Time-frequency volatility transmission among energy commodities and financial markets during the COVID-19 pandemic: A Novel TVP-VAR frequency connectedness approach

Abstract

This paper investigates the dynamic volatility spillover among energy commodities and financial markets in pre-and mid-COVID-19 periods by utilizing a novel TVP-VAR frequency connectedness approach and the QMLE-based realized volatility data. Our findings indicate that the volatility spillover is mainly driven by long-term components and prominently time-varying with a remarkable but short-lived surge during the COVID-19 outbreak. We further spot that WTI and NGS are prevailingly transmitting and being exposed to the system volatility simultaneously, especially during the global pandemic, suggesting the energy commodity market becoming more integrated with, more influential and meanwhile vulnerable to global financial markets.

1. Introduction

The consistently growing interconnectedness of drastic volatilities in energy commodities and fluctuations in non-energy commodities and other financial assets attracts much attention of financial investors, policymakers, and academic researchers (Adekoya and Oliyide, 2021; Balcilar et al., 2021; Shah and Dar, 2021; Bagheri et al., 2022; Farid et al., 2022). The return and volatility transmission among energy commodities and global financial assets is significantly strengthened and increasingly complex due to the globalization, technological development and the financialization of commodity markets. It is widely acknowledged that the global market integration and financialization not only result in increased liquidity and ease of trading in energy commodity markets, but also tend to foster speculation and thus increasing market volatilities, which may serve as the channel for the time-varying and asymmetric volatility spillovers across energy commodities and non-commodity markets (Balli et al., 2019; Gong et al., 2021; Ma et al., 2021; Farid et al., 2022; Gong and Xu, 2022), as well as other financial markets (Naeem et al., 2020; Huang et al., 2021; Mensi et al., 2022). For example, Gong et al. (2022) document that the global commodity financialization significantly contributes to stronger pure contagion effects among energy and non-energy commodity markets, highlighting the central role of energy market in volatility transmission, while Gong and Xu (2022) further investigate the asymmetric effect of Geopolitical risk on the volatility spillover between energy commodities and other financial markets. Mensi et al. (2022) provide further evidence on the asymmetric spillover between gold, oil and EU subsector markets under extreme events such as the 2008 GFC and Covid-19, while Wang (2022) addresses the issue from the perspective of market efficiency and spots stronger spillover from more efficient markets (industrial metal and energy) to less efficient ones (EU stock markets).

The COVID-19 pandemic was first declared as a “public health emergency of international concern.” in late January 2020 by World Health Organization (WHO) who later adjusted the risk level of global outbreak to “extremely high” as the epidemic spread drastically around the world causing exceptional global turbulences in late February 2020. The global outbreak of the COVID-19 pandemic and the subsequent large scale quarantine measures and economic activity restrictions not only directly brought about significant shocks to the real economy with worsening demand and production stagnation, but also triggered tremendous volatilities and further risk transmission among financial markets with the contagion of decreased liquidity and pessimistic market expectations.1 The spread of market panic caused by the pandemic may lead to a prominent decrease in market value and an increase in market volatility thus significantly reducing the possibility of risk diversification. (Goodell, 2020; Disli et al., 2021; Le et al., 2021; Umar et al., 2021; Maghyereh and Abdoh, 2022; Wang et al., 2022).

Given the profound and lasting impact of the COVID-19 pandemic on the global economy and financial markets, many recent researches emerge to examine how the global pandemic drives and shapes the risk contagion, or return/volatility spillovers between energy commodities and global financial markets, where one prevailing type of approach is the DY connectedness method utilizing VAR based generalized forecast error variance decomposition (GFEVD) proposed by Diebold and Yilmaz (2012, 2014) and its extended models in the frequency domain (Baruník and Krehlík, 2018), the time-varying domain (Antonakakis et al., 2020) and the quantile domain (Chatziantoniou et al., 2022; Farid et al., 2022).2 In light of return spillovers, for example, Mensi et al. (2021) employ the Diebold and Yilmaz (2012) approach (DY-12 hereafter) to spot that the asymmetry spillover between financial markets and hedging effectiveness are influenced by the global health crisis. Some scholars further apply the TVP-VAR based connectedness index of Antonakakis et al. (2020) and discover that the impact of COVID-19 strengthening cross-market linkages is quite short-lived (Bouri et al., 2021; Lin and Su, 2021), whereas Farid et al. (2022) propose the quantile VAR based connectedness index and show that the return spillover transmission is stronger in the extreme tails. As for the volatility spillover transmission, Ding et al. (2021) utilize the Baruník and Krehlík (2018) time-frequency spillover framework (BK-18 hereafter) to confirm the short-term spillover effect being the major source of risk transmission and significantly stronger during the COVID-19 pandemic. However, Wang et al. (2022) suggest the contrary as the total spillovers are mainly driven by low frequency components.

