The Dynamic Cointegration Relationship between International Crude Oil, Natural Gas, and Coal Price (2024)

1. Introduction

The acceleration of globalization has increasingly turned the international energy market into a significant driver of world economic development. Energy security, environmental protection, and sustainable development have emerged as common concerns within the international community. In recent years, global investment in clean energy rose from USD 1.074 trillion in 2015 to USD 1.740 trillion in 2023, whereas investment in fossil fuels declined from USD 1.319 trillion in 2015 to USD 1.050 trillion in 2023. The international energy market covers various fields such as oil, natural gas, coal, nuclear energy, and renewable energy. Notably, the oil and natural gas markets dominate, while the coal market is gradually diminishing, and the nuclear energy and renewable energy markets have experienced rapid growth. Global primary energy consumption reached 604.04 exajoules in 2022, an increase of 1.1% from the previous year, which is about 3% higher than the pre-pandemic level in 2019. Renewable energy (excluding hydropower) comprised 7.5% of primary energy consumption in 2022, representing a near 1% increase from 2021. Meanwhile, fossil fuels accounted for 81.8% of primary energy consumption in 2022, with oil, natural gas, and coal constituting approximately 31.57%, 23.49%, and 26.73%, respectively. This shows that fossil fuels have remained an important energy source for world economic development in recent years. It is worth noting that, as the cleanest fossil fuel, natural gas use is becoming more and more common in world primary energy [1]. However, since 2020, challenges such as the global COVID-19 pandemic, geopolitical conflicts, and energy crises in regions including Europe have led to renewed concerns about energy security. Ensuring energy supply has become a priority for many nations. During the pandemic, major oil-supplying countries such as Saudi Arabia and Russia did not immediately reach production reduction agreements, and even intended to further expand production to seize the international energy market at low prices. This, coupled with already high oil inventories at the pandemic’s outset, resulted in a supply surplus [2]. The ongoing Russia–Ukraine conflict has had a significant impact on energy markets in the EU and globally, with different emergency policy measures being implemented at the EU and national levels to address high energy prices [3]. Most countries give priority to avoiding the use of natural gas from Russia, even if the actions taken result in higher emissions than would be created by natural gas alternatives [4]. However, it is difficult to immediately decouple the EU’s energy imports from Russia [5]. The Russia–Ukraine conflict exacerbated the global energy crisis, affecting the pattern of international energy trade [6]; also, the sanctions imposed on Russia by the United States and Western countries severely damaged global energy exports, reshaping global interests and political patterns at multiple levels [7]. Both crises have had a significant impact on the energy market [8].

Under the combined influence of multiple factors such as the COVID-19 pandemic, the Russia–Ukraine conflict, great power competition, and the transition to a low-carbon economy, the international energy landscape has been undergoing rapid transformation. Notably, there is a growing trend towards “politicization” and “bloc formation” in energy cooperation. With the widespread development of anti-globalization sentiment and the interplay of old and new factors in the global political and economic landscape, both long-term trends and short-term phenomena are affecting energy development, leading to increasing complexity in global energy security. Energy security is being impacted by various economic and political events, tensions between nations, social unrest, and sanctions [9]. In the future, the concept of energy security will continue to evolve, and its scope will expand further. Energy cooperation and the transition to a low-carbon economy will progress amidst setbacks and repetitions, presenting both challenges and opportunities. Looking ahead beyond 2024, the geopolitical situation is expected to remain complex and severe. The macroeconomy faces the prospect of slower growth amid high inflation and high interest rates in the United States, while the oil market continues to grapple with macroeconomic weakness and strong willingness among OPEC members to cut production [10]. It is imperative for the international community to promptly identify and manage new risks, balancing energy security with development. Therefore, examining the relationship between prices of international crude oil, natural gas, and coal holds significant importance for nations seeking to ensure their energy security [11].

This article aims to provide an in-depth analysis of the relationship between the prices of international crude oil, natural gas, and coal, as well as a comparative analysis of their cointegration relationships after the COVID-19 pandemic and the Russia–Ukraine conflict in recent years. This study further reveals the operational patterns and pricing mechanisms of the energy market. It not only helps countries better understand the changes in the energy market caused by global events, but also provides crucial reference for countries to formulate scientific energy policies.

