Simmering tensions on the Russia–Ukraine border and natural gas futures prices: identifying the impact using new hybrid GARCH

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Reviewing the recent studies on market prices for natural gas, Wang et al. (2022) investigated the volatilities of oil rents and natural gas rents.Akcora and Kocaaslan (2023) examined the price bubbles in European natural gas markets.Dastan (2023) investigated the Turkish natural gas market during the supply disruptions that occurred in early 2022.Further, Liu et al. (2023) analyzed the impacts of oil and natural gas prices on China's carbon efficiency.Su et al. (2023) examined whether there were multiple bubbles in the European natural gas market.Mensi et al. (2024) investigated the connectedness between bond, oil, and natural gas prices.Although not focused on natural gas, many studies have argued for the importance of analyzing market prices for energy (e.g., Ju et al. 2014;Tsuji 2018Tsuji , 2020;;van Eyden et al. 2019;Dutta et al. 2021;Puig-Gamero et al. 2021;Jingjian et al. 2023).
Given this and considering the importance of actual natural gas market prices, our current analysis focuses on the impact of the the Russia-Ukraine war on natural gas futures prices.We suggest that in doing such new analysis, it is vital to incorporate fattailed distributions and structural breaks.However, in existing studies, although either fat-tailed distributions (e.g., Geweke 1993;Fan et al. 2008) or structural breaks (e.g., Wang and Moore 2009;Malik 2022) are considered, both components are not incorporated into a single model simultaneously.Therefore, our approach of newly considering and incorporating both in this study fills a research gap left by existing literature.
Hence, we examine natural gas futures volatilities of the period around the Russian invasion of Ukraine, with a particular focus on the impact of simmering tensions on the Russia-Ukraine border on natural gas futures volatilities, especially because at that time, a very large volatility spike was observed.We consider that this spike was due to heightened concerns about the imbalance in natural gas supply and demand, particularly at that time.
Our research questions are as follows."How do we well capture the natural gas futures volatility spike when the tensions simmered on the Russia-Ukraine border by considering both fat-tailed errors and structural breaks as hybrid models?" and "What is the beneficial characteristic of the volatility estimated by our superior hybrid models?"To clarify these matters is the goal of this study.For this purpose, we develop new hybrid generalized autoregressive conditional heteroscedasticity (GARCH) models, which incorporate not only fat-tailed distribution errors but also structural breaks.To the best of our knowledge, no existing study has analyzed the points raised in our research questions mentioned above.
As a result of our careful examinations applying four new hybrid models, i.e., the autoregressive (AR) mean Student's t (T) distribution error and structural break (SB) incorporated GARCH model (hereinafter, the AR-T-SB-GARCH model), the AR-generalized error distribution (GED)-SB-GARCH model (hereinafter, the AR-GED-SB-GARCH model), the AR-T-SB-exponential GARCH (EGARCH) model (hereinafter, the AR-T-SB-EGARCH model), and the AR-GED-SB-EGARCH model, we have uncovered the following findings as new contributions to this topic area.
First, we reveal that it was not when Russia invaded Ukraine in February 2022, but rather when tensions on the Russia-Ukraine border simmered in January 2022, that the Russia-Ukraine war had the most impact on natural gas futures prices.Second, we clarify that our hybrid models, which incorporate both fat-tailed distribution errors-either Student's t distribution or GED errors-and structural breaks, are effective to capture the natural gas futures volatility jump in a timely manner when tensions on the Russia-Ukraine border simmered.
Third, we also reveal that our hybrid modeling approach-incorporating both fattailed errors and structural breaks-is effective not only in GARCH but also in EGARCH modeling.This proves the robustness of our hybrid approach for modeling natural gas futures volatilities.Fourth, we further uncover that the volatility estimates from our hybrid models of the AR-T-SB-GARCH and the AR-GED-SB-GARCH models have predictive power for the volatilities of nonhybrid versions of the AR-T-GARCH and AR-GED-GARCH models.We consider that because of this characteristic, our two hybrid GARCH models well capture the natural gas futures volatility spike when the tensions on the Russia-Ukraine border simmered.
Fifth, we further reveal that the volatility estimates from our exponential-form hybrid AR-T-SB-EGARCH and AR-GED-SB-EGARCH models also have forecast power for the volatilities of nonhybrid AR-T-EGARCH and AR-GED-EGARCH models.This indicates the robustness of both (i) the volatility predictive power and (ii) our perspective that because of this timely characteristic of the volatility estimates from our hybrid models, our four hybrid GARCH and EGARCH models well capture the jump in natural gas futures volatility when tensions on the Russia-Ukraine border simmered.
Sixth, we suggest that our new evidence of the predictive power of hybrid models for the volatilities of nonhybrid models means that the volatilities from the nonhybrid models lag behind those of the hybrid models.We consider that because of this characteristic, the nonhybrid models are inferior to hybrid models in capturing the natural gas futures volatility spike in a timely manner when tensions on the Russia-Ukraine border simmered.Seventh, in addition to the points mentioned above, we also derive many valuable interpretations, implications, and innovative perspectives for future energy informatics research and risk management in the energy industries through the use of artificial intelligence (AI).This emphasizes the significance of our work.
The remainder of the paper is organized as follows.Section "Data and the characteristics" details the data and the characteristics, Section "Hybrid GARCH models" presents our hybrid GARCH models, and Section "Why are hybrid GARCHs effective?"tests the predictive power of their volatility estimates.Afterwards, Section "Hybrid EGARCH models" provides our hybrid EGARCH models, Section "Why are hybrid EGARCHs effective?"examines the predictive power of their volatility estimates, and Section "Implications and perspectives" provides the implications and our innovative perspectives.Finally, Section "Contributions and conclusions" concludes the paper.