In this paper, we investigate the complex dynamics of volatility spillover transmission among energy commodities and global financial indicators and this paper may contribute to or complement the existing literature in twofold. Firstly, we utilize the novel TVP-VAR frequency connectedness approach proposed by Chatziantoniou et al. (2021) which efficiently takes advantage of the essence of the previous work of Baruník and Krehlík (2018) and Antonakakis et al. (2020). The TVP-VAR frequency connectedness approach makes it possible for the decomposition of volatility connectedness into short-run and long-run components when taking into account the time-varying coefficient and variance-covariance structure simultaneously without losing observations as no arbitrary rolling-window is employed, and meanwhile there is no need for the concern of outliers or erratic parameters. Secondly, we employ market volatility data of the novel QMLE based realized volatility proposed by Da and Xiu (2021) who exploit high frequency market trade data and make sure the realized volatility uniformly valid over a various noise processes varying with different sample sizes.

This paper is organized as follows. Section 2 shows the preliminary analysis of the realized volatility data.3 Section 3 resents the empirical results, whereas conclusions and implications are presented in Section 4.

2. Data descriptions

2.1. Data

In this paper, we utilize the daily data for the realized volatility (RV) of energy commodities and the other four major asset classes of gold, stocks, bonds and cryptocurrencies. The data set on realized volatilities (RV) is obtained from Da and Xiu (2021) who collect the trade data from Tick Data Inc with the highest frequency available (up to every millisecond) and rinse the data by utilzing the prevalent national best bid and offer (NBBO) after which the realized volatility of nonzero returns for prices is estimated with quasi-maximum likelihood estimates (QMLE) of volatility built on moving-average models MA(q) (Da and Xiu, 2021).4

The realized volatility data set ranges from January 2, 2018, to April 27, 2022, which cover a pre-pandemic period (January 2018 to December 2019) and mid-pandemic period (January 2020 to April 2022). The RV data for COMEX Gold futures, Soybean futures, SP500 futures, Dollar index futures, US 10-year T-note futures and CME bitcoin futures are chosen to proxy for volatilities in the non-energy commodity markets (gold for precious metal commodity and soybean for agriculture commodity), stock market, forex market, bond market, and the cryptocurrency market, respectively. We utilize the light sweet crude oil (WTI) futures and NYMEX natural gas futures to investigate the energy commodity market.

2.2. Preliminary analysis

All the daily realized volatility series are strictly stationary as indicated by the PP test results as shown in Table 1 , which satisfies the demand of TVP-VAR specification. Fig. 1 shows that the Bitcoin market experiences highest volatility throughout the data sample. WTI encounters a prominent but not lasting volatility spike during the outbreak of the global pandemic, while the natural gas market, gold and SP500 present a relatively mild but also short-lived increase in volatility level with the outbreak of the COVID-19.

Table 1.

Descriptive statistics results for the daily RV data.