2. Literature Review and Hypotheses Development

2.1. Literature Review

The relationship between international energy prices has always been a topic of keen interest in academic circles. The prevalent view is that there is a long-term cointegration relationship between natural gas and crude oil prices. Bachmeier and Griffin (2006) examined the relationship between US crude oil and natural gas prices through the establishment of an error correction model (ECM) and found a long-term positive equilibrium relationship between the two [12]. Mohammadi (2011) utilized the threshold autoregressive model (TAR) and momentum threshold autoregressive model (MTAR) approaches to test the long-term relationship between actual natural gas and crude oil prices in the US energy market from 1976 to 2008, revealing that the long-term relationship between the two was asymmetric [13]. Ramberg and Parsons (2010) used a vector error correction model (VECM) and conditional ECM to investigate the long-term cointegration relationship between natural gas and crude oil, finding that combined cycle power generation technology may have contributed to the cointegration of natural gas and crude oil in 2006 [14]. Erdős (2012) employed the VECM model to study the price relationship between North American natural gas and crude oil and the results indicate that, due to the oversupply of shale gas production in North America and the decrease in equipment that can switch between natural gas and fuel oil production, a “decoupling” occurred in the price relationship between the two in 2009 [15].

Brigida (2014) applied Markov switching cointegration equations and found that there are two states in the long-term cointegration relationship between natural gas and crude oil prices. The first is before the 21st century, when crude oil prices rose and natural gas price increases were lower than expected. The second is from August 2000 to May 2009 when crude oil prices rose, natural gas prices increased more than expected and, afterwards, this relationship returned to its original state [16]. Stavroyiannis (2020) used autoregressive distributed lag cointegration techniques (ARDL) to examine the long-term relationship between Dubai crude oil prices from 1992 to 2018 and US natural gas prices. The experimental results revealed a one-way, causal, long-term relationship between Dubai crude oil prices and US natural gas prices [17]. Li et al. (2017) conducted research on the import prices of natural gas and coal in Europe from 1994 to 2014. A long-term positive relationship persisted between natural gas and coal import prices, with the impact of natural gas import prices on coal import prices being stronger than the opposite relationship [18]. Chiappini et al. (2019) demonstrated that, due to the development of the global natural gas trade, the trend towards the integration of natural gas markets in various regions of the world is gradually increasing. Studying the relationship between international crude oil, natural gas, and coal prices has played a guiding role in the global energy market [19]. Renjin et al. (2020) investigated the relationship between crude oil, natural gas, and coal price indexes through cointegration tests and VAR models. They concluded that there is a positive correlation between crude oil and natural gas prices in the long term, as well as a long-term equilibrium relationship involving crude oil, natural gas, and coal prices [20].

Alzate-Ortega et al. (2024) investigated the impact of supply–demand oil shocks on emerging energy markets, emerging market stocks, and gold, revealing that the volatility spillover effects on emerging markets during the COVID-19 pandemic were unprecedented compared to the 2008 financial crisis [21]. Mothana Obadi et al. (2024) analyzed the developments, events, and turning points experienced in each oil and gas market since the early 1970s, discovering that geopolitical events played an active and prominent role in shaping the trends of oil and gas prices in global markets. Their research found that the indirect impacts of geopolitical events on prices outweighed the direct impacts, with supply factors being the primary driver of oil and gas price increases [22]. Khan Khalid et al. (2023) evaluated the causal relationship between geopolitical risks and energy security from 2004 to 2022, indicating that geopolitical risks significantly impacted energy security across different sub-samples, driven by factors such as economic crises, interstate conflicts, political instability, and terrorism. These factors led to disruptions in energy supply and exacerbated energy insecurity [23]. Yang et al. (2023) found that the Ukraine war evolved from a geopolitical crisis into an energy crisis, with the estimation results of direct and indirect impacts clearly showing spillover effects across various energy sources [24]. Rokicki et al. (2023) assessed the COVID-19 and Ukraine war’s impact on the EU’s reliance on Russian energy imports. Prior to the Ukraine conflict, the EU heavily relied on Russian energy. Post-conflict, oil, oil products, and solid fossil fuel imports dropped sharply due to sanctions, while natural gas restrictions were less severe [25]. Jingyu et al. (2022) investigated the impact of the COVID-19 pandemic on the time–frequency volatility spillovers in the international crude oil market and China’s major energy futures markets. The outbreak of COVID-19 intensified the overall volatility spillover effects in these markets. Policymakers and investors should be cautious about the pandemic’s varying impacts at multiple levels [26].