Data and the characteristics
This study uses the natural gas futures price data from the New York Mercantile Exchange.Using the price series, denoted p NG , we compute the daily log-difference percentage return series as dlr NG t = ln(p NG t /p NG t−1 ) × 100 .To examine the effectiveness of our hybrid models described later for capturing the natural gas futures price fluctuations during the Russian invasion of Ukraine, we analyze the returns, dlr NG t , from January 3, 2020, to January 5, 2024.We note that this sample period also means the post-COVID-19 outbreak period.
Figures 1 and 2 plot the price and return evolution of the natural gas futures, respectively.From these figures, we can see that before the Russian invasion of Ukraine in February 2022, natural gas futures prices had jumped (Fig. 1), and the returns largely and sharply fluctuated (Fig. 2).Referring to the return data, the largest return jump took place on January 27, 2022, which was when tensions on the Russia-Ukraine border simmered as discussed.
Table 1 provides summary statistics of the daily natural gas futures returns.The high kurtosis values and large Jarque-Bera statistics in this period indicate that it is meaningful to employ non-normal distribution errors in quantitative models like our present analysis.Moreover, the augmented Dickey-Fuller test statistic indicates that natural gas futures returns are stationary, suggesting that applying GARCH models like ours is applicable.
Furthermore, as Fig. 2 also indicates, there clearly exist structural breaks in natural gas futures returns.Hence, we investigated and identified the structural break points using the iterated cumulative sums of squares (ICSS) algorithm (Inclán and Tiao 1994).We stress that this ICSS algorithm is a very effective algorithm for identifying structural breaks in time series data (e.g., Wang and Moore 2009;Malik 2022).More precisely, as the Bayesian information criterion (BIC) suggests the appropriate autoregressive lag order for the natural gas futures returns is one, we estimate the regression, dlr NG t = µ + κdlr NG t−1 + τ t , and identify the structural break points in the return residu- als, τ t .
As shown in Table 2, we find that the number of break points for our analysis period is seven.We also show these break points with the bands of ±3 standard deviations of the return residuals in Fig. 3.We consider that this figure clearly indicates the importance of taking structural breaks into consideration to analyze the impact of the Russia-Ukraine war on natural gas futures prices.

