Regular article 常规文章The role of higher moments in predicting China's oil futures volatility: Evidence from machine learning models
中国石油期货波动预测中高阶矩的作用：来自机器学习模型的证据
Abstract 摘要
This paper expands the emerging literature on volatility forecasting for China's oil market by exploring the predictive ability of higher-order moments (skewness, kurtosis, hyperskewness, and hyperkurtosis) based on high-frequency data. Our investigation is originally based on the heterogeneous autoregressive (HAR) framework, but considering the possible multicollinearity and nonlinearity, it is extended to various machine learning (ML) models and combination forecasting models. The results reveal that higher-order moments, including the two highest moments, always significantly improve predictive performance for the COVID-19 crisis. We further examine the interpretability of ML models and each factor's contribution to the prediction, finding that odd and even moments contain short- and long-term prediction information, respectively. This paper also highlights the effectiveness of ML models for capturing trends in oil futures volatility with higher-order moments and the satisfactory performance of combination forecasting models. Finally, we investigate the predictability of asymmetric risk patterns and obtain identical results. Our study has important implications for financial risk management, asset pricing, and portfolio allocation.
本文通过基于高频数据探索高阶矩（偏度、峰度、超偏度和超峰度）的预测能力，扩展了关于中国石油市场波动预测的现有文献。我们的研究最初基于异质自回归（HAR）框架，但考虑到可能的多重共线性和非线性，将其扩展到各种机器学习（ML）模型和组合预测模型。结果表明，包括两个最高阶矩在内的高阶矩始终显著提高了对 COVID-19 危机的预测性能。我们进一步检验了 ML 模型的可解释性和每个因素对预测的贡献，发现奇数和偶数矩分别包含短期和长期预测信息。本文还突出了 ML 模型利用高阶矩捕捉石油期货波动趋势的有效性以及组合预测模型的令人满意的性能。最后，我们研究了非对称风险模式的可预测性，并获得了相同的结果。 我们的研究对金融风险管理、资产定价和投资组合配置具有重要意义。
本文通过基于高频数据探索高阶矩（偏度、峰度、超偏度和超峰度）的预测能力，扩展了关于中国石油市场波动预测的现有文献。我们的研究最初基于异质自回归（HAR）框架，但考虑到可能的多重共线性和非线性，将其扩展到各种机器学习（ML）模型和组合预测模型。结果表明，包括两个最高阶矩在内的高阶矩始终显著提高了对 COVID-19 危机的预测性能。我们进一步检验了 ML 模型的可解释性和每个因素对预测的贡献，发现奇数和偶数矩分别包含短期和长期预测信息。本文还突出了 ML 模型利用高阶矩捕捉石油期货波动趋势的有效性以及组合预测模型的令人满意的性能。最后，我们研究了非对称风险模式的可预测性，并获得了相同的结果。 我们的研究对金融风险管理、资产定价和投资组合配置具有重要意义。
Keywords 关键词
China's oil futures
COVID-19
Higher-order moments
Machine learning
Combination forecasting
中国的石油期货 COVID-19 高阶矩机器学习组合预测
1. Introduction 1. 引言
Oil is a major commodity in financial markets, and volatility forecasting of oil prices is crucial for financial modeling and decision-making (Chiang et al., 2015; Hamilton, 1983; Kilian and Park, 2009; Pan et al., 2017). A growing number of studies have made significant gains in volatility prediction by applying realized variance measures built on returns, such as jumps, bipower variation, realized semi-variance, etc. Higher-order moments are no exception (Bollerslev et al., 2016; Ma et al., 2019; Patton and Sheppard, 2015; Prokopczuk et al., 2016; Sévi, 2014). As an integral consideration in risk management, portfolio allocation, and asset pricing, it is worth examining whether higher-order moments are relevant to oil volatility prediction.
石油是金融市场的主要商品，油价波动预测对金融建模和决策至关重要（Chiang 等，2015；Hamilton，1983；Kilian 和 Park，2009；Pan 等，2017）。越来越多的研究通过应用基于收益的实实现金收益度量，如跳跃、双幂变化、实实现半方差等，在波动预测方面取得了显著进展。高阶矩也不例外（Bollerslev 等，2016；Ma 等，2019；Patton 和 Sheppard，2015；Prokopczuk 等，2016；Sévi，2014）。作为风险管理、投资组合分配和资产定价的一个基本考虑因素，值得探讨高阶矩是否与石油波动预测相关。
This study is the first to explore the role of higher-order moments in influencing realized volatility (RV) forecasting in the oil market.
这项研究首次探讨了高阶矩在影响石油市场实现波动率（RV）预测中的作用。
石油是金融市场的主要商品，油价波动预测对金融建模和决策至关重要（Chiang 等，2015；Hamilton，1983；Kilian 和 Park，2009；Pan 等，2017）。越来越多的研究通过应用基于收益的实实现金收益度量，如跳跃、双幂变化、实实现半方差等，在波动预测方面取得了显著进展。高阶矩也不例外（Bollerslev 等，2016；Ma 等，2019；Patton 和 Sheppard，2015；Prokopczuk 等，2016；Sévi，2014）。作为风险管理、投资组合分配和资产定价的一个基本考虑因素，值得探讨高阶矩是否与石油波动预测相关。
This study is the first to explore the role of higher-order moments in influencing realized volatility (RV) forecasting in the oil market.
这项研究首次探讨了高阶矩在影响石油市场实现波动率（RV）预测中的作用。
The inspiration for focusing on the predictive role of higher-order moments in oil futures RV stems from a review of the extensive theoretical literature dating back to Kraus and Litzenberger (1976), including macroeconomic disaster studies by Barro (2006), Longstaff and Piazzesi (2004), and Rietz (1988), who argue that heavy-tailed shocks in general, and left-tailed events in particular, significantly influence asset-price behavior.
对石油期货波动率风险（RV）中高阶矩预测作用的关注灵感来源于对自 Kraus 和 Litzenberger（1976 年）以来的广泛理论文献的回顾，包括 Barro（2006 年）的宏观经济灾难研究、Longstaff 和 Piazzesi（2004 年）以及 Rietz（1988 年）的研究，他们认为，在一般情况下，重尾冲击，尤其是左尾事件，对资产价格行为有显著影响。
Throughout the process, higher-order moments aim to capture RV asymmetries and extreme fluctuations, which are easily connected to the COVID-19 pandemic and the ensuing profound shifts in the world's financial markets (Kostakis et al., 2012). Thus, higher-order moments are recognized as essential indicators of market-wide risk, with the capability to enhance RV predictions.