	Mean	SD	Skewness	Kurtosis	Jarque-Bera	PP
WTI	0.3756	0.2942	4.9163	34.1605	4.3e+04***	−59.41***
NGS	0.4374	0.1989	0.9391	4.0630	187.3***	−65.10***
Gold	0.1335	0.0563	2.4541	14.1193	5940***	−174.94***
Soybean	0.1754	0.0634	1.5548	7.4469	1184***	−367.13***
SP500	0.1114	0.0687	2.9187	19.5601	1.2e+04***	−125.07***
US Bond	0.0406	0.0168	2.0484	10.1706	2742***	−220.43***
US Dollar	0.0528	0.0159	1.4111	7.8682	1273***	−336.05***
BTC	0.5915	0.3305	10.9422	33.165	3291***	−326.30***

Notes: PP are test statistics to test the null hypotheses of unit root. *, **, *** denotes statistical significance at the 10%, 5% and 1% level respectively. PP tests shows that all series of financial market realized volatility are strictly stationary.

Fig. 1.

Daily realized volatility of each market.

3. Empirical results and discussion

3.1. Total averaged volatility connectedness

Table 2 presents the results of the total average connectedness among energy commodities and global financial markets, whereas Tables 3 and 4 presents the short-run (1–5 traded days) and long-run (5-infinite days) components. Firstly, the total averaged connectedness index TACI is 47.02, which implies that on average, 47.02% of the forecast error variance in this network of global financial markets can be attributed to the shock transmission among these markets while the remaining 53% is captured by the idiosyncratic factor in each market. Such significant intensity of volatility transmission during the COVID-19 is largely consistent with Ding et al. (2021) and Wang et al. (2022) who spot that the total volatility transmission among energy and financial markets exceeds 60% with the COVID-19 outbreak and remains above 40% afterwards, highlighting the important role of the global epidemic in cross-market risk contagion. Moreover, from a frequency decomposition perspective, intermarket volatility connectedness is mainly driven by volatility transmission in the long term (37.23%) rather than that in the short term (9.79%). Wang et al. (2022) as well as Zhang and Hamori (2021) also document the long-term pattern of volatility transmission, indicating that the global pandemic may cause a fundamental change in investor expectations and increase the long-term uncertainty and systemic risk.5

Table 2.

Averaged volatility connectedness.

	WTI	NGS	Gold	Soybean	SP500	US Bond	US Dollar	BTC	FROM
WTI	45.3	14.36	9.58	3.16	11.29	7.12	7.39	1.79	54.7
NGS	9.64	56.15	7.62	2.39	8.29	4.43	9.36	2.1	43.84
Gold	9.17	8.9	42.67	2.55	7.88	11.27	14.76	2.79	57.32
Soybean	4.01	3.75	4.21	67.62	4.72	3.88	5.44	6.36	32.38
SP500	10.37	7.08	5.83	2.28	55.21	8.95	8.52	1.75	44.79
US Bond	6.71	4.89	14.67	2.32	11.24	44.96	14.08	1.15	55.05
US Dollar	5.82	5.15	15.59	3.32	12.35	13.32	42.39	2.06	57.61
BTC	3.87	5.96	5.91	4.48	3.44	2.99	3.84	69.52	30.49
TO	49.6	50.07	63.42	20.5	59.21	51.98	63.39	17.99	TACI
NET	−5.09	6.24	6.1	−11.87	14.42	−3.07	5.78	−12.49	47.02

Notes: Results are based on a TVP-VAR model with lag length of order one (BIC) and a 100-step-ahead generalized forecast error variance decomposition.

Table 3.

Averaged volatility connectedness in the short run (1–5 traded days).

	WTI	NGS	Gold	Soybean	SP500	US Bond	US Dollar	BTC	FROM
WTI	7.93	0.32	0.63	0.28	0.83	0.7	0.31	0.13	3.2
NGS	0.33	8.91	0.48	0.43	0.42	0.5	0.62	0.19	2.98
Gold	1.07	0.48	13.2	0.69	1.48	4.14	4.01	0.55	12.42
Soybean	0.63	0.61	1.09	30.42	0.67	1.16	1.31	1.11	6.59
SP500	1.47	0.32	1.48	0.38	13.12	2.22	1.54	0.53	7.94
US Bond	1.55	0.83	5.73	1.08	2.96	18.52	6.41	0.42	18.97
US Dollar	0.62	1.15	7.85	1.23	2.19	7.81	22.08	0.55	21.39
BTC	0.35	0.34	1.03	0.77	1.1	0.61	0.64	30.58	4.84
TO	6.02	4.03	18.3	4.85	9.66	17.15	14.84	3.47	TACI
NET	2.83	1.06	5.88	−1.74	1.71	−1.82	−6.55	−1.36	9.79

Notes: Results are based on a TVP-VAR model based generalized forecast error variance decomposition and its frequency spectral presentation by BK-18 approach.