Model constructions
This section constructs our hybrid volatility estimation models.Note that we use 'hybrid' to refer to the incorporation of both non-normal fat-tailed errors and structural breaks throughout this study.Due to the non-normality of natural gas futures returns shown in Table 1, we employ either Student's t distribution or GED errors in our models.This is because existing literature suggests the effectiveness of modeling fat-tailed asset returns using Student's t or the GED distribution (e.g., Geweke 1993;Fan et al. 2008).As stated, the BIC suggests the adequate autoregressive lag order for natural gas futures returns is one, and we thus include only the first autoregressive variable in the mean equations of all our models.Accordingly, the base models used to construct our hybrid GARCH models are the following AR-T-GARCH and AR-GED-GARCH models: and where dlr NG t denotes the return of natural gas futures at time t; τ t,t ( τ g,t ) denotes the Stu- dent's t distribution (GED) errors at time t with ν (k) being the distribution shape param- eter; and h t,t ( h g,t ) is the variance to be estimated by model (1) (model ( 2)) at time t.In addition, µ denotes the mean equation intercept; κ denotes the autoregressive term coef- ficient; ψ is the variance equation intercept; χ is the ARCH-term coefficient; and ξ is the GARCH-term coefficient.
(1) As the next step, we construct two hybrid GARCH models by including structural break effects into models (1) and (2).Namely, our hybrid GARCH models are the following AR-T-SB-GARCH and AR-GED-SB-GARCH models: and where τ tsb,t ( τ gsb,t ) denotes the Student's t distribution (GED) errors in model (3) (model ( 4)) at time t; and h tsb,t ( h gsb,t ) is the variance to be estimated by model (3) (model ( 4)) at time t.
In addition, D SB i,t denotes the i-th structural break dummy variable, n is the number of structural breaks, and φ i is the i-th structural break dummy variable's coefficient.The i- th structural break dummy variable takes a value of zero until the break point identified for the return residuals by the ICSS algorithm, and one thereafter.This study employs the ICSS algorithm because existing studies suggest its effectiveness in capturing structural breaks in asset returns (e.g., Wang and Moore 2009;Malik 2022).For details of the ICSS algorithm, see Inclán and Tiao (1994).Note that including the parameters of the non-normal fat-tailed distributions, ν and k, the other notations of models (3) and (4) described above are the same as those of models (1)-(2).

Verifications
Table 3 provides the estimation results of models (1)-( 4).We note that all model parameters are estimated using the maximum likelihood method throughout the paper.Panels A-D present that all the model parameters including the shape parameters of the fat-tailed distribution errors are generally well estimated.In addition, as Panels B and D show, most of the coefficients of our structural break dummy variables-φ 1 to φ 7 -are statistically significant.
It is noteworthy that when incorporating structural breaks, the ARCH effect disappears (The significant coefficients, χ , in Panels A and C of Table 3 become insignificant in Panels B and D.) and the GARCH effect is weakened (The significant values of the coefficients, ξ , in Panels A and C of Table 3 become smaller in Panels B and D.).This also indicates the effectiveness of structural breaks in explaining the evolution of natural gas futures volatilities.
We also conduct likelihood ratio (LR) tests to compare the performance of models (1)-( 4) and present the results in Table 4.As shown in Table 4, (Panel C), and the AR-GED-GARCH model is superior to the AR-GED-SB-GARCH model (Panel D)-are all strongly rejected.Moreover, the values of the Akaike's information criterion (AIC) for models (1)-( 4) are 5, 784.032, 5,794.934, 5,716.146, and 5,716.906,respectively.That is, the LR test results and the AIC values clearly indicate the superiority of our hybrid models, the AR-T-SB-GARCH model and the AR-GED-SB-GARCH model, and demonstrate the effectiveness for taking both non-normal fat-tail errors and structural breaks into account in our analysis for natural gas futures volatilities.