在整个过程中，高阶矩旨在捕捉 RV 的对称性和极端波动，这些波动与 COVID-19 大流行及其引发的全球金融市场深刻变化密切相关（Kostakis 等，2012）。因此，高阶矩被视为市场风险的重要指标，具有增强 RV 预测的能力。
Empirically, higher-order moments can improve investment performance when incorporated into portfolio strategies and can properly define returns distribution when employed in model frameworks (Dittmar, 2002; Ghysels et al., 2016; Jensen et al., 2000; Liu et al., 2020). Notably, Mei et al. (2017) are the first to reveal the predictive ability of higher-order moments for stock market volatility in the US and China. Building on this study, Gkillas et al. (2019) conclude that higher-order moments can improve our comprehension of exchange-rate volatility dynamics based on six major currencies relative to the US dollar. Bonato et al. (2022) provide further evidence of the predictive ability of higher-order moments for the RV of international real estate investment trusts, which exceeds that of jumps. However, these articles only study skewness and kurtosis. So far, few researchers have investigated the potential predictive importance of the fifth and sixth moments. Kinateder and Papavassiliou (2019) discover that the two highest moments contribute to sovereign bond returns forecasts. Driven by the assumption that agents prefer higher odd moments and dislike even ones, Khademalomoom et al. (2019) certificate that the incorporation of odd/even moments (third and fifth moments/fourth and sixth moments) into return/variance modeling improves the performance of generalized autoregressive conditional heteroskedasticity (GARCH) models in predicting exchange-rate returns. However, we still discover that these papers neglect the oil market.
经验表明，将高阶矩纳入投资组合策略可以改善投资业绩，并在模型框架中使用时可以正确定义收益分布（Dittmar，2002；Ghysels 等，2016；Jensen 等，2000；Liu 等，2020）。值得注意的是，Mei 等（2017）是第一个揭示高阶矩对美国和中国股市波动预测能力的学者。在此基础上，Gkillas 等（2019）得出结论，高阶矩可以提高我们对相对于美元的六种主要货币汇率波动动态的理解。Bonato 等（2022）提供了进一步证据，表明高阶矩对国际房地产投资信托的回报率波动率预测能力超过跳跃。然而，这些文章仅研究了偏度和峰度。迄今为止，很少有研究人员调查第五和第六矩的潜在预测重要性。Kinateder 和 Papavassiliou（2019）发现，两个最高矩对主权债券回报率预测有贡献。 受制于代理机构偏好更高的奇数矩和不喜欢偶数矩的假设，Khademalomoom 等人（2019）证明，将奇数/偶数矩（第三和第五矩/第四和第六矩）纳入回报/方差建模，可以提高广义自回归条件异方差（GARCH）模型在预测汇率回报方面的性能。然而，我们仍然发现这些论文忽略了石油市场。
Given the importance of higher-order moments and the oil market, for the first time, this study examines the impact of hyper order moments on oil futures volatility forecasting.
鉴于高阶矩和石油市场的重要性，本研究首次考察了超高阶矩对石油期货波动率预测的影响。
We also differentiate between odd and even moments and takes the lead in measuring the effect of higher-order moments on positive and negative volatility separately.
我们还将奇数和偶数时刻区分开来，并率先分别测量高阶矩对正负波动率的影响。
对石油期货波动率风险（RV）中高阶矩预测作用的关注灵感来源于对自 Kraus 和 Litzenberger（1976 年）以来的广泛理论文献的回顾，包括 Barro（2006 年）的宏观经济灾难研究、Longstaff 和 Piazzesi（2004 年）以及 Rietz（1988 年）的研究，他们认为，在一般情况下，重尾冲击，尤其是左尾事件，对资产价格行为有显著影响。
Throughout the process, higher-order moments aim to capture RV asymmetries and extreme fluctuations, which are easily connected to the COVID-19 pandemic and the ensuing profound shifts in the world's financial markets (Kostakis et al., 2012). Thus, higher-order moments are recognized as essential indicators of market-wide risk, with the capability to enhance RV predictions.
在整个过程中，高阶矩旨在捕捉 RV 的对称性和极端波动，这些波动与 COVID-19 大流行及其引发的全球金融市场深刻变化密切相关（Kostakis 等，2012）。因此，高阶矩被视为市场风险的重要指标，具有增强 RV 预测的能力。
Empirically, higher-order moments can improve investment performance when incorporated into portfolio strategies and can properly define returns distribution when employed in model frameworks (Dittmar, 2002; Ghysels et al., 2016; Jensen et al., 2000; Liu et al., 2020). Notably, Mei et al. (2017) are the first to reveal the predictive ability of higher-order moments for stock market volatility in the US and China. Building on this study, Gkillas et al. (2019) conclude that higher-order moments can improve our comprehension of exchange-rate volatility dynamics based on six major currencies relative to the US dollar. Bonato et al. (2022) provide further evidence of the predictive ability of higher-order moments for the RV of international real estate investment trusts, which exceeds that of jumps. However, these articles only study skewness and kurtosis. So far, few researchers have investigated the potential predictive importance of the fifth and sixth moments. Kinateder and Papavassiliou (2019) discover that the two highest moments contribute to sovereign bond returns forecasts. Driven by the assumption that agents prefer higher odd moments and dislike even ones, Khademalomoom et al. (2019) certificate that the incorporation of odd/even moments (third and fifth moments/fourth and sixth moments) into return/variance modeling improves the performance of generalized autoregressive conditional heteroskedasticity (GARCH) models in predicting exchange-rate returns. However, we still discover that these papers neglect the oil market.
经验表明，将高阶矩纳入投资组合策略可以改善投资业绩，并在模型框架中使用时可以正确定义收益分布（Dittmar，2002；Ghysels 等，2016；Jensen 等，2000；Liu 等，2020）。值得注意的是，Mei 等（2017）是第一个揭示高阶矩对美国和中国股市波动预测能力的学者。在此基础上，Gkillas 等（2019）得出结论，高阶矩可以提高我们对相对于美元的六种主要货币汇率波动动态的理解。Bonato 等（2022）提供了进一步证据，表明高阶矩对国际房地产投资信托的回报率波动率预测能力超过跳跃。然而，这些文章仅研究了偏度和峰度。迄今为止，很少有研究人员调查第五和第六矩的潜在预测重要性。Kinateder 和 Papavassiliou（2019）发现，两个最高矩对主权债券回报率预测有贡献。 受制于代理机构偏好更高的奇数矩和不喜欢偶数矩的假设，Khademalomoom 等人（2019）证明，将奇数/偶数矩（第三和第五矩/第四和第六矩）纳入回报/方差建模，可以提高广义自回归条件异方差（GARCH）模型在预测汇率回报方面的性能。然而，我们仍然发现这些论文忽略了石油市场。
Given the importance of higher-order moments and the oil market, for the first time, this study examines the impact of hyper order moments on oil futures volatility forecasting.
鉴于高阶矩和石油市场的重要性，本研究首次考察了超高阶矩对石油期货波动率预测的影响。
We also differentiate between odd and even moments and takes the lead in measuring the effect of higher-order moments on positive and negative volatility separately.
我们还将奇数和偶数时刻区分开来，并率先分别测量高阶矩对正负波动率的影响。
It is worth noting that this paper focus on the RV of China's oil market during the COVID-19 pandemic, paying particular attention to four considerations: 1)The importance of China's crude oil futures.