Table 4.

Averaged volatility connectedness in the long run (5-infinite traded days).

	WTI	NGS	Gold	Soybean	SP500	US Bond	US Dollar	BTC	FROM
WTI	37.37	14.04	8.95	2.88	10.46	6.42	7.08	1.66	51.5
NGS	9.31	47.24	7.14	1.96	7.87	3.93	8.74	1.91	40.86
Gold	8.1	8.42	29.47	1.86	6.4	7.13	10.75	2.24	44.9
Soybean	3.38	3.14	3.12	37.2	4.05	2.72	4.13	5.25	25.79
SP500	8.9	6.76	4.35	1.9	42.09	6.73	6.98	1.22	36.85
US Bond	5.16	4.06	8.94	1.24	8.28	26.44	7.67	0.73	36.08
US Dollar	5.2	4	7.74	2.09	10.16	5.51	20.31	1.51	36.22
BTC	3.52	5.62	4.88	3.71	2.34	2.38	3.2	38.94	25.65
TO	43.58	46.04	45.12	15.65	49.55	34.83	48.55	14.52	TACI
Net	−7.92	5.18	0.22	−10.13	12.71	−1.25	12.33	−11.13	37.23

Notes: Results are based on a TVP-VAR model based generalized forecast error variance decomposition and its frequency spectral presentation by BK-18 approach.

In light of the net total directional connectedness NET of each asset, it is noticeable that, on average, SP500 is the main net-transmitter of volatility, followed by the natural gas market (NGS), gold and US Dollar, while Bitcoin is the main net-recipient of volatility, followed by soybean, WTI and US Bond. With reference to frequency bands, it indicates that volatility transmission related to SP500, NGS and US Dollar is mainly driven by long-run factors while gold presents mainly short-run volatility transmission. Bitcoin, soybean and WTI also present to be mostly affected in the long run. However, it is worth-noted that US Dollar demonstrates heterogeneous volatility transmission characteristic as being a net-recipient of volatility in the short term and a prominent net-transmitter in the long term.

3.2. Total dynamic volatility connectedness

We proceed by investigating dynamic measures of volatility connectedness. Fig. 2 demonstrates not only the overall dynamic evolution of the total average connectedness index TACI (i.e., black-shaded area), but also the decomposition into components in the short-run (i.e., red-shaded area) and the long-run (i.e., green-shaded area). From Fig. 2 we can spot several noticeable characteristics of the total volatility transmission dynamics. Firstly, there is a drastic increase in TACI in late February 2020 when the COVID-19 epidemic spreads drastically around the world, which is unsurprisingly consistent with four consecutive meltdowns in U.S. stock market and the subsequent oil market slump that are believed to cause the swift transmission of market turmoil in global financial markets (Adekoya and Oliyide, 2021; Bouri et al., 2021; Lin and Su, 2021; Wang et al., 2022). Secondly, while the COVID-19 continues to develop with several virus mutations and still poses a severe threat to human health and social economy discovery in the present, there is an apparent downward trend of the total volatility transmission after its peak in March 2020. Such asynchrony is aligned with the observation of Lin and Su (2021) and Wang et al. (2022) whereas the latter provides an explanation that it may be due to the governments’ quick response to the global financial risk but a relatively inactive attitude towards controlling the COVID-19 with the herd immunity principle (except for China though). Thirdly, with respect to the decomposition of the total volatility spillover in the frequency domain, it shows that total spillovers during the outbreak of the global epidemic is driven mainly by the long-term component, which indicates that shocks are processed and transmitted over a longer period and investors’ expectations may be fundamentally changed (Wang et al., 2022). That the total volatility spillover is driven mainly by the low-frequency response to shocks contradicts to the observation of Ding et al. (2021) but is again coincident with that of Wang et al. (2022), which, according to Baruník and Krehlík (2018), may indicate an increase in long term uncertainty and systemic risk. Finally, a noticeable upward trend in the dynamic volatility transmission can be spotted since February 2022, which might be related to the recent Russia-Ukraine military conflict triggering a growing intense atmosphere and consequently market uncertainty around the globe.