Testing models
This section examines the predictive power of the hybrid model volatility estimates for the nonhybrid model volatilities (precisely, variances).Specifically, to test the forecast power of the volatility estimates from the AR-T-SB-GARCH model (3), we first use the following AR-T-GARCH-X model with ĥtsb,t−1 and the AR-GED- GARCH-X model with ĥtsb,t−1 : where ĥtsb,t−1 denotes the one-day lagged volatility estimates from our hybrid model (3), and is the coefficient.Hence, the statistically significant positive values indicate the predictive power of the volatility estimates from our hybrid model, the AR-T-SB-GARCH model (3).Note that the other notations of models ( 5)-( 6) are the same as those of models ( 1)-( 2).
Moreover, to test the forecast power of the volatility estimates from the AR-GED-SB-GARCH model ( 4) for the nonhybrid model volatilities (variances), we use the following AR-T-GARCH-X model with ĥgsb,t−1 and the AR-GED-GARCH-X model with ĥgsb,t−1 : and where ĥgsb,t−1 denotes the one-day lagged volatility estimates from our hybrid model (4), and is the coefficient.Therefore, the statistically significant positive values again mean the predictive power of the volatility estimates from our hybrid model, the AR-GED-SB-GARCH model (4).Note that the remaining notations of models ( 7)-( 8) are the same as those of models (1)-(2).

Results
Table 5 shows the results for the predictive power of our hybrid model estimates.Panel A shows the results for the volatility estimates from the AR-T-SB-GARCH model, and Panel B presents those for the volatility estimates from the AR-GED-SB-GARCH model.
In these two panels, the estimated coefficients, s, are always statistically significant with positive signs.This indicates that the volatility estimates from our hybrid models, the AR-T-SB-GARCH and the AR-GED-SB-GARCH models, have predictive power for the volatilities (variances) of the nonhybrid AR-T-GARCH and AR-GED-GARCH models. (5) We consider that these new findings mean that volatility estimates from our hybrid GARCH models (3)-( 4) lead the volatilities of nonhybrid GARCH models (1)-( 2).We thus believe that because of this favorable characteristic, the two hybrid GARCH models well captured the volatility jump of the natural gas futures when the tensions simmered on the Russia-Ukraine border.

Model constructions
This section formulates alternative hybrid volatility estimation models using exponential-form GARCH (Nelson 1991).As before, we employ the Student's t distribution or GED errors, and we include the first autoregressive variable in the mean equations of our models.
The base models for building our alternative hybrid models are the following AR-T-EGARCH and AR-GED-EGARCH models:  10)) at time t.Further, µ e denotes the mean equation intercept; κ e denotes the autoregressive term coefficient; ψ e is the variance equation intercept; χ e is the ARCH- term coefficient; and ξ e is the GARCH-term coefficient.
As before, we construct the hybrid EGARCH models by including structural break effects into models ( 9) and (10).Our exponential-form hybrid models are the following AR-T-SB-EGARCH and AR-GED-SB-EGARCH models: and where τ e tsb,t ( τ e gsb,t ) denotes the Student's t distribution (GED) errors in model (11) (model ( 12)) at time t; h e tsb,t ( h e gsb,t ) means the variance to be estimated by model (11) (model ( 12)) at time t.
Furthermore, D SB i,t denotes the i-th structural break dummy variable and n is the number of the structural breaks as before, and φ e,i is the coefficient of the i-th break dummy variable.The dummy variable is the same as in models (3) and (4).We also note that including the parameters of fat-tailed distributions, ν e and k e , the other notations of models ( 11) and ( 12) described above are the same as those of models ( 9)-( 10).