Scholars are also concerned about the mechanism of price fluctuations of oil, natural gas, and coal in the international market. For example, Han et al. discussed the relationship between international crude oil prices and China’s refined oil prices based on a structural VAR model [27]. Chen et al. studied the asymmetric effect between China’s refined oil prices and international crude oil Prices [28]. Ederington et al. discussed the relationship between petroleum product prices and crude oil prices [29]. As for natural gas price fluctuation, Brown et al. discussed the North American energy system responses to natural gas price shocks [30]. Singh and Kumar conducted statistical modeling of natural gas prices in relation to demand, supply and economic growth indicators [31]. However, discussion about the relationship among prices of oil, gas and coal is rare.

Although previous studies have extensively explored the cointegration relationship between international crude oil, natural gas, and coal, there has been a paucity of research in recent years, rarely dividing the time periods into detailed segments for comparison. This study particularly focuses on three time periods: January 2009 to December 2023, January 2009 to December 2019, and January 2020 to December 2023. Notably, existing research indicates that COVID-19 and the Ukraine–Russia conflict have had significant impacts on the energy sector in recent years [26]. During this period, the world has experienced the COVID-19 pandemic and significant fluctuations in the energy market. By conducting an in-depth study of this specific period, we can more accurately capture the impact of these events on the cointegration relationship in the energy market.

2.2. Hypotheses Development

Based on previous literature studies, we hypothesize that there exists a cointegration relationship between the prices of international crude oil, natural gas, and coal during the period from January 2009 to December 2023, and this cointegration relationship can be expressed through a linear combination. Additionally, the recent COVID-19 pandemic and the Russia–Ukraine conflict may have imposed shocks on the relationship between these three commodities.

3. Methodology and Data

3.1. Methodology

The vector autoregression (VAR) model is a type of model that is similar to simultaneous equations but, unlike simultaneous equations, there are no exogenous variables in the VAR model. Instead, all variables in the model are explained by their own lagged terms, lagged terms of other endogenous variables, and random error terms. In this study, we choose the VAR model to examine the relationship between international crude oil, natural gas, and coal prices. A notable characteristic of the VAR model is that it does not rely on strict economic theory as a basis for modeling, allowing for a significant degree of freedom in adding explanatory variables. The flexibility and extensibility of this model help us more fully understand the dynamic relationships between variables, thereby enhancing the scientific rigor and reliability of empirical research results. Through the application of the VAR model, we can more accurately reveal the interaction and influence mechanisms between international energy prices, providing a solid theoretical foundation for relevant decision-making and policy formulation. The general mathematical expression for the VAR(p) model is presented in Equation (1):

$Y_{t} = C + Θ_{1} Y_{t - 1} + \dots + Θ_{p} Y_{t - p} + ε_{t}$

(1)

Here, $E (ε_{t}) = 0$ , $E (ε_{t} {ε^{'}}_{t}) = \{\begin{matrix} Ω (t = τ) \\ 0 (t \neq τ) \end{matrix}$ . When the random error terms $ε_{t}$ are mutually independent and follow a normal distribution at different times, the expression represents a p-order vector autoregression (VAR) model. The stochastic process that satisfies this model is known as a p-order VAR process and is denoted as VAR(p).

3.2. Data

3.2.1. Variable Selection and Data Processing

This study utilizes the nominal spot price indices for world crude oil, natural gas and coal, published by the International Monetary Fund (IMF), as proxies for the respective spot prices, denoted as OILP, GASP, and COALP (https://www.imf.org/en/Research/commodity-prices, accessed on 1 May 2024). This dataset offers several advantages. Firstly, its monthly frequency provides a detailed and comprehensive overview. Secondly, by using 2016 as the base year, it facilitates an intuitive longitudinal analysis of monthly variations. The indices comprehensively consider crude oil price data from key benchmarks such as Dated Brent, West Texas Intermediate (WTI), and Dubai Fateh; natural gas prices from Europe, Japan, and the United States; and coal prices from Australia and South Africa, ensuring broad representativeness. Furthermore, these nominal spot prices serve as a straightforward reference for importing nations. Recognizing the significant and unique impact of the 2008 global financial crisis on various sectors, including the economy and energy, this study deliberately selects 2009 as the starting point. The research period spans from January 2009 to December 2023, encompassing 15 years with 180 data points. To enhance data stability and mitigate potential issues of collinearity and heteroscedasticity, the time series for OILP, GASP, and COALP have undergone logarithmic transformations. If the b power of a (a > 0, and a ≠ 1) is equal to N, i.e., a^b = N, then the number b is called the logarithm of N with base a, denoted as log_aN = b (where a is called the base of the logarithm and N is called the true number), and this is the logarithm transformation. Logarithmic transformations are a common technique in economic and financial data analysis, particularly when dealing with variables that exhibit exponential growth or significant differences in scale. These transformations preserve the inherent properties and relationships of the data while effectively compressing the variable scales. As a result, the final sequences obtained are LNOILP, LNGASP, and LNCOALP.