值得注意的是，本文聚焦于 COVID-19 疫情期间中国石油市场的 RV，特别关注以下四个方面：1）中国原油期货的重要性。
The first RMD-dominated crude oil futures were formally launched on the Shanghai International Energy Exchange (INE) on March 26, 2018, opening up to international participants, which had theoretical and practical significance for China (Ji and Zhang, 2019). China is rapidly advancing its position in the global market as the second-biggest consumer of crude oil and the world's largest importer of commodities (Liu and Lee, 2021). 2) The uniqueness of China's crude oil futures. China holds the distinction of being the first price benchmark in Asia, setting it apart from other countries such as Japan and Singapore, which have made unsuccessful attempts to establish their own benchmarks. Furthermore, in just three months since its launch, the INE ranked first in Asia and among the top three worldwide in terms of trading volume (Sun et al., 2022), which reveals the exceptional appeal of China's crude oil futures. 3) The limited research on China's crude oil futures. Compared to the large amount of research on global benchmarks, the literature on China's crude oil futures is scant (Duan et al., 2023; Huang and Huang, 2020; Niu et al., 2021; Yang and Zhou, 2020). 4) The noteworthy issues in China's crude oil futures. The INE has faced various extreme domestic and international challenges over the three years since its launch, including the severe shock of the COVID-19 pandemic, which triggered huge fluctuations in oil prices.
2018 年 3 月 26 日，上海国际能源交易所（INE）正式推出首只 RMD 主导的原油期货，向国际参与者开放，这对中国具有理论和实践意义（Ji 和 Zhang，2019）。中国作为全球第二大原油消费国和最大商品进口国，在全球市场中的地位迅速提升（Liu 和 Lee，2021）。2）中国原油期货的独特性。中国拥有亚洲首个价格基准的称号，与日本和新加坡等国家试图建立自己的基准的努力形成鲜明对比。此外，自其推出以来仅三个月，INE 在亚洲排名首位，在全球交易量方面位居前三（Sun 等人，2022），这揭示了我国原油期货的非凡吸引力。3）对中国原油期货的研究有限。与全球基准的大量研究相比，关于中国原油期货的文献相对较少（Duan 等人，2023；Huang 和 Huang，2020；Niu 等人，2021；Yang 和 Zhou，2020）。 4) 中国原油期货值得关注的问题。INE 自推出以来，已面临各种极端的国内外挑战，包括 COVID-19 大流行带来的严重冲击，这引发了油价的大幅波动。
Considering the crucial ability of higher-order moments to cope with extreme situations, it makes sense for us to undertake the first exploration of the relationship between higher-order moments and the volatility of China's oil market during the COVID-19 crisis.
考虑到高阶矩应对极端情况的关键能力，在我们对 COVID-19 危机期间中国石油市场波动与高阶矩之间的关系进行首次探索时，这是有意义的。
值得注意的是，本文聚焦于 COVID-19 疫情期间中国石油市场的 RV，特别关注以下四个方面：1）中国原油期货的重要性。
The first RMD-dominated crude oil futures were formally launched on the Shanghai International Energy Exchange (INE) on March 26, 2018, opening up to international participants, which had theoretical and practical significance for China (Ji and Zhang, 2019). China is rapidly advancing its position in the global market as the second-biggest consumer of crude oil and the world's largest importer of commodities (Liu and Lee, 2021). 2) The uniqueness of China's crude oil futures. China holds the distinction of being the first price benchmark in Asia, setting it apart from other countries such as Japan and Singapore, which have made unsuccessful attempts to establish their own benchmarks. Furthermore, in just three months since its launch, the INE ranked first in Asia and among the top three worldwide in terms of trading volume (Sun et al., 2022), which reveals the exceptional appeal of China's crude oil futures. 3) The limited research on China's crude oil futures. Compared to the large amount of research on global benchmarks, the literature on China's crude oil futures is scant (Duan et al., 2023; Huang and Huang, 2020; Niu et al., 2021; Yang and Zhou, 2020). 4) The noteworthy issues in China's crude oil futures. The INE has faced various extreme domestic and international challenges over the three years since its launch, including the severe shock of the COVID-19 pandemic, which triggered huge fluctuations in oil prices.
2018 年 3 月 26 日，上海国际能源交易所（INE）正式推出首只 RMD 主导的原油期货，向国际参与者开放，这对中国具有理论和实践意义（Ji 和 Zhang，2019）。中国作为全球第二大原油消费国和最大商品进口国，在全球市场中的地位迅速提升（Liu 和 Lee，2021）。2）中国原油期货的独特性。中国拥有亚洲首个价格基准的称号，与日本和新加坡等国家试图建立自己的基准的努力形成鲜明对比。此外，自其推出以来仅三个月，INE 在亚洲排名首位，在全球交易量方面位居前三（Sun 等人，2022），这揭示了我国原油期货的非凡吸引力。3）对中国原油期货的研究有限。与全球基准的大量研究相比，关于中国原油期货的文献相对较少（Duan 等人，2023；Huang 和 Huang，2020；Niu 等人，2021；Yang 和 Zhou，2020）。 4) 中国原油期货值得关注的问题。INE 自推出以来，已面临各种极端的国内外挑战，包括 COVID-19 大流行带来的严重冲击，这引发了油价的大幅波动。
Considering the crucial ability of higher-order moments to cope with extreme situations, it makes sense for us to undertake the first exploration of the relationship between higher-order moments and the volatility of China's oil market during the COVID-19 crisis.
考虑到高阶矩应对极端情况的关键能力，在我们对 COVID-19 危机期间中国石油市场波动与高阶矩之间的关系进行首次探索时，这是有意义的。
Building on scarce empirical literature that predicting volatility using higher-order moments (Bonato et al., 2022; Gkillas et al., 2019), we investigate the impact of different higher-order moments by incorporating third- and fourth-order moments, odd moments, even moments, and all higher-order moments (including fifth- and sixth-order moments) into the benchmark heterogeneous autoregressive (HAR) models developed by Corsi (2009). The out-of-sample results show that third-to sixth-order moments all provide useful information for oil futures volatility forecasts in China.
基于预测波动性使用高阶矩的稀缺实证文献（Bonato 等，2022；Gkillas 等，2019），我们通过将三阶和四阶矩、奇数矩、偶数矩以及所有高阶矩（包括五阶和六阶矩）纳入 Corsi（2009）开发的基准异质自回归（HAR）模型，研究了不同高阶矩的影响。样本外结果表明，三阶至六阶矩均为中国石油期货波动性预测提供了有用的信息。
We also discover that odd moments are more important than even moments for short-term forecasting, whereas even moments are more critical for long-term forecasting, which agrees with an earlier finding that odd and even moments have different effects (Khademalomoom et al., 2019). These results also align with forecasts of positive and negative volatility. Moreover, based on alternative evaluation approaches, forecasting horizons, and rolling schemes, the robustness of these conclusions is confirmed.