Fig. 2.

Dynamic total volatility connectedness.

Notes: Fig. 2 presents the decomposition of total volatility spillovers into frequency up to 1 week ($d=\left(\frac{\pi }{5},\pi \right)$) and more than 1 week ($d=\left(0,\frac{\pi }{5}\right)$) corresponding to short-term (red-shaded area) and long-term (green-shaded area) volatility spillovers, respectively.

3.3. Net total directional volatility connectedness

Fig. 3(a), (b),(c) demonstrates the dynamic evolution of the directional volatility spillover TO others, FROM others and their difference, that is, the net total directional connectedness, respectively. Firstly, Fig. 3(a) shows that spillovers from WTI, SP500 and gold to other markets increase significantly after the outbreak but again swiftly decrease afterwards, and mainly demonstrate as long-term volatility spillovers, while the natural gas market shows a moderately declining trend of volatility transmission to other assets in the mid-COVID 19 period. And Bitcoin presents to have only very faint influence on traditional financial markets but presents to be relatively vulnerable to external market instability, which lay doubts on its role as a safe-haven asset. It is also worth noticing that the volatility transmission from US bond to other markets changes substantially with respect to frequency reference after the outbreak, with the long-run component dominating the risk transmission. Secondly, as all assets including WTI and NGS experience a surge in volatility transmission from other markets, results of volatility spillovers FROM others present largely similar features for each market with US bond and US Dollar being the exception as they present to be affected mainly by the long-run volatility after the outbreak and then gradually swift back to the short-run volatility exposure in a more recent period. Finally, the net directional volatility connectedness is explicitly time-varying. Particularly, WTI is a net-recipient of volatility spillovers before 2019 and transform into a net-transmitter afterwards, especially after the global pandemic outbreak with WTI's prominently increasing volatility spillover to other market, and a similar trend is also spotted in SP500, which is consistent with the pervious observation. On the other hand, the natural gas market mainly plays a role as a moderate net-transmitter before the pandemic and then a prevalent net-recipient during the pandemic. Again, the frequency decomposition results show that the long-term components serve as the major source of the directional volatility spillover, which indicates that the volatility transmission among energy commodities and financial assets tend to be processed over a longer horizon.6

Fig. 3a.

Dynamic total directional volatility connectedness TO others.

Fig. 3b.

Dynamic total directional volatility connectedness FROM others.

Fig. 3c.

Dynamic net total volatility directional connectedness.

3.4. Discussion

Overall, the TVP-VAR frequency connectedness model utilized in this paper provides a comprehensive picture of the volatility transmission among energy commodity markets and financial markets during the global pandemic era. Firstly, our findings indicate that a large source of the average total spillover among energy commodities and major financial markets comes from the network volatility transmission throughout the examined period, which confirms a more integrated global financial system with the speeding trend of energy commodity financialization (Ma et al., 2021), as well as the increasing likely systemic risk contagion due to the global pandemic. Secondly, the total volatility transmission pattern is prominently time-varying, with a remarkable peak reached during the outbreak of COVID-19 and a swift decline afterwards, which might be due to the distinct government attitudes toward pandemic controlling and financial stability (Lin and Su, 2021; Wang et al., 2022). Thirdly, the decomposition of the volatility spillover (both in net terms and in total terms) in the frequency domain, shows that the total spillover is driven mainly by the long-term component throughout the period, which points to the COVID-19 causing a prominent change in market expectations and increasing the long-term uncertainty and systemic risk (Baruník and Krehlík, 2018). Furthermore, we show that whether WTI and NGS are net transmitters or net recipients in the volatility transmission process depend on different periods, with both of them prevailingly transmitting and being exposed to the market volatility simultaneously, especially during the global pandemic, which implies that the energy commodity market is becoming more integrated with global financial markets and tends to be more influential and meanwhile vulnerable to other financial assets.