3.2.2. Data Description

Figure 1 illustrates the changes in the nominal spot price indices for crude oil, natural gas, and coal from January 2009 to December 2023. As can be seen in the figure, the trends in the prices of crude oil, natural gas, and coal were generally consistent during this period, especially from January 2009 to December 2019, when the trends of crude oil and natural gas were even more aligned. However, starting from 2020, the trends of the three commodities began to diverge, primarily due to recent events such as the COVID-19 pandemic, significant downside risks being met by the global economy, and prominent geopolitical issues like the Russia–Ukraine conflict. Nevertheless, in the long run, the trends among the three commodities remained similar. Further cointegration analysis was conducted for the periods of January 2009–December 2023, January 2009–December 2019, and January 2020–December 2023.

4. Results

4.1. Model Construction

The long-term relationship among crude oil, coal, and natural gas prices was analyzed using the cointegration equation. Based on the previous research hypothesis, we have constructed a linear relationship between international crude oil, natural gas, and coal, and represented the difference between the actual observed values and the predicted values from the cointegration relationship using the residual term ε. This residual term measures the deviation of the actual values from the predicted cointegration relationship. The constructed long-term equilibrium equation is presented in Equation (2):

$L N G A S P + α_{1} L N O I L P + α_{2} L N C O A L P + β_{1} = ε_{1}$

(2)

In the cointegration equation, $α_{i}$ represents the cointegration coefficient, $β_{1}$ is the constant term, and $ε_{1}$ denotes the residual term, while LNGASP, LNOILP, and LNCOALP stand for the natural gas price index, crude oil price index, and coal price index, respectively. The long-term equilibrium equation is then constructed as follows below.

4.2. Cointegration Analysis from January 2009 to December 2023

Firstly, we conducted a cointegration analysis on the international crude oil, natural gas, and coal price indices from January 2009 to December 2023, aiming to explore the long-term cointegration relationship among the three commodities since the onset of the financial crisis.

4.2.1. Stationarity Test of the Sequence

As presented in Table 1, we performed the ADF (augmented Dickey–Fuller) unit root test on the three time series of LNGASP, LNOILP, and LNCOALP. The results indicated that all three series contained unit roots at the 5% and 1% confidence levels, indicating non-stationarity. Consequently, a first-order differencing was applied. After differencing, the resulting series DLNGASP, DLNOILP, and DLNCOAL exhibited stationarity at the 1% significance level with p-values less than 0.05. This suggests that these three differenced series are first-order integrated, enabling cointegration analysis to be conducted on the original series.

4.2.2. VAR Model and Johansen Cointegration Test

(1) Optimal Lag Selection for VAR Model

As shown in Table 2, based on the LR, FPE, and AIC criteria, the optimal lag order we selected was 2, which is indicated by the asterisk (*).

(2) Johansen Cointegration Test

To conduct a cointegration test on the price index sequences of crude oil, natural gas, and coal, the most commonly used cointegration test methods include the EG (Engle–Granger) two-step method and the Johansen maximum likelihood method. However, the results of the EG two-step method rely on the order of variables, and are not suitable for testing the existence of cointegration relationships among multiple variables. Therefore, the Johansen maximum likelihood method was adopted for the cointegration test.

The Johansen test results presented in Table 3 show the statistics for varying numbers of cointegration relationships and their respective 5% critical values. At rank 1, the null hypothesis of the presence of at least one cointegration relationship was not rejected.

4.2.3. Vector Error Correction Model

Through the cointegration test, it is possible to determine whether the relationship among the variable sequences remains stationary over an extended period of time. By constructing a vector error correction model (VECM), we explored the impacts of short-term deviations in the variables on natural gas prices within the context of the long-term equilibrium relationship between crude oil, natural gas, and coal. The results of the cointegration equation are presented in Table 4.