我们还发现，对于短期预测来说，奇数时刻比偶数时刻更重要，而对于长期预测来说，偶数时刻则更为关键，这与早期发现奇数和偶数时刻具有不同影响（Khademalomoom 等，2019 年）相一致。这些结果也与正负波动率的预测相一致。此外，基于不同的评估方法、预测范围和滚动方案，这些结论的稳健性得到了证实。
基于预测波动性使用高阶矩的稀缺实证文献（Bonato 等，2022；Gkillas 等，2019），我们通过将三阶和四阶矩、奇数矩、偶数矩以及所有高阶矩（包括五阶和六阶矩）纳入 Corsi（2009）开发的基准异质自回归（HAR）模型，研究了不同高阶矩的影响。样本外结果表明，三阶至六阶矩均为中国石油期货波动性预测提供了有用的信息。
We also discover that odd moments are more important than even moments for short-term forecasting, whereas even moments are more critical for long-term forecasting, which agrees with an earlier finding that odd and even moments have different effects (Khademalomoom et al., 2019). These results also align with forecasts of positive and negative volatility. Moreover, based on alternative evaluation approaches, forecasting horizons, and rolling schemes, the robustness of these conclusions is confirmed.
我们还发现，对于短期预测来说，奇数时刻比偶数时刻更重要，而对于长期预测来说，偶数时刻则更为关键，这与早期发现奇数和偶数时刻具有不同影响（Khademalomoom 等，2019 年）相一致。这些结果也与正负波动率的预测相一致。此外，基于不同的评估方法、预测范围和滚动方案，这些结论的稳健性得到了证实。
Given the inclusion of higher-order moments and their lags as explanatory variables in the forecasting models, which are all realized variance measures constructed from returns, it is reasonable to suspect correlations among them.
考虑到在预测模型中包含了高阶矩及其滞后项作为解释变量，这些变量都是基于回报率构建的实实现方差度量，因此怀疑它们之间存在相关性是合理的。
In such cases, conventional linear HAR models may falter. Additionally, the presence of nonlinearity contributing to volatility forecasting among higher-order moments cannot be identified using conventional linear models.
在这种情况下，传统的线性 HAR 模型可能失效。此外，使用传统的线性模型无法识别出对高阶矩波动预测有贡献的非线性因素。
However, machine learning (ML) methods have proven to be effective in addressing multicollinearity and accommodating more sophisticated functional forms, enabling the exploration of complex nonlinear relationships among variables This advantage holds promise for better approximating the unknown and likely complex data generating process underlying oil price volatility.
然而，机器学习（ML）方法已被证明在解决多重共线性并适应更复杂的函数形式方面是有效的，这使人们能够探索变量之间复杂的非线性关系。这一优势有望更好地近似油价波动背后可能复杂的未知数据生成过程。
Furthermore, some empirical research has supported the claim that ML models are superior to other models. For example, Ding et al. (2021) show that the least absolute shrinkage and selection operator (LASSO) method outperforms the benchmark HAR model in forecasting the RV of stocks in eight countries. Christensen et al. (2021) demonstrate that a wide range of ML (regularization, tree-based regression, and neural network) models outperform the HAR model for volatility prediction.
此外，一些实证研究支持了机器学习模型优于其他模型的论断。例如，Ding 等人（2021 年）表明，最小绝对收缩和选择算子（LASSO）方法在预测八个国家的股票回报率波动（RV）方面优于基准 HAR 模型。Christensen 等人（2021 年）证明，广泛的机器学习模型（正则化、基于树的回归和神经网络）在波动率预测方面优于 HAR 模型。
Therefore, we investigate the influence of higher-order moments on oil futures volatility using ML models, including LASSO, elastic net (EN), gradient boosting (GBDT), and random forests (RF).
因此，我们使用机器学习模型，包括 LASSO、弹性网络（EN）、梯度提升（GBDT）和随机森林（RF），研究高阶矩对石油期货波动率的影响。
考虑到在预测模型中包含了高阶矩及其滞后项作为解释变量，这些变量都是基于回报率构建的实实现方差度量，因此怀疑它们之间存在相关性是合理的。
In such cases, conventional linear HAR models may falter. Additionally, the presence of nonlinearity contributing to volatility forecasting among higher-order moments cannot be identified using conventional linear models.
在这种情况下，传统的线性 HAR 模型可能失效。此外，使用传统的线性模型无法识别出对高阶矩波动预测有贡献的非线性因素。
However, machine learning (ML) methods have proven to be effective in addressing multicollinearity and accommodating more sophisticated functional forms, enabling the exploration of complex nonlinear relationships among variables This advantage holds promise for better approximating the unknown and likely complex data generating process underlying oil price volatility.
然而，机器学习（ML）方法已被证明在解决多重共线性并适应更复杂的函数形式方面是有效的，这使人们能够探索变量之间复杂的非线性关系。这一优势有望更好地近似油价波动背后可能复杂的未知数据生成过程。
Furthermore, some empirical research has supported the claim that ML models are superior to other models. For example, Ding et al. (2021) show that the least absolute shrinkage and selection operator (LASSO) method outperforms the benchmark HAR model in forecasting the RV of stocks in eight countries. Christensen et al. (2021) demonstrate that a wide range of ML (regularization, tree-based regression, and neural network) models outperform the HAR model for volatility prediction.
此外，一些实证研究支持了机器学习模型优于其他模型的论断。例如，Ding 等人（2021 年）表明，最小绝对收缩和选择算子（LASSO）方法在预测八个国家的股票回报率波动（RV）方面优于基准 HAR 模型。Christensen 等人（2021 年）证明，广泛的机器学习模型（正则化、基于树的回归和神经网络）在波动率预测方面优于 HAR 模型。
Therefore, we investigate the influence of higher-order moments on oil futures volatility using ML models, including LASSO, elastic net (EN), gradient boosting (GBDT), and random forests (RF).
因此，我们使用机器学习模型，包括 LASSO、弹性网络（EN）、梯度提升（GBDT）和随机森林（RF），研究高阶矩对石油期货波动率的影响。
Interestingly, individual forecasts may react differently to structural disruptions or fluctuating market conditions, but combination forecasts can balance bias and estimation variance to achieve prediction efficiency. (Elliott and Timmermann, 2016; Smith and Wallis, 2009; Timmermann, 2006). Many papers have shown that combination methods are the most effective in most cases (Claeskens et al., 2016; Clemen, 1989; Makridakis et al., 2018). Thus, we base our combination forecasts on ML models. To the best of our knowledge, this study is the first to offer empirical proof of the correlation between higher-order moments and volatility based on ML models and combination forecasting models.
有趣的是，个别预测可能对结构性中断或波动的市场条件有不同的反应，但组合预测可以通过平衡偏差和估计方差来实现预测效率。（Elliott 和 Timmermann，2016；Smith 和 Wallis，2009；Timmermann，2006）。许多论文表明，在大多数情况下，组合方法是最高效的（Claeskens 等，2016；Clemen，1989；Makridakis 等，2018）。因此，我们基于机器学习模型进行组合预测。据我们所知，这项研究是首次基于机器学习模型和组合预测模型提供高级矩与波动性之间相关性的实证证据。
有趣的是，个别预测可能对结构性中断或波动的市场条件有不同的反应，但组合预测可以通过平衡偏差和估计方差来实现预测效率。（Elliott 和 Timmermann，2016；Smith 和 Wallis，2009；Timmermann，2006）。许多论文表明，在大多数情况下，组合方法是最高效的（Claeskens 等，2016；Clemen，1989；Makridakis 等，2018）。因此，我们基于机器学习模型进行组合预测。据我们所知，这项研究是首次基于机器学习模型和组合预测模型提供高级矩与波动性之间相关性的实证证据。
Our paper makes several contributions. First, no study has investigated the role of higher (up to sixth-order) moments in predicting RV in the oil market. Devpura and Narayan (2020) assert that the COVID-19 pandemic led to nearly 10-fold more volatility in oil prices during its early stages, contributing between 8% and 22% per day to oil price volatility.
