher predicted probabilities correspond to lower costs and are retained preferentially. “Level” denotes the distance from the root (Level 1 = Solar’s direct branches; Level 2 = their children). Edges are ordered by level and, within each level, by predicted probability.

.

As the root of the spanning tree, Solar forms five direct branches (to Wind, Nuclear, WTI, NG, and Brent), of which the branches through Wind, Nuclear, WTI, and NG connect further to Water, Green, INE, and IPE, while Brent is directly attached to the root as a leaf. This structure is shown in Figure 7.

The rooted directed spanning tree distills the predicted network into a minimum-cost directed skeleton. Solar occupies the central position, with direct links to Wind, Nuclear, WTI, natural gas, and Brent. Wind and Nuclear further connect to Water and Green, respectively, while WTI and natural gas extend to INE and IPE, respectively. This structure reveals that within-new-energy and cross-market spillover paths coexist at the first level of the predicted network, highlighting Solar’s potential role in linking new-energy equities with traditional-energy futures. The Solar–Brent edge has a relatively low predicted probability (p = 0.014), suggesting that its inclusion is mainly associated with the spanning-tree requirement of connecting all markets.

5.4. Sensitivity Analysis of XGBoost Parameters

To examine whether the prediction results depend on a particular parameter configuration, we conduct a one-factor-at-a-time sensitivity analysis, adjusting the learning rate, the maximum tree depth, the minimum child weight, and the positive-sample weight in turn while holding the other parameters fixed. Each configuration is retrained under 10 random seeds, and the results are reported in

Table A9. Parameter sensitivity of the XGBoost link-prediction model.

Parameter	Value	AUPRC (Mean ± SD)	AUROC (Mean ± SD)	F1 (Mean ± SD)	P@10 (Mean ± SD)	Brier (Mean ± SD)	Best Round (Mean ± SD)	Val. AUPRC
learning_rate	0.03	0.4247 ± 0.0050	0.9647 ± 0.0001	0.4756 ± 0.0085	0.1210 ± 0.0004	0.01299 ± 0.00006	180.0 ± 25.2	0.3964
learning_rate	0.05 †	0.4227 ± 0.0072	0.9645 ± 0.0004	0.4788 ± 0.0065	0.1213 ± 0.0005	0.01303 ± 0.00010	104.3 ± 26.2	0.3982
learning_rate	0.10	0.4253 ± 0.0065	0.9647 ± 0.0006	0.4748 ± 0.0152	0.1211 ± 0.0008	0.01300 ± 0.00012	62.3 ± 14.0	0.3957
max_depth	2 †	0.4227 ± 0.0072	0.9645 ± 0.0004	0.4788 ± 0.0065	0.1213 ± 0.0005	0.01303 ± 0.00010	104.3 ± 26.2	0.3982
max_depth	4	0.4086 ± 0.0088	0.9630 ± 0.0011	0.4565 ± 0.0146	0.1192 ± 0.0008	0.01320 ± 0.00016	89.2 ± 25.9	0.3775
max_depth	6	0.3683 ± 0.0097	0.9601 ± 0.0008	0.4014 ± 0.0175	0.1191 ± 0.0008	0.01381 ± 0.00013	37.4 ± 10.9	0.3712
minchildweight	1 †	0.4227 ± 0.0072	0.9645 ± 0.0004	0.4788 ± 0.0065	0.1213 ± 0.0005	0.01303 ± 0.00010	104.3 ± 26.2	0.3982
minchildweight	5	0.4229 ± 0.0073	0.9645 ± 0.0004	0.4769 ± 0.0112	0.1212 ± 0.0004	0.01304 ± 0.00012	100.0 ± 23.1	0.3967
minchildweight	10	0.4241 ± 0.0047	0.9646 ± 0.0003	0.4790 ± 0.0141	0.1212 ± 0.0006	0.01302 ± 0.00008	100.6 ± 18.5	0.3985
scaleposweight multiplier	0.5	0.4282 ± 0.0084	0.9651 ± 0.0008	0.4753 ± 0.0105	0.1215 ± 0.0009	0.01302 ± 0.00013	102.2 ± 31.5	0.3999
scaleposweight multiplier	1.0 †	0.4227 ± 0.0072	0.9645 ± 0.0004	0.4788 ± 0.0065	0.1213 ± 0.0005	0.01303 ± 0.00010	104.3 ± 26.2	0.3982
scaleposweight multiplier	1.5	0.4188 ± 0.0045	0.9642 ± 0.0003	0.4681 ± 0.0074	0.1210 ± 0.0006	0.01310 ± 0.00007	90.9 ± 14.3	0.3891

Notes: Each panel varies one hyperparameter around its baseline value while holding the others fixed at the baseline configuration; † marks the baseline value (learning_rate = 0.05, max_depth = 2, min_child_weight = 1, scale_pos_weight multiplier = 1.0). All statistics are computed and reported over n = 10 random seeds.

in Appendix D.

The results indicate that the model is stable with respect to changes in the learning rate and the minimum child weight. For learning rates between 0.03 and 0.10, AUPRC remains between 0.4227 and 0.4253; raising the minimum child weight from 1 to 10 likewise produces no material change in any metric. Adjusting the positive-sample weight introduces small fluctuations but does not alter the model’s overall predictive performance.

By contrast, the maximum tree depth has a more pronounced effect on the results. As the tree depth increases from 2 to 4 and 6, AUPRC declines from 0.4227 to 0.4086 and 0.3683, respectively, and F1 and the Brier score deteriorate accordingly. This indicates that deeper trees are more prone to fitting local fluctuations in the training sample, whereas the shallow tree with depth 2 offers better out-of-sample generalization. Overall, apart from tree depth, XGBoost is robust to moderate adjustments of the other parameters, indicating that the main results of this paper are not driven by a particular parameter configuration.