4. Concluding remarks

This paper empirically investigates the dynamic volatility spillover among energy commodities and financial markets in pre-and mid-COVID-19 periods by utilizing the novel TVP-VAR frequency connectedness approach and the QMLE-MA model based realized volatility data of each market. Our findings indicate that the global pandemic significantly intensifies the cross-market volatility transmission with a prominently time-varying pattern. Moreover, the volatility spillover decomposition in the frequency domain shows a long-term pattern throughout the period, suggesting a prominent change in market expectations as well as a larger possibility of long-term uncertainty and systemic risk. We further spot that the role of energy commodities in volatility transmission depends on different periods, with both of them prevailingly transmitting and being exposed to the market volatility simultaneously, especially during the global pandemic, suggesting the energy commodity market becoming more integrated with, more influential and meanwhile vulnerable to global financial markets.

The role of the volatility transmission between different assets is growingly emphasized in portfolio diversification and risk hedging. With the increasing important role of energy commodity markets in the global financial market, policy makers and investors are suggested to take into account the distinctively time-varying characteristics of the volatility transmission connectedness among energy commodities and various financial assets in the time-frequency domain from a systemwide perspective.

CRediT authorship contribution statement

Jionghao Huang: Conceptualization, Methodology, Investigation, Data curation, Software, Formal analysis, Writing – original draft, Writing – review & editing, Visualization, Validation. Baifan Chen: Investigation, Data curation, Software, Formal analysis, Writing – original draft, Writing – review & editing. Yushi Xu: Investigation, Data curation, Software, Formal analysis, Writing – original draft, Writing – review & editing. Xiaohua Xia: Conceptualization, Methodology, Resources, Formal analysis, Validation, Supervision, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix

A1. Relevant methodology

A1.1. TVP-VAR based connectedness approach

This paper utilizes the novel TVP-VAR frequency connectedness approach proposed by Chatziantoniou et al. (2021) which efficiently takes advantage of the essence of the previous work of Baruník and Krehlík (2018) and Antonakakis et al. (2020).7 In this section, we first give a brief introduction of the TVP-VAR based connectedness approach of Antonakakis et al. (2020) which efficiently integrates the connectedness index of Diebold and Yilmaz (2012) and the TVP-VAR model of Koop and Korobilis (2014). The TVP-VAR(p) can be presented as:

$${\mathit{y}}_t={\mathbf{\Phi }}_{1t}{\mathit{y}}_{t-1}+{\mathbf{\Phi }}_{2t}{\mathit{y}}_{t-2}+\dots +{\mathbf{\Phi }}_{pt}{\mathit{y}}_{t-p}+{ϵ}_t{ϵ}_t\sim N\left(0,{\mathbf{\Sigma }}_t\right)$$ (1)

where y_t and ε_t are N × 1 vectors, Σ _t the N × N time-varying variance-covariance matrix and Φ _it, i = 1, …, p represents the N × N time-varying VAR coefficient. With the matrix lag-polynomial Φ(L) = [I_N − Φ _1t L − … − Φ _pt L^p] and the Wold representation theorem, the stationary TVP-VAR process can be rewritten as a TVP-VMA(∞): x_t =  Ψ(L)ε_t where Φ(L) = [Ψ(L)]⁻¹. As Ψ(L) includes infinite lags, it is approximated by computed Ψ _h at h = 1, …, H horizons (Chatziantoniou et al., 2021).

With the TVP-VMA coefficients Ψ _h, we can compute the generalized forecast error variance decomposition (GFEVD) which can be interpreted as the effect that a shock in variable j has on variable i in terms of its forecast error variance and can be written as:

$$C_{ijt}\left(H\right)=\frac{{\left({\mathbf{\Sigma }}_t\right)}_{jj}^{-1}{\sum }_{h=0}^H{\left({\left({\mathbf{\Psi }}_h{\mathbf{\Sigma }}_t\right)}_{ijt}\right)}^2}{{\sum }_{h=0}^H{\left({\mathbf{\Psi }}_h{\mathbf{\Sigma }}_{\mathit{t}}{\mathbf{\Psi }}_h^{{}^{\prime }}\right)}_{ii}}$$ (2)