我们的论文做出了几个贡献。首先，没有研究调查过在石油市场中预测 RV 时，高阶（高达六阶）矩的作用。Devpura 和 Narayan（2020）断言，COVID-19 大流行在其早期阶段导致石油价格波动增加了近 10 倍，每天对石油价格波动贡献了 8%至 22%。
Due to such oil price volatility characteristics, third- and fourth-order moments may be inadequate for understanding volatility dynamics. In fact, empirical research on hyperskewness and hyperkurtosis has offered important predictive information on sovereign bond returns, cryptocurrency returns, and exchange-rate behavior but not oil price volatility. Therefore, we fill a gap in the existing literature and discover that higher (third-to sixth-order) moments are effective for oil futures volatility forecasting.
由于这种油价波动特性，三阶和四阶矩可能不足以理解波动动态。事实上，关于超偏度和超峰度的实证研究已经为国债收益、加密货币收益和汇率行为提供了重要的预测信息，但并非油价波动。因此，我们填补了现有文献的空白，并发现更高阶（三阶至六阶）矩对于石油期货波动预测是有效的。
Second, previous papers have studied higher-order moments using GARCH, HAR, and MIDAS models (Khademalomoom et al., 2019; Mei et al., 2017). Considering the complexity of the oil futures market and the variable correlations among higher-order moments, this paper is the first to employ ML models to explore the relevance of higher-order moments to oil futures volatility.
其次，先前的研究使用 GARCH、HAR 和 MIDAS 模型研究了高阶矩（Khademalomoom 等，2019；Mei 等，2017）。鉴于石油期货市场的复杂性和高阶矩之间的变量相关性，本文首次采用机器学习（ML）模型来探讨高阶矩对石油期货波动性的相关性。
Also, since we further use combination forecasting models for volatility prediction, the evidence indicates that combination forecasting models are superior to other models, demonstrating that all the ML techniques applied to higher-order moment predictions in this study have specific capabilities and inevitable restrictions.
此外，鉴于我们进一步使用组合预测模型进行波动率预测，证据表明组合预测模型优于其他模型，表明本研究中应用于高阶矩预测的所有机器学习技术都具有特定的能力和不可避免的限制。
我们的论文做出了几个贡献。首先，没有研究调查过在石油市场中预测 RV 时，高阶（高达六阶）矩的作用。Devpura 和 Narayan（2020）断言，COVID-19 大流行在其早期阶段导致石油价格波动增加了近 10 倍，每天对石油价格波动贡献了 8%至 22%。
Due to such oil price volatility characteristics, third- and fourth-order moments may be inadequate for understanding volatility dynamics. In fact, empirical research on hyperskewness and hyperkurtosis has offered important predictive information on sovereign bond returns, cryptocurrency returns, and exchange-rate behavior but not oil price volatility. Therefore, we fill a gap in the existing literature and discover that higher (third-to sixth-order) moments are effective for oil futures volatility forecasting.
由于这种油价波动特性，三阶和四阶矩可能不足以理解波动动态。事实上，关于超偏度和超峰度的实证研究已经为国债收益、加密货币收益和汇率行为提供了重要的预测信息，但并非油价波动。因此，我们填补了现有文献的空白，并发现更高阶（三阶至六阶）矩对于石油期货波动预测是有效的。
Second, previous papers have studied higher-order moments using GARCH, HAR, and MIDAS models (Khademalomoom et al., 2019; Mei et al., 2017). Considering the complexity of the oil futures market and the variable correlations among higher-order moments, this paper is the first to employ ML models to explore the relevance of higher-order moments to oil futures volatility.
其次，先前的研究使用 GARCH、HAR 和 MIDAS 模型研究了高阶矩（Khademalomoom 等，2019；Mei 等，2017）。鉴于石油期货市场的复杂性和高阶矩之间的变量相关性，本文首次采用机器学习（ML）模型来探讨高阶矩对石油期货波动性的相关性。
Also, since we further use combination forecasting models for volatility prediction, the evidence indicates that combination forecasting models are superior to other models, demonstrating that all the ML techniques applied to higher-order moment predictions in this study have specific capabilities and inevitable restrictions.
此外，鉴于我们进一步使用组合预测模型进行波动率预测，证据表明组合预测模型优于其他模型，表明本研究中应用于高阶矩预测的所有机器学习技术都具有特定的能力和不可避免的限制。
Third, we pay attention to the interpretability of ML models and the contribution of each higher-order moment to forecasting, measured using three different tools.
第三，我们关注机器学习模型的可解释性和每个高阶矩对预测的贡献，使用三种不同的工具进行衡量。
Moreover, considering out-of-sample prediction results produced by HAR-type models collectively, we conclude that odd moments outperform even moments for short-term forecasts, whereas even moments contain more predictive information for long-term forecasts.
此外，综合考虑 HAR 类型模型产生的样本外预测结果，我们得出结论：对于短期预测，奇数矩优于偶数矩，而对于长期预测，偶数矩包含更多的预测信息。
Fourth, we consider the special case of the COVID-19 crisis and further measure the impact of higher-order moments on positive and negative volatility forecasts.
第四，我们考虑 COVID-19 危机的特殊情况，并进一步测量高阶矩对正负波动率预测的影响。
Additionally, given the scarcity of previous studies on China's oil market, our study concentrates on the China's oil futures market to explore the relationship between higher-order moments and volatility forecasting, and utilize 5-min high-frequency data to better reflect the volatility information (Andersen and Bollerslev, 1998; Haugom et al., 2014; Liu et al., 2015).
此外，鉴于以往关于中国石油市场的研究较少，本研究聚焦于中国石油期货市场，以探讨高阶矩与波动率预测之间的关系，并利用 5 分钟高频数据更好地反映波动率信息（Andersen 和 Bollerslev，1998；Haugom 等，2014；Liu 等，2015）。
第三，我们关注机器学习模型的可解释性和每个高阶矩对预测的贡献，使用三种不同的工具进行衡量。
Moreover, considering out-of-sample prediction results produced by HAR-type models collectively, we conclude that odd moments outperform even moments for short-term forecasts, whereas even moments contain more predictive information for long-term forecasts.
此外，综合考虑 HAR 类型模型产生的样本外预测结果，我们得出结论：对于短期预测，奇数矩优于偶数矩，而对于长期预测，偶数矩包含更多的预测信息。
Fourth, we consider the special case of the COVID-19 crisis and further measure the impact of higher-order moments on positive and negative volatility forecasts.
第四，我们考虑 COVID-19 危机的特殊情况，并进一步测量高阶矩对正负波动率预测的影响。
Additionally, given the scarcity of previous studies on China's oil market, our study concentrates on the China's oil futures market to explore the relationship between higher-order moments and volatility forecasting, and utilize 5-min high-frequency data to better reflect the volatility information (Andersen and Bollerslev, 1998; Haugom et al., 2014; Liu et al., 2015).