The beta values represent the coefficient estimates of the cointegration equation. Specifically, lngasp, lnoilp, and lncoalp denote the coefficient estimates for natural gas prices, crude oil prices, and coal price indices, respectively. The _ce1 indicates the coefficient estimate for the constant term.

Based on the results of the cointegration equation presented in Table 4, we can identify a long-term equilibrium relationship between natural gas prices (lngasp), crude oil prices (lnoilp), and coal prices (lncoalp) during the examined time period. Specifically, the coefficient estimate for crude oil prices was −1.454857, which is statistically significant (p < 0.000), indicating a significant negative correlation between crude oil prices and natural gas prices. In other words, when crude oil prices rise, natural gas prices tend to fall, and vice versa. However, the coefficient estimate for coal prices was −0.2709987, but it is not statistically significant (p = 0.103), suggesting a weak or insignificant negative correlation between coal prices and natural gas prices. Additionally, the estimated value of the constant term (_cons) was 3.6424.

The cointegration equation representing the long-term equilibrium relationship between LNGASP, LNOILP, and LNCOALP is presented in Equation (3):

$L N G A S P = 1.454857 \times L N O I L P - 3.6424$

(3)

Further AR root testing was here conducted to ensure the stability of the VECM model. As shown in Figure 2, which illustrates the model stability test under the AR root unit circle, it can be seen that the reciprocals of the model’s characteristic roots were all within the unit circle, indicating that the constructed VECM model is stable. Therefore, from January 2009 to December 2023, we can identify a long-term cointegration relationship among the international crude oil and natural gas indices.

4.3. Cointegration Analysis from January 2009 to December 2019

After conducting an in-depth study on the data of international crude oil, natural gas, and coal price indices from January 2009 to December 2023, we found a long-term cointegration relationship between crude oil and natural gas, while the results for coal were not significant. However, upon further analysis of the data it became apparent that, starting from January 2020, the COVID-19 pandemic, coupled with the ongoing Russia–Ukraine war, had a significant impact on the energy markets of the EU and globally, leading to a distinct divergence in the price trends of these three commodities [21]. To more accurately assess the impacts of these events on the relationship between international crude oil, natural gas, and coal markets, we conducted further cointegration analysis for the periods of January 2009 to December 2019 and January 2020 to December 2023, respectively.

4.3.1. Stationarity Test of the Sequence

We tested the stationarity of crude oil (LNGASP1), natural gas (LNOILP1), and coal (LNCOALP1) price indices from 2009–2019. As Table 5 shows, their ADF values exceeded critical values at the 5% and 1% levels, with p-values > 0.05, indicating non-stationarity. After first-order differencing, the new series (DLNGASP1, DLNOILP1, DLNCOALP1) were stationary at the 1% level, confirming first-order integration. This allows for cointegration analysis on the original series.

4.3.2. VAR Model and Johansen Cointegration Test

(1) Optimal Lag Selection for VAR Model

As shown in Table 6, based on the LR, FPE, and AIC criteria, the optimal lag order we selected was 1.

(2) Johansen Cointegration Test

The Johansen maximum likelihood method was used to conduct a cointegration test.

4.3.3. Vector Error Correction Model

By constructing a vector error correction model (VECM), we explored the impacts of short-term changes in various variables on natural gas prices under a long-term equilibrium relationship among crude oil, natural gas, and coal. The results of the cointegration equation are presented in Table 8.

Based on the results presented in Table 8 from the cointegration equation, we can identify a long-term equilibrium relationship between natural gas prices (lngasp), crude oil prices (lnoilp), and coal prices (lncoalp) during the examined time period. Specifically, the coefficient estimate for crude oil prices was −1.79829, which is statistically significant (p < 0.000), indicating a significant negative correlation between crude oil prices and natural gas prices. In other words, when crude oil prices rise, natural gas prices tend to decline, and vice versa. Meanwhile, the coefficient estimate for coal prices was 1.132257, which is also statistically significant (p = 0.001), suggesting a significant positive correlation between coal prices and natural gas prices. Additionally, the estimated value for the constant term (_cons) was −1.364981.

The cointegration equation representing the long-term equilibrium relationship between LNGASP, LNOILP, and LNCOALP is presented in Equation (4):

$L N G A S P = 1.79829 \times L N O I L P - 1.132257 \times L N C O A L P + 1.364981$

(4)

Further AR root testing was conducted to ensure the stability. As shown in Figure 3, which illustrates the model stability test under the AR root unit circle, it can be seen that the reciprocals of the model’s characteristic roots were all within the unit circle, indicating that the constructed VECM model is stable. Therefore, from January 2009 to December 2019, there was a long-term cointegration relationship among the international crude oil, natural gas, and coal price indices.