$${\tilde{C}}_{ijt}\left(H\right)=\frac{C_{ijt}\left(H\right)}{{\sum }_{k=1}^N{C}_{ijt}\left(H\right)}$$ (3)

where ${\tilde{C}}_{ijt}\left(H\right)$ represents the contribution of the jth variable to the variance of the forecast error of the ith variable at horizon H. With row normalization of ${\tilde{C}}_{ijt}\left(H\right)$ we have $\sum _{i=1}^N{\tilde{C}}_{ijt}\left(H\right)=1$ and $\sum _{j=1}^N\sum _{i=1}^N{\tilde{C}}_{ijt}\left(H\right)=N$ .

With Eqs. (2) and (3) we are able to compute all the connectedness measures including:

Net pairwise directional connectedness:

$$NPD{C}_{ijt}\left(H\right)={\tilde{C}}_{ijt}\left(H\right)-{\tilde{C}}_{jit}\left(H\right)$$ (4)

It means that variable j influences variable i more (less) than vice versa with NPDC_ijt(H) > ( < )0.

Total directional connectedness TO others:

$$T{O}_{it}\left(H\right)=\sum _{i=1,i\ne j}^N{\tilde{C}}_{jit}\left(H\right)$$ (5)

It measures how much of a shock in variable i is transmitted to all other variables j.

Total directional connectedness FROM others:

$$FRO{M}_{it}\left(H\right)=\sum _{j=1,i\ne j}^N{\tilde{C}}_{ijt}\left(H\right)$$ (6)

It measures how much variable i is receiving from shocks in all other variables j.

Net total directional connectedness:

$$NE{T}_{it}\left(H\right)=T{O}_{it}\left(H\right)-FRO{M}_{it}\left(H\right)$$ (7)

It represents the difference between the total directional connectedness TO others TO_it(H) and the total directional connectedness FROM others FROM_it(H), which can be interpreted as the net influence variable i has on the corresponding volatility transmission network. It is regarded as a net transmitter (receiver) of shocks with NET_it(H) > ( < )0 indicating that variable i influences all others j more (less) than being influenced by them.

Total averaged connectedness index:

$$TAC{I}_t\left(H\right)=N^{-1}\sum _{i=1}^{N}T{O}_{it}\left(H\right)=N^{-1}\sum _{i=1}^{N}FRO{M}_{it}\left(H\right)$$ (8)

It depicts the average impact a shock in one variable has on all others, thus measuring the degree of network interconnectedness and the market risk (Chatziantoniou et al., 2021).

A1.2. Connectedness in the frequency domain

Unifying the TVP-VAR connectedness framework with the spectral representation of variance decompositions introduced by the BK-18 model, we can explore the volatility connectedness between variables of interest in the frequency domain. With the frequency response function, $\mathbf{\Psi }\left(e^{-i\omega }\right)=\sum _{h=0}^{\infty }{e}^{-i\omega h}{\mathbf{\Psi }}_h$, where $i=\sqrt{-1}$ and ω represents the frequency to continue with the spectral density of y_t at frequency ω. The spectral density of y_t over ω can be defined as a Fourier transformation of the TVP-VMA(∞):

$${\mathit{S}}_y\left(\omega \right)=\sum _{h=-\infty }^{\infty }E\left({\mathit{y}}_t{\mathit{y}}_{t-h}^{{}^{\prime }}\right)e^{-i\omega h}=\Psi \left(e^{-i\omega h}\right){\mathbf{\Sigma }}_t{\mathbf{\Psi }}^{\prime }\left(e^{+i\omega h}\right)$$ (9)