此外，鉴于以往关于中国石油市场的研究较少，本研究聚焦于中国石油期货市场，以探讨高阶矩与波动率预测之间的关系，并利用 5 分钟高频数据更好地反映波动率信息（Andersen 和 Bollerslev，1998；Haugom 等，2014；Liu 等，2015）。
The remainder of the paper is organized as follows. Next part provides a definition of higher-order moments. Section 3 describes the data. Section 4 introduces the models used to forecast oil price volatility. Section 5 presents the empirical prediction analysis and results. Section 6 explains the robustness testing. Section 7 provides a further discussion of positive and negative volatility forecasts, and the final section presents the conclusions.
本文其余部分组织如下。下文部分提供了高阶矩的定义。第 3 节描述了数据。第 4 节介绍了用于预测油价波动的模型。第 5 节展示了实证预测分析和结果。第 6 节解释了稳健性测试。第 7 节进一步讨论了正负波动预测，最后部分提出了结论。
本文其余部分组织如下。下文部分提供了高阶矩的定义。第 3 节描述了数据。第 4 节介绍了用于预测油价波动的模型。第 5 节展示了实证预测分析和结果。第 6 节解释了稳健性测试。第 7 节进一步讨论了正负波动预测，最后部分提出了结论。
2. Volatility estimation and higher-order moments
2. 波动率估计和高阶矩
2.1. Realized volatility 2.1. 实现波动率
The purpose of this study is to forecast the realized volatility of China's oil prices as a natural estimator for the quadratic variation of a process. Following Andersen and Bollerslev (1998), we define day t's realized volatility as:(1)where is the intraday return on day , and is the sampling frequency. Thus, , where is the total trading days, is the number of intraday intervals, and is the log oil price.
本研究旨在预测中国石油价格的实现波动率，将其作为过程二次变差的自然估计量。遵循 Andersen 和 Bollerslev（1998）的定义，我们定义第 t 天的实现波动率 为： (1) ，其中 是第 天的日内回报率 ， 是采样频率。因此， ，其中 是总交易日数， 是日内间隔数， 是石油价格的对数。
本研究旨在预测中国石油价格的实现波动率，将其作为过程二次变差的自然估计量。遵循 Andersen 和 Bollerslev（1998）的定义，我们定义第 t 天的实现波动率 为： (1) ，其中 是第 天的日内回报率 ， 是采样频率。因此， ，其中 是总交易日数， 是日内间隔数， 是石油价格的对数。
2.2. Higher-order moment methods
2.2. 高阶矩方法
Moments are scalar values that provide a summary of the underlying data distribution's univariate features. They reflect the average of a unimodal distribution's multiple powers of deviation from the mean (Pillai, 2019). The third-to sixth-order moments (skewness, kurtosis, hyperskewness, and hyperkurtosis, respectively) have different meanings, and we define higher-order moments in the following paragraphs.
瞬间是标量值，提供了底层数据分布的单变量特征的总结。它们反映了单峰分布从均值出发的多个偏差幂的平均值（Pillai，2019）。三至六阶矩（分别为偏度、峰度、超偏度和超峰度）有不同的含义，我们将在下一段中定义更高阶的矩。
瞬间是标量值，提供了底层数据分布的单变量特征的总结。它们反映了单峰分布从均值出发的多个偏差幂的平均值（Pillai，2019）。三至六阶矩（分别为偏度、峰度、超偏度和超峰度）有不同的含义，我们将在下一段中定义更高阶的矩。
According to Amaya et al. (2015), skewness (the third moment) explains the asymmetry of daily returns distribution, and kurtosis (the fourth moment) reflects extreme occurrences. We compute the realized third moment, , and realized fourth moment, as:(2)(3)
根据 Amaya 等人（2015 年）的研究，偏度（三阶矩）解释了日收益分布的不对称性，峰度（四阶矩）反映了极端事件。我们计算实现的三阶矩， ，和实现的四阶矩， ，如下： (2) (3)
根据 Amaya 等人（2015 年）的研究，偏度（三阶矩）解释了日收益分布的不对称性，峰度（四阶矩）反映了极端事件。我们计算实现的三阶矩， ，和实现的四阶矩， ，如下： (2) (3)
Furthermore, following Khademalomoom et al. (2019) and Kinateder and Papavassiliou (2019), hyperskewness (the fifth moment) is the asymmetric sensitivity of the kurtosis, and hyperkurtosis (the sixth moment) measures the level of peakedness and tailedness of the normal distribution. We define the realized fifth moment, , and realized sixth moment, , as follows:(4)(5)
此外，根据 Khademalomoom 等人（2019 年）和 Kinateder 与 Papavassiliou（2019 年）的研究，超偏度（第五矩）是峰度的非对称敏感性，超峰度（第六矩）衡量正态分布的尖峰程度和尾部程度。我们定义实现第五矩 和实现第六矩 如下： (4) (5)
此外，根据 Khademalomoom 等人（2019 年）和 Kinateder 与 Papavassiliou（2019 年）的研究，超偏度（第五矩）是峰度的非对称敏感性，超峰度（第六矩）衡量正态分布的尖峰程度和尾部程度。我们定义实现第五矩 和实现第六矩 如下： (4) (5)
3. Methodology 3. 研究方法
3.1. HAR-type models 3.1. HAR 型模型
Benchmark forecasting (HAR) model. Our research is based on the HAR framework originally developed by Corsi (2009), which become popular due to its ease of implementation and effective performance.
基准预测（HAR）模型。我们的研究基于 Corsi（2009）最初开发的 HAR 框架，该框架因其易于实现和有效性能而变得流行。
The idea is to blend daily (past 1 day), weekly (past 5 days), and monthly (past 22 days) average volatility elements to capture the empirical properties of volatility series, such as multi-scaling behavior, long memory, and fat tails. The HAR model is defined as follows(6)where represents the 1-day, 1-week, and 1-month-ahead RV, respectively; is the daily defined in Eq. (1); and denote the average daily over lags 1 to 5 and lags 1 to 22, representing the weekly and monthly .
将每日（过去 1 天）、每周（过去 5 天）和每月（过去 22 天）的平均波动性元素混合，以捕捉波动性序列的经验性质，如多尺度行为、长记忆和厚尾。HAR 模型定义为如下 (6) 其中 分别代表 1 天、1 周和 1 个月的预期回报率； 是方程（1）中定义的每日 ； 和 表示滞后 1 到 5 和滞后 1 到 22 的平均每日 ，代表每周和每月的 。
基准预测（HAR）模型。我们的研究基于 Corsi（2009）最初开发的 HAR 框架，该框架因其易于实现和有效性能而变得流行。
The idea is to blend daily (past 1 day), weekly (past 5 days), and monthly (past 22 days) average volatility elements to capture the empirical properties of volatility series, such as multi-scaling behavior, long memory, and fat tails. The HAR model is defined as follows(6)where represents the 1-day, 1-week, and 1-month-ahead RV, respectively; is the daily defined in Eq. (1); and denote the average daily over lags 1 to 5 and lags 1 to 22, representing the weekly and monthly .