Verifications
Table 6 presents the estimation results of models ( 9)-( 12).Panels A-D present that all the model parameters including the shape parameters of the fat-tailed distribution errors are well estimated.Further, as Panels B and D show, the coefficients of our structural break dummy variables-φ e,1 to φ e,7 -are all statistically significant.We note that when incorporating structural breaks, the ARCH effect again disappears (The significant coefficients, χ e , in Panels A and C of Table 6 become insignificant in Panels B and D.) and the GARCH effect is again weakened (The significant values of the coefficients, ξ e , in Panels A and C of Table 6 become smaller in Panels B and D.).Therefore, we understand that this again shows the effectiveness of structural breaks in explaining the evolution of natural gas futures volatilities.
Moreover, we performed the LR tests to compare the performance of models ( 9)-( 12) as before, and showed the results in Table 7.As Table 7 indicates, the four null hypotheses-the AR-EGARCH model, which has normal distribution errors,  Furthermore, the AIC values for models (9)-( 12) are 5775.172,5789.563,5718.041, and 5718.860,respectively.That is, the LR test results and the AIC values clearly show the superiority of our alternative hybrid models, the AR-T-SB-EGARCH model and the AR-GED-SB-EGARCH model, and prove the effectiveness for taking both nonnormal fat-tail errors and structural breaks into consideration in our natural gas futures volatility analyses.

Why are hybrid EGARCHs effective?
This section considers why our hybrid EGARCH models are effective.For this purpose, we plot the volatility estimates from our alternative hybrid models ( 11) and ( 12 From Figs. 6 and 7, we can see that when volatilities jump, the volatility estimates from the AR-T-SB-EGARCH and the AR-GED-SB-EGARCH models lead the volatility estimates from the AR-T-EGARCH and the AR-GED-EGARCH models, respectively.Considering this and to clarify why our hybrid EGARCH models are effective, we below test whether the volatility estimates from our two hybrid EGARCH models have predictive power for the volatilities of the nonhybrid EGARCH models.

Testing models
This section examines the forecast power of our hybrid EGARCH model estimates for the nonhybrid EGARCH model volatilities (precisely, variances).Specifically, to test the predictive power of the volatility estimates from the AR-T-SB-EGARCH model ( 11), we first use the following AR-T-EGARCH-X model with ĥe tsb,t−1 and the AR-GED- EGARCH-X model with ĥe tsb,t−1 : and where ĥe tsb,t−1 denotes the one-day lagged volatility estimates from our hybrid model ( 11), and e is the coefficient.Hence, the statistically significant positive e values mean the predictive power of the volatility estimates from our hybrid EGARCH model, the AR-T-SB-EGARCH model.Note that the other notations of models ( 13)-( 14) are the same as those of models ( 9)-( 10).
Moreover, also to test the forecast power of the volatility estimates from the AR-GED-SB-EGARCH model (12), we use the following AR-T-EGARCH-X model with ĥe gsb,t−1 and the AR-GED-EGARCH-X model with ĥe gsb,t−1 : and where ĥe gsb,t−1 denotes the one-day lagged volatility estimates from our hybrid EGARCH model ( 12), and e is the coefficient.Thus, the statistically significant positive e again indicates the predictive power of the volatility estimates from our alternative hybrid model, the AR-GED-SB-EGARCH model.Note that the other notations of models ( 15)-( 16) are the same as those of models ( 9)-( 10).( 13)

Results
Table 8 presents the results for the predictive power of our hybrid EGARCH model estimates.Panel A exhibits the results for the volatility estimates from the AR-T-SB-EGARCH model, and Panel B shows those for the volatility estimates from the AR-GED-SB-EGARCH model.In these two panels, the estimated coefficients, e s, are always statistically significant with positive signs; and thus this means that the volatility estimates from our hybrid EGARCH models, the AR-T-SB-EGARCH and the AR-GED-SB-EGARCH models, have predictive power for the volatilities (variances) of the nonhybrid models, the AR-T-EGARCH and the AR-GED-EGARCH models.
We consider that these new findings mean that volatility estimates from our hybrid EGARCH models ( 11)-( 12) lead the volatilities of nonhybrid EGARCH models ( 9)-( 10).Hence as before, we consider that because of this favorable characteristic, the two hybrid EGARCH models timely captured the volatility spile of the natural gas futures when the tension summered in the Ukraine-Russia border.