4.4. Cointegration Analysis from January 2020 to December 2023

Below, we describe the conduction of a cointegration test for January 2020 to December 2023.

4.4.1. Stationarity Test of the Sequence

Tested crude oil (LNGASP2), natural gas (LNOILP2), and coal (LNCOALP2) price indices from 2020–2023. ADF values in Table 9 exceeded critical values at the 5% and 1% levels, with p-values > 0.05, indicating non-stationarity. After first-order differencing, DLNGASP2, DLNOILP2, and DLNCOALP2 were stationary at the 1% level, confirming first-order integration. This allowed for cointegration analysis on the original series.

4.4.2. VAR Model and Johansen Cointegration Test

(1) Optimal Lag Selection for VAR Model

As shown in Table 10, according to the LR, FPE, and AIC criteria, the optimal lag order selected by us was 1.

(2) Johansen Cointegration Test

The Johansen maximum likelihood method was still used to carry out the cointegration test.

The results of the Johansen test in Table 11 display the statistics for different numbers of cointegration relationships and their corresponding 5% critical values. Under the condition of rank 0, the trace statistic was less than the critical value at the 5% significance level, indicating that the null hypothesis of no cointegration relationship could not be rejected. Therefore, during the period from January 2020 to December 2023, there was no cointegration relationship among international crude oil, natural gas, and coal prices.

5. Discussion

In this study, we conducted an in-depth analysis of the price indices of international crude oil, natural gas, and coal across three time periods: from January 2009 to December 2023, from January 2009 to December 2019, and from January 2020 to December 2023. The aim was to explore the potential cointegration relationships among them.

The research findings reveal significant cointegration between the price indices of crude oil and natural gas for the entire study period (January 2009 to December 2023), with a notable negative correlation between crude oil and natural gas prices. However, the cointegration between coal and natural gas price indices was not significant during the same period.

In the analysis of the sub-period (January 2009 to December 2019), a significant cointegration relationship was observed among the price indices of all three energy sources. Specifically, crude oil prices exhibited a significant negative correlation with natural gas prices, while coal prices showed a significant positive correlation with natural gas prices. Nevertheless, in the more recent sub-period (January 2020 to December 2023), this cointegration relationship appeared to be disrupted and no longer significant. The study hypothesizes that this change is primarily attributed to the impact of unexpected major events such as the COVID-19 pandemic and the Russia–Ukraine conflict, which are often referred to as “black swan” events in the academic literature.

Furthermore, it is worth noting that, although there was evidence of cointegration among international crude oil, natural gas, and coal in both the sub-period (January 2009 to December 2019) and the entire study period (January 2009 to December 2023), the cointegration coefficients for crude oil and coal underwent significant changes. In particular, the cointegration relationship between coal and natural gas shifted from a significant positive correlation to non-significance. This transition merits further investigation and discussion.

6. Conclusions

This study has evaluated the cointegration relationships among international crude oil, natural gas, and coal price indices across different timeframes, revealing significant shifts influenced by global events and policy changes. Long-term analysis from 2009 to 2019 indicates a stable cointegration among these energy prices, reflecting a balanced interaction of supply and demand forces, macroeconomic conditions, and geopolitical situations.

However, since 2020, this cointegration has been notably disrupted, primarily due to the COVID-19 pandemic and geopolitical tensions, exemplified by conflicts such as the Russia–Ukraine crisis. These events have introduced unprecedented levels of volatility and uncertainty into the global energy markets, affecting both supply chains and demand patterns.

Additionally, since 2021, many countries have announced ambitious climate goals, committing to significant reductions in greenhouse gas emissions and the adoption of cleaner energy sources. This transition towards a low-carbon future has had profound implications for fossil fuel markets, particularly crude oil and coal, due to their higher carbon emissions. Natural gas, being a relatively cleaner energy source with lower carbon dioxide emissions during combustion, has emerged as a significant factor contributing to the changing cointegration relationship between coal and natural gas in recent years.