将每日（过去 1 天）、每周（过去 5 天）和每月（过去 22 天）的平均波动性元素混合，以捕捉波动性序列的经验性质，如多尺度行为、长记忆和厚尾。HAR 模型定义为如下 (6) 其中 分别代表 1 天、1 周和 1 个月的预期回报率； 是方程（1）中定义的每日 ； 和 表示滞后 1 到 5 和滞后 1 到 22 的平均每日 ，代表每周和每月的 。
Relatively low (third- and fourth-order) moment-based forecasting model (HAR-LOW): To check whether the two highest moments (hyperskewness and hyperkurtosis) contain information for oil price volatility forecasting, we include only third- and fourth-order moments in the benchmark model. The model is expressed by(7)where and are vectors of the daily, weekly, and monthly skewness and kurtosis, respectively, and and are the coefficients for and .
相对较低阶（三阶和四阶）的基于矩的预测模型（HAR-LOW）：为了检验两个最高阶矩（超偏度和超峰度）是否包含油价波动预测的信息，我们在基准模型中仅包括三阶和四阶矩。该模型表示为 (7) ，其中 和 分别是每日、每周和每月的偏度和峰度向量， 、 和 是 和 的系数。
相对较低阶（三阶和四阶）的基于矩的预测模型（HAR-LOW）：为了检验两个最高阶矩（超偏度和超峰度）是否包含油价波动预测的信息，我们在基准模型中仅包括三阶和四阶矩。该模型表示为 (7) ，其中 和 分别是每日、每周和每月的偏度和峰度向量， 、 和 是 和 的系数。
Odd (third- and fifth-order) moment-based forecasting model (HAR-ODD): To explore the importance of odd-order moments, we expand the HAR model by incorporating the third-order moment (skewness) and the fifth-order moment (hyperskewness). The HAR-ODD model is given by(8)where and are vectors of the daily, weekly, and monthly skewness and hyperskewness, respectively, and and are the coefficients for and .
奇数阶矩（三阶和五阶）预测模型（HAR-ODD）：为了探讨奇数阶矩的重要性，我们通过引入三阶矩（偏度）和五阶矩（超偏度）来扩展 HAR 模型。HAR-ODD 模型由 (8) 给出，其中 和 分别是每日、每周和每月的偏度和超偏度向量， 和 是 和 的系数。
奇数阶矩（三阶和五阶）预测模型（HAR-ODD）：为了探讨奇数阶矩的重要性，我们通过引入三阶矩（偏度）和五阶矩（超偏度）来扩展 HAR 模型。HAR-ODD 模型由 (8) 给出，其中 和 分别是每日、每周和每月的偏度和超偏度向量， 和 是 和 的系数。
Even (fourth- and sixth-order) moment-based forecasting model (HAR-EVEN): To investigate the difference between even and odd moments, we extend the model by inserting the fourth-order moment (kurtosis) and sixth-order moment (hyperkurtosis). We termed the extended model HAR-ODD, which is expressed by(9)where and are vectors of the daily, weekly, and monthly kurtosis and hyperkurtosis, respectively, and and are the coefficients for and .
即使（四阶和六阶）矩预测模型（HAR-EVEN）：为了研究偶数阶和奇数阶矩之间的差异，我们通过插入四阶矩（峰度）和六阶矩（超峰度）来扩展模型。我们将扩展后的模型称为 HAR-ODD，其表达式为 (9) ，其中 和 分别是每日、每周和每月的峰度和超峰度向量， 和 是 和 的系数。
即使（四阶和六阶）矩预测模型（HAR-EVEN）：为了研究偶数阶和奇数阶矩之间的差异，我们通过插入四阶矩（峰度）和六阶矩（超峰度）来扩展模型。我们将扩展后的模型称为 HAR-ODD，其表达式为 (9) ，其中 和 分别是每日、每周和每月的峰度和超峰度向量， 和 是 和 的系数。
All-order (third, fourth, fifth, and sixth) moment-based forecasting model (HAR-ALL): We add all higher order moments into the benchmark model and constructed the HAR-ALL model as:(10)
所有阶次（第三、第四、第五和第六）矩预测模型（HAR-ALL）：我们将所有高阶矩添加到基准模型中，构建了 HAR-ALL 模型： (10)
所有阶次（第三、第四、第五和第六）矩预测模型（HAR-ALL）：我们将所有高阶矩添加到基准模型中，构建了 HAR-ALL 模型： (10)
3.2. Machine learning models
3.2. 机器学习模型
3.2.1. Least absolute shrinkage and selection operator
3.2.1. 最小绝对收缩与选择算子
The problem of overfitting tends to occur when the number of indicators increases in low signal-to-noise environments. Instead of fitting the relevant information, linear models tend to fit the noise.
过拟合问题通常发生在低信噪比环境中指标数量增加时。线性模型倾向于拟合噪声，而不是拟合相关信息。
LASSO regression is an efficient technique for avoiding overfitting, increasing out-of-sample performance, and selecting predictors that are computationally efficient (Ding et al., 2021).
LASSO 回归是一种有效的避免过拟合、提高样本外性能以及选择计算效率高的预测因子的技术（Ding 等，2021）。
过拟合问题通常发生在低信噪比环境中指标数量增加时。线性模型倾向于拟合噪声，而不是拟合相关信息。
LASSO regression is an efficient technique for avoiding overfitting, increasing out-of-sample performance, and selecting predictors that are computationally efficient (Ding et al., 2021).
LASSO 回归是一种有效的避免过拟合、提高样本外性能以及选择计算效率高的预测因子的技术（Ding 等，2021）。
The LASSO technique is essentially a restricted least squares regression in which regression coefficients are shrunk by imposing a penalty term. The LASSO equation for forecasting oil price volatility is given as(11)and(12)where denotes the predictor on day , is the total number of predictors in the regression, is the regression coefficients' shrinkage estimator, and is the tuning parameter that controls the shrinkage predictor in terms of penalty strictness. The first part of Eq. (12) is the least squares criterion, and the second part is the penalty term for the regression parameters. Increasing causes more coefficients from the LASSO regression to be penalized to zero, with stricter coefficient selection. All the coefficients are set to zero when . The LASSO model includes all predictors from the HAR-ALL model.
LASSO 技术本质上是一种限制性最小二乘回归，通过施加惩罚项来缩小回归系数。预测油价波动的 LASSO 方程为 (11) 和 (12) ，其中 表示第 天的 预测因子， 是回归中的预测因子总数， 是回归系数的收缩估计量， 是控制收缩预测因子的惩罚严格程度的调整参数。方程（12）的第一部分是最小二乘准则，第二部分是回归参数的惩罚项。增加 会导致更多 LASSO 回归系数被惩罚为零，从而进行更严格的系数选择。当 时，所有系数都设置为零。LASSO 模型包括 HAR-ALL 模型中的所有预测因子。
LASSO 技术本质上是一种限制性最小二乘回归，通过施加惩罚项来缩小回归系数。预测油价波动的 LASSO 方程为 (11) 和 (12) ，其中 表示第 天的 预测因子， 是回归中的预测因子总数， 是回归系数的收缩估计量， 是控制收缩预测因子的惩罚严格程度的调整参数。方程（12）的第一部分是最小二乘准则，第二部分是回归参数的惩罚项。增加 会导致更多 LASSO 回归系数被惩罚为零，从而进行更严格的系数选择。当 时，所有系数都设置为零。LASSO 模型包括 HAR-ALL 模型中的所有预测因子。
Equivalently, Eq. (12) can also be written as an OLS estimator:(13)with an penalty function of(14)where the parameter plays the same role as , controlling for the amount of shrinkage.