Implications and perspectives
This section discusses how we can interpret our results and derive significant implications and perspectives for academic energy research and practical industries in the post-COVID-19 era.As in Ahmed et al. (2022), Chaklader et al. (2023), andDirican (2015), AI will play a more significant role in various industries in the future.The implications of our study along with the innovative perspectives derived from our research are as follows.
Firstly, as we have demonstrated, to accurately capture the impact of significant events, natural gas futures volatilities should be estimated by incorporating fat-tailed errors and structural breaks.We believe that capturing structural breaks in a timely manner is particularly crucial for volatility modeling and future risk management in the energy markets and industries.This is because, as we found in Figs. 4, 5, 6, and 7, the volatilities derived from nonhybrid models lag behind the volatilities from our four hybrid models, which incorporate structural breaks.The lag is especially evident during the time when tensions simmered on the Russia-Ukraine border.Therefore, it is crucial for us to accurately forecast energy price volatilities without lagging behind significant events by utilizing innovative AI techniques such as machine learning.This will lead to more effective risk management in the energy markets and industries in the post-COVID-19 era.
Secondly, given the increasing importance of structural breaks in energy markets that we have observed, it is now more crucial than ever for us to accurately predict these breaks.We should note that these breaks are caused by information related to the energy markets.We should strive to accurately predict these breaks by using cutting-edge AI technologies.This will improve risk management in the energy markets and industries, particularly in the unpredictable post-COVID-19 world.We also believe that accurately predicting structural breaks allows us to forecast volatilities without falling behind sudden changes in energy prices caused by significant events.
Thirdly, as a policy implication, we suggest that policy makers and energy-related industry practitioners need to pay closer attention not only to the demand and supply balance for energy, but also to various news and its impact on energy market prices, even during normal circumstances.These efforts and close observation should help prevent potential imbalances in energy markets in the future.Furthermore, our results suggest that policy making and risk management in the energy industry should be conducted more swiftly than in the past.This is due to the fact that energy prices now respond more quickly to various events and news than ever before.One deeper interpretation of our results is that market efficiency in energy markets may have shifted.We believe that we should take this viewpoint into consideration in policy making and risk management in the energy industry.
Fourth, an interesting implication for future research would be to utilize machine learning approaches, such as reinforcement learning or deep learning for predicting energy price fluctuations.We consider that comparing traditional econometric models with machine learning approaches in our context would be a valuable direction for future research.We believe that our methods can be applied to other real-world assets, such as fuel and other energy commodities.
While this study does not provide methods for forecasting structural breaks, the comprehensive inspections conducted in this paper provide many insightful interpretations, implications, and innovative perspectives.Collectively, these demonstrate the importance of our current work for future energy research and practice.Therefore, we believe that our empirical findings, along with the significant interpretations, implications, and innovative viewpoints discussed above, all signify our substantial contribution to current and future research on energy informatics.