In conclusion, the proposed hypothesis of this study has been verified by the results. However, our research findings also reveal that, while unexpected events may temporarily impact the energy markets, disrupting the existing cointegration relationships, in the long run, the markets are able to self-adjust and restore these relationships. This demonstrates the resilience and self-healing capability of the energy markets, providing market participants with confidence that the markets can maintain their fundamental structures and interconnectedness despite uncertainties. This discovery also serves as a significant reference for policymakers. Based on the long-term cointegration of the markets, policymakers can formulate more stable and long-term energy policies. In the face of unexpected events, they can intervene with policies to smooth out market fluctuations without excessive concern for the long-term stability of the markets.

For energy market participants, understanding both short-term fluctuations and long-term stability is crucial. They can devise risk management strategies based on the cointegration relationships in the market, such as making adjustments to portfolios or utilizing financial instruments for hedging purposes after unexpected events, to mitigate risks brought about by market volatility.

In the international energy market, countries need to consider various potential risk factors, including the impact of unexpected events. However, this finding indicates that, although unexpected events may lead to short-term market fluctuations, the fundamental structures and interconnectedness of the energy market remain unchanged in the long term. Therefore, when formulating energy security strategies, countries can focus more on long-term market trends and stability, rather than being distracted by short-term market fluctuations.

7. Limitations and Further Research Directions

While this study has investigated the changes in the cointegration relationship between international crude oil, natural gas, and coal after the COVID-19 pandemic and the Ukraine–Russia conflict, compared to the periods from January 2009 to December 2019 and January 2009 to December 2023, future research could further explore the short-term impact of these two events on the cointegration relationship between these commodities using the time-varying parameter vector autoregression (TVP-VAR) model.

Author Contributions

Writing—original draft, L.C. and K.Z.; Writing—review and editing, L.P. All authors have read and agreed to the published version of the manuscript.

Funding

The authors gratefully acknowledge the funding from the Science and Technology Commission of Shanghai Municipality (Grant No. 23ZR1444300, 21692105000) and National Natural Science Foundation of China (Program NO. 71704110).

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. International crude oil, natural gas, and coal price index, 2009–2023. Source: IMF.

Figure 2. Model stability testing.

Figure 3. AR root testing.

Table 1. Unit root test results for the series.

Time Series	ADF Value	5% Critical Value	1% Critical Value	p-Value	Stationary
LNGASP	−2.171	−2.885	−3.486	0.217	NO
LNOILP	−2.265	−2.885	−3.486	0.1836	NO
LNCOALP	−1.821	−2.885	−3.486	0.3701	NO
DLNGASP	−10.410	−2.885	−3.484	0	YES
DLNOILP	−9.596	−2.885	−3.484	0	YES
DLNCOALP	−9.911	−2.885	−3.484	0	YES

Table 2. Optimal lag selection for VAR model.

Lag	LL	LR	df	p	FPE	AIC	HQIC	SBIC
0	483.991				8.20 × 10⁷	−5.49704	−5.47503	−5.44278
1	506.968	45.954	9	0.000	7.00 × 10⁷	−5.65677	−5.5687 *	−5.43976
2	521.74	29.545 *	9	0.001	6.60 × 10⁷ *	−5.72275 *	−5.5687	−5.34297
3	528.448	13.416	9	0.145	6.70 × 10⁷	−5.69655	−5.47649	−5.15402 *
4	532.469	8.0417	9	0.530	7.10 × 10⁷	−5.63965	−5.35356	−4.93435

Note: LL represents the log-likelihood function; LR stands for the likelihood ratio statistic; FPE is the final prediction error criterion; AIC denotes the Akaike information criterion; HQIC refers to the Hannan–Quinn information criterion; SBIC signifies the Schwarz information criterion; and the asterisk (*) indicates the optimal choice based on the respective information criterion.

Table 3. Johansen cointegration test results.

Maximum Rank	Params	LL	Eigenvalue	Trace Statistic	Critical Value 5%
0	12	511.72659	.	38.1086	29.68
1	17	526.43342	0.15231	8.6949 *	15.41
2	20	528.93144	0.02768	3.6989	3.76
3	21	530.78087	0.02057

Note: The asterisk (*) indicates that the selected number of cointegration relationships is 1, confirming the existence of a cointegration relationship among the three variables.

Table 4. Cointegration equation.

Beta	Coefficient	Std. Err.	z	p > \|z\|	(95% Confidence Interval)
_ce1
lngasp	1	.	.	.	.	.
lnoilp	−1.454857	0.2348158	−6.20	0.000	−1.915087	−0.9946263
lncoalp	−0.2709987	0.1663088	−1.63	0.103	−0.596958	0.0549606
_cons	3.6424	.	.	.	.	.