6. Conclusions

Using a quantile time-frequency connectedness framework, this study examines risk transmission among three crude-oil futures, two natural-gas futures, and five Chinese new-energy sector indices. Total connectedness varies over time, is concentrated in the short-term frequency band, and rises sharply during crisis episodes. Asymmetry across quantiles is also primarily concentrated in the short-term band. INE crude-oil futures and IPE natural-gas futures are identified as net risk receivers. Under extreme market conditions, Brent emerges as the dominant net transmitter in the network. As the investment horizon lengthens, the Solar sector shifts from a net risk receiver to a net risk transmitter. By contrast, the Nuclear sector is a strong net transmitter in the short-term band, whereas its average long-term net position is substantially weaker and close to neutral.

Furthermore, an out-of-sample link-prediction exercise, validated over an independent test period in 2023, identifies Solar as the market with the highest predicted net transmission score and as the root of the directed minimum spanning arborescence (Chu & Liu, 1965; Edmonds, 1967). The selected XGBoost model achieves an AUPRC of 0.4227, compared with a random baseline of 0.01836. These results identify Solar as the market most likely to initiate new short-term risk spillover links. Notably, although Solar acts primarily as a net receiver in the realized network, it emerges as the leading initiator of newly forming links in the predicted network. This finding characterizes the aggregate predicted network over the holdout period and represents a prospective, model-implied association rather than evidence of a causal relationship.

Taken together, the findings show that average system-wide connectedness is dominated by the short-horizon component. The substantially higher short-term TCI suggests that shocks are transmitted primarily within the 1–5-day horizon. Connectedness also increases during the COVID-19 pandemic and the Russia–Ukraine conflict, indicating heightened short-horizon risk transmission during major crisis episodes. The directional results identify INE, IPE, and NG primarily as net receivers, whereas WTI and Brent generally act as net transmitters, particularly under central market conditions. Among the new-energy sectors, Solar and Green generally shift from net receivers in the short-term band to net transmitters in the long-term band, whereas Nuclear exhibits its strongest net transmission role over the short horizon. The predicted network adds a prospective dimension to the analysis: Solar shifts from a realized short-term net receiver to the leading predicted initiator of newly forming links. These findings capture directional associations and predictive relationships rather than causal mechanisms.

The empirical findings and their interpretations provide the basis for the risk management and policy considerations discussed below.

6.1. Policy Implications

Because the connectedness and link-prediction results capture directional associations and predictive relationships rather than causal mechanisms, the following discussion should be interpreted as risk-monitoring, portfolio-management, and hedging considerations conditional on the estimated network. These considerations are not intended as causal policy prescriptions or point-in-time forecasts. They are particularly relevant in the context of the COVID-19 pandemic, the Russia–Ukraine conflict, and the associated disruptions to financial markets.

Policymakers may wish to pay closer attention to the extreme risks arising from large fluctuations in energy futures prices, given the potential for shocks originating in newly established energy futures markets to spill over to the broader capital market. During periods of extreme market volatility, closer short-term monitoring of the risks associated with WTI and Brent crude-oil futures may be warranted. Risk-management attention and, where appropriate, economic support could be directed toward the markets identified by the network as net risk receivers, including INE crude-oil futures, IPE natural-gas futures, and the Water Conservancy sector. Over longer horizons, attention should focus on Brent crude-oil futures and the Solar sector, which tend to act as net transmitters, as well as on natural-gas futures markets, which generally remain net receivers. In parallel, promoting energy conservation and emission reduction, gradually developing alternative energy sources, advancing new-energy technologies, reducing dependence on conventional energy, and mitigating the effects of international energy price volatility remain appropriate long-term policy directions.

New-energy enterprises, particularly those in the wind and solar sectors, may benefit from remaining alert to fluctuations in domestic and international energy prices. Comprehensive planning and risk management, including the use of derivatives such as futures for hedging, could help mitigate their exposure to such fluctuations. In particular, enterprises may wish to closely monitor fluctuations in Brent crude-oil futures prices and incorporate government support policies for the new-energy industry into their sustainable development strategies.

Investors may benefit from understanding patterns of risk transmission across energy markets when managing investment risk, particularly during disruptions to the global energy supply. New-energy stocks could be evaluated in conjunction with trends in international energy prices. During periods of substantial market volatility, investors may wish to account for the associations between Brent crude-oil futures and the solar and hydro sectors. Investors should also consider the horizon-dependent roles of the Solar 50 Index and the Green Leading Index, which generally act as net receivers over the short term but as net transmitters over the long term. Short-horizon investors should pay particular attention to the Nuclear sector because of its pronounced short-term net transmission role. More generally, investors are encouraged to evaluate the determinants of stock prices from multiple perspectives and adopt risk-management strategies appropriate to their investment horizons.

6.2. Limitations and Future Work

Despite these findings, this study has several limitations, particularly its relatively short sample period. The data covers the period from 26 March 2018 to 28 December 2023. A sample of this length may assign a disproportionate weight to the effects of two exceptional events—the COVID-19 pandemic and the Russia–Ukraine conflict—while potentially overlooking other important factors that may emerge over a longer period.

In addition, although this study evaluates link prediction over an independent out-of-sample period and benchmarks several models, including logistic regression, random forest, XGBoost, and a directed graph autoencoder, the analysis is still based on a relatively small set of ten markets and a limited feature space. Future research could incorporate richer node attributes, larger cross-market panels, and more expressive graph-learning architectures. These extensions may further improve the accuracy and reliability of the predictions and provide more robust evidence to support policy development.