Contributions and conclusions
Paying keen attention to the Russia-Ukraine war, and particularly its impact on natural gas futures prices when tensions simmered on the Russia-Ukraine border, this paper empirically examined natural gas futures volatilities.Using the four hybrid models, the AR-T-SB-GARCH, the AR-GED-SB-GARCH, the AR-T-SB-EGARCH, and the AR-GED-SB-EGARCH models, which incorporated both fat-tailed errors and structural breaks, we derived the following findings as new contributions.
• First, we found that very interestingly, it was not when Russia invaded Ukraine in February 2022, but rather when tensions on the border simmered in January 2022, that the matter of the Russia-Ukraine war had the most impact on natural gas futures prices.• Second, we revealed that our hybrid models, not only with fat-tailed errors-Student's t distribution or GED errors-but also with structural breaks, effectively capture the natural gas futures volatility spike when tensions on the Russia-Ukraine border simmered in January 2022.• Third, we also uncovered that our hybrid modeling approach-incorporating both fat-tailed errors and structural breaks-is highly effective, not only in GARCH models but also in EGARCH models.This demonstrates the robustness of the effectiveness of our hybrid modeling approach in capturing natural gas futures volatilities.• Fourth, we also newly found that the volatility estimates from our hybrid models of the AR-T-SB-GARCH and the AR-GED-SB-GARCH models have predictive power for the volatilities of nonhybrid AR-T-GARCH and AR-GED-GARCH models.We consider that because of this beneficial characteristic, our hybrid GARCH models well captured the natural gas futures volatility jump when the tensions on the Russia-Ukraine border simmered.• Fifth, we also uncovered that the volatility estimates from our exponential-form hybrid models of the AR-T-SB-EGARCH and the AR-GED-SB-EGARCH models also have forecast power for the volatilities of nonhybrid AR-T-EGARCH and AR-GED-EGARCH models.This indicates the robustness of the predictive power, and again, we suggest that because of this characteristic, our hybrid EGARCH models also timely captured the natural gas futures volatility spike when the tensions in Russia-Ukraine border simmered.• Sixth, we further elucidated that the evidence of the predictive power of our hybrid models for the volatilities of nonhybrid models means that the volatility estimates from the nonhybrid models lag behind the volatilities of the hybrid models.We con-sider that because of this characteristic, the nonhybrid models are inferior to the hybrid models in capturing sudden changes in natural gas futures volatilities.• Seventh, in addition to the above, we have also developed and shared numerous valuable interpretations, implications, and innovative perspectives for future risk management, policy-making practices, and related research in energy informatics.This represents another significant contribution of our work.These interpretations, implications, and innovative perspectives, wihch include the effective use of AI, should be valuable not only for academic researchers but also for industry practitioners.
This study examined the impact of war as a manmade disaster on natural gas futures prices, which is of great importance for research in energy informatics.Additionally, the findings from our study indicate the growing significance of structural breaks in energy markets.This highlights the need to closely monitor the immediate news impact on contemporary energy markets.
As a broader implication for future research, studies that explore similar models in different energy-related markets or under various geopolitical scenarios would be interesting.Further research into the adaptability and scalability of the proposed hybrid models in our current study would provide a pathway for ongoing contributions in the field of energy informatics.Given this, we believe that the many new findings derived from our present study make a significant contribution not only to the body of academic energy informatics research but also to actual practice in energy-related industries.

Fig. 1 Fig. 2
Fig. 1 Daily price evolution of natural gas futures, January 2020 to January 2024.Natural gas futures prices in US dollars

Fig. 3
Fig. 3 Structural breaks in natural gas futures return residuals.Bands of ±3 standard deviations and change points identified by the ICSS algorithm the null hypotheses-the AR-GARCH model, which has normal distribution errors, is superior to the AR-T-GARCH model (Panel A), the AR-T-GARCH model is superior to the AR-T-SB-GARCH model (Panel B), the AR-GARCH model is superior to the AR-GED-GARCH model

Fig. 6
Fig. 6 Estimated volatilities from AR-T-EGARCH and AR-T-SB-EGARCH models ) in Figs. 6 and 7 to compare the model estimates.Specifically, Fig. 6 exhibits the volatility estimates from the AR-T-EGARCH model with those from the AR-T-SB-EGARCH model, and Fig. 7 displays the volatility estimates from the AR-GED-EGARCH model with those from the AR-GED-SB-EGARCH model.

Table 1
Summary statistics for natural gas futures returns, January 2020 to January 2024

Table 2
Structural break points for natural gas futures return residuals, January 2020 to January 2024

Table 3
Estimation results for hybrid GARCH models LL: log-likelihood value.*** and * denote the 1% and 10% significance levels, respectively

Table 4
LR tests for GARCH model discrimination *** denotes the 1% significance level.The test statistic follows a χ 2

Table 5
Results for predictive power tests of hybrid GARCH model volatility estimates

Predictive power of volatility estimates from the AR-GED-SB-GARCH model
9)

Table 6
Estimation results for hybrid EGARCH models LL: log-likelihood value.*** denotes the 1% significance level

Table 8
Results for predictive power tests of hybrid EGARCH model volatility estimatesLL: log-likelihood value.*** and * denote the 1% and 10% significance levels, respectively