Table 5. Staionarity test results for the series from 2009–2019.

Time Series	ADF Value	5% Critical Value	1% Critical Value	p-Value	Stationary
LNGASP1	0.177	−2.888	−3.499	0.9709	NO
LNOILP1	−1.963	−2.888	−3.499	0.3031	NO
LNCOALP1	−1.408	−2.888	−3.499	0.5783	NO
DLNGASP1	−7.300	−2.888	−3.499	0	YES
DLNOILP1	−8.760	−2.888	−3.499	0	YES
DLNCOALP1	−8.763	−2.888	−3.499	0	YES

Table 6. Optimal lag selection for VAR model of period 2009–2019.

Lag	LL	LR	df	p	FPE	AIC	HQIC	SBIC
0	523.48				6.3 × 10⁸	−8.06946	−8.04244	−8.00296 *
1	542.727	38.494 *	9	0.000	5.4 × 10⁸ *	−8.22833 *	−8.12024 *	−7.9623
2	549.455	13.456	9	0.143	5.6 × 10⁸	−8.1931	−8.00394	−7.72755
3	552.62	6.3296	9	0.707	6.1 × 10⁸	−8.10264	−7.8324	−7.43756
4	556.26	7.2797	9	0.608	6.6 × 10⁸	−8.01953	−7.66823	−7.15494

Note: Asterisks (*) are used to mark model choices that are considered “optimal” according to some information criterion.

Table 7. Johansen cointegration test results of period 2009–2019.

Maximum Rank	Params	LL	Eigenvalue	Trace Statistic	Critical Value 5%
0	3	525.96213	.	33.9660	29.68
1	8	535.44547	0.13290	14.9994 *	15.41
2	11	540.67894	0.07568	4.5324	3.76
3	12	542.94515	0.03350

Note: The asterisk (*) indicates that the selected number of cointegration relationships is 1, confirming the existence of a cointegration relationship among the three variables.

Table 8. Cointegration equation of period 2009–2019.

Beta	Coefficient	Std. Err.	z	p > \|z\|	(95% Confidence Interval)
_ce1
lngasp	1	.	.	.	.	.
lnoilp	−1.79829	0.278009	−6.47	0.000	−2.343178	−1.253402
lncoalp	1.132257	0.34694	3.26	0.001	−0.4522673	1.812247
_cons	−1.364981	.	.	.	.	.

Table 9. Stationarity test results for the series from 2020–2023.

Time Series	ADF Value	5% Critical Value	1% Critical Value	p-Value	Stationary
LNGASP2	−1.321	−2.938	−3.600	0.6195	NO
LNOILP2	−1.150	−2.938	−3.600	0.6945	NO
LNCOALP2	−1.023	−2.938	−3.600	0.7447	NO
DLNGASP2	−5.504	−2.941	−3.628	0	YES
DLNOILP2	−4.522	−2.941	−3.607	0	YES
DLNCOALP2	−4.906	−2.941	−3.607	0	YES

Table 10. Optimal lag selection for VAR model of period 2020–2023.

Lag	LL	LR	df	p	FPE	AIC	HQIC	SBIC
0	83.2925				4.8 × 10⁶	−3.73453	−3.68922 *	−3.61166 *
1	94.1277	21.67 *	9	0.010	4.4 × 10⁶ *	−3.81989 *	−3.63864	−3.32839
2	99.7402	11.225	9	0.261	5.2 × 10⁶	−3.66233	−3.34515	−2.80221
3	102.939	6.3971	9	0.700	6.9 × 10⁶	−3.3925	−2.93938	−2.16376
4	107.971	10.065	9	0.345	8.6 × 10⁶	−3.20797	−2.61891	−1.61061

Note: The asterisk (*) indicates that the selected number of cointegration relationships is 1, confirming the existence of a cointegration relationship among the three variables.

Table 11. Johansen cointegration test results of period 2020–2023.

Maximum Rank	Params	LL	Eigenvalue	Trace Statistic	Critical Value 5%
0	3	67.851163	.	24.1579 *	29.68
1	8	75.477384	0.27686	8.9055	15.41
2	11	79.22267	0.14732	1.4149	3.76
3	12	79.93013	0.02966

Note: The asterisk (*) indicates that the selected number of cointegration relationships is 1, confirming the existence of a cointegration relationship among the three variables.

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