The frequency GFEVD, as the combination of the spectral density and the GFEVD, therefore, can be computed as:

$$C_{ijt}\left(\omega \right)=\frac{{\left({\mathbf{\Sigma }}_t\right)}_{jj}^{-1}{\left|{\sum }_{h=0}^{\infty }{\left(\mathbf{\Psi }\left(e^{-i\omega h}\right){\mathbf{\Sigma }}_t\right)}_{ijt}\right|}^2}{{\sum }_{h=0}^{\infty }{\left(\mathbf{\Psi }\left(e^{-i\omega h}\right){\mathbf{\Sigma }}_t\mathbf{\Psi }\left(e^{i\omega h}\right)\right)}_{ii}}$$ (10)

$${\tilde{C}}_{ijt}\left(\omega \right)=\frac{C_{ijt}\left(\omega \right)}{{\sum }_{k=1}^N{C}_{ijt}\left(\omega \right)}$$ (11)

We further aggregate all frequencies within a range of interest, ${\tilde{\theta }}_{ijt}\left(d\right)=\underset{a}{\overset{b}{\int }}{\tilde{\theta }}_{ijt}\left(\omega \right)d\omega$, where d = (a, b): a, b ∈ ( − π, π), a < b, and then we can calculate all the frequency connectedness measures that provide information about spillovers in a certain frequency range d:

$$NPD{C}_{ijt}\left(d\right)={\tilde{C}}_{ijt}\left(d\right)-{\tilde{C}}_{jit}\left(d\right)$$ (12)

$$T{O}_{it}\left(d\right)=\sum _{i=1,i\ne j}^N{\tilde{C}}_{jit}\left(d\right)$$ (13)

$$FRO{M}_{it}\left(d\right)=\sum _{j=1,i\ne j}^N{\tilde{C}}_{ijt}\left(d\right)$$ (14)

$$NE{T}_{it}\left(d\right)=T{O}_{it}\left(d\right)-FRO{M}_{it}\left(d\right)$$ (15)

$$TAC{I}_t\left(d\right)=N^{-1}\sum _{i=1}^{N}T{O}_{it}\left(d\right)=N^{-1}\sum _{i=1}^{N}FRO{M}_{it}\left(d\right)$$ (16)

And we have

$$CN\left(H\right)=\sum _{d}CN\left(d\right)$$ (17)

where CN( · ) = [NPDC, TO, FROM, NET, TACI] represent all the connectedness measures discussed above, indicating that the aggregation of all the frequencies of the frequency connectedness measure equals the corresponding connectedness in the time domain.

A2. Net pairwise volatility connectedness

In this section we further focus on the pairwise volatility spillovers between energy commodities and other financial markets in time-frequency domain as presented in Fig. A1. In light of the directional volatility spillover between WTI and other financial assets, WTI receives from the natural gas market on net terms until 2019 and then transmits to the natural gas market after the outbreak of COVID-19, where a similar trend is also spotted in WTI-Gold and WTI-US Dollar volatility nexus. Moreover, WTI seems to dominate the volatility transmission to Bitcoin and the soybean market for most of the time. Meanwhile, WTI transmits to SP 500 and to US Bond (though to a lesser extent) before the pandemic outbreak and then presents to be a net recipient from the two markets afterwards. However, it needs to be pointed out that the net pairwise volatility spillover between WTI and other financial assets reaches a peak as the COVID-19 spreads drastically around March 2020 but soon declines with an exception of the WTI-SP500 nexus. Turning to the directional volatility spillover between the natural gas market and other financial assets, NGS transmits to the gold market on net terms until 2019 and then presents to be a net recipient from the gold market after the outbreak of COVID-19, where a similar reversal is also spotted in the NGS-SP 500 and NGS-US Bond nexus before and after the pandemic outbreak. Although NGS transmits significantly to Bitcoin before the pandemic, it presents only a very moderate degree of volatility transmission during the global epidemic outbreak, and the NGS-Soybean volatility nexus is largely insignificant throughout the mid-COVID 19 period. Overall, however, the net pairwise volatility between energy commodities and major financial markets is short-lived, which again confirms the results of previous analysis.

Fig. A1.

Dynamic net pairwise directional volatility connectedness among energy commodities and financial markets.

Notes: We focus on the first variable on the title of each panel. Explicitly, i − j means that the positive net pairwise directional volatility connectedness indicates net volatility spillover from i to j, suggesting that i has a stronger effect on j than vice versa.

Data availability

• Data will be made available on request.