等效地，方程（12）也可以写成带有惩罚函数 (14) 的 OLS 估计量 (13) ，其中参数 与 扮演相同角色，控制收缩量。
等效地，方程（12）也可以写成带有惩罚函数 (14) 的 OLS 估计量 (13) ，其中参数 与 扮演相同角色，控制收缩量。
3.2.2. Elastic net 3.2.2. 弹性网络
Introduced by Zou and Hastie (2005), EN is another widely used shrinkage method to overcome overfitting. Besides the penalty of the norm in the LASSO method, the elastic net method introduces penalties simultaneously. Like the LASSO model, the elastic model forecasts oil price volatility, computed as follows:(15)and(16)where is a constant ranging from 0 to 1. When , the EN becomes LASSO, and it transforms to the ridge regression when . The predictor variables of the EN model are the same as those included in the HAR-ALL model.
由 Zou 和 Hastie（2005）提出，EN 是另一种广泛使用的收缩方法，用于克服过拟合问题。除了 LASSO 方法中的 范数惩罚外，弹性网络方法同时引入了 惩罚。与 LASSO 模型类似，弹性模型预测油价波动性，计算如下： (15) 和 (16) ，其中 是一个介于 0 到 1 之间的常数。当 时，EN 变为 LASSO，而当 时，它转变为岭回归。EN 模型的预测变量与 HAR-ALL 模型中包含的变量相同。
由 Zou 和 Hastie（2005）提出，EN 是另一种广泛使用的收缩方法，用于克服过拟合问题。除了 LASSO 方法中的 范数惩罚外，弹性网络方法同时引入了 惩罚。与 LASSO 模型类似，弹性模型预测油价波动性，计算如下： (15) 和 (16) ，其中 是一个介于 0 到 1 之间的常数。当 时，EN 变为 LASSO，而当 时，它转变为岭回归。EN 模型的预测变量与 HAR-ALL 模型中包含的变量相同。
Eq. (16) can also be described as an OLS predictor:(17)with and penalty functions of(18)where the parameter plays the same role as , controlling for the amount of shrinkage.
式(16)也可以描述为一个 OLS 预测器：带有 (18) 的惩罚函数 和 的 (17) ，其中参数 与 扮演相同角色，控制收缩量。
式(16)也可以描述为一个 OLS 预测器：带有 (18) 的惩罚函数 和 的 (17) ，其中参数 与 扮演相同角色，控制收缩量。
3.2.3. Gradient boosting 3.2.3. 梯度提升
Possible nonlinear relationships between dependent variables and predictors, and the interactions among predictors, cannot be captured by linear models. In contrast, regression trees allow for nonlinearity and consider interactions among predictors using a fully nonparametric approach. On this basis, Friedman (2001) introduce GBDT, which grows each tree depending on information extracted from the preceding tree. The essential idea behind gradient boosting is the tree-boosting model.
可能无法用线性模型捕捉因变量与预测变量之间的非线性关系以及预测变量之间的交互作用。相比之下，回归树允许非线性，并采用完全非参数方法考虑预测变量之间的交互作用。基于此，弗里德曼（2001 年）引入了 GBDT，该算法根据前一棵树提取的信息来生长每一棵树。梯度提升背后的基本思想是树提升模型。
To minimize the errors caused by the current models, new models are added until no further enhancement is achieved. This process is known as “boosting” (Christensen et al., 2021) and proceeds as follows:
为了最小化当前模型造成的错误，添加新的模型直到不再获得进一步改进。这个过程被称为“提升”（Christensen 等人，2021 年）并按以下方式进行：
对于 ，其中 是学习率， 是最终预测。
可能无法用线性模型捕捉因变量与预测变量之间的非线性关系以及预测变量之间的交互作用。相比之下，回归树允许非线性，并采用完全非参数方法考虑预测变量之间的交互作用。基于此，弗里德曼（2001 年）引入了 GBDT，该算法根据前一棵树提取的信息来生长每一棵树。梯度提升背后的基本思想是树提升模型。
To minimize the errors caused by the current models, new models are added until no further enhancement is achieved. This process is known as “boosting” (Christensen et al., 2021) and proceeds as follows:
为了最小化当前模型造成的错误，添加新的模型直到不再获得进一步改进。这个过程被称为“提升”（Christensen 等人，2021 年）并按以下方式进行：
- •is set as a constant that depends on . The prediction correlates with the average when using the mean squared error(MSE) as the loss function.
被设定为依赖于 的常数。当使用均方误差（MSE）作为损失函数时，预测与平均值相关。 - •The loss function's negative gradient (equivalent to the residuals) relative to the prediction value is calculated.
损失函数相对于预测值的负梯度（相当于残差）被计算。 - •The residuals based on a shallow tree are fitted and yield a group of terminal nodes , where and indicate the leaf and tree, respectively.
基于浅层树的残差拟合产生了一组终端节点 ，其中 和 分别表示叶子和树。 - •
- •Iteratively, is modified as follows:
迭代地， 被修改如下：
对于 ，其中 是学习率， 是最终预测。
3.2.4. Random forests 3.2.4. 随机森林
The essence of RF is a combination prediction model, which randomly builds a forest containing many single regression trees (Breiman, 2001) with weak correlations or even irrelevance. Subsequently, many regression trees use aggregate statistics to jointly predict the value of the newly observed output variable.
射频（RF）的本质是一种组合预测模型，它随机构建一个包含许多单个回归树（Breiman，2001）的森林，这些回归树之间具有弱相关性甚至无关。随后，许多回归树使用汇总统计量共同预测新观察到的输出变量的值。
The input variables of the RF model in this study include the predictors of the standard HAR (RVD, RVW, and RVM), all higher-order moments, and their lags, which are the same as in the HAR-ALL model.
本研究中 RF 模型的输入变量包括标准 HAR（RVD、RVW 和 RVM）的预测因子、所有高阶矩及其滞后，与 HAR-ALL 模型相同。
射频（RF）的本质是一种组合预测模型，它随机构建一个包含许多单个回归树（Breiman，2001）的森林，这些回归树之间具有弱相关性甚至无关。随后，许多回归树使用汇总统计量共同预测新观察到的输出变量的值。
The input variables of the RF model in this study include the predictors of the standard HAR (RVD, RVW, and RVM), all higher-order moments, and their lags, which are the same as in the HAR-ALL model.
本研究中 RF 模型的输入变量包括标准 HAR（RVD、RVW 和 RVM）的预测因子、所有高阶矩及其滞后，与 HAR-ALL 模型相同。
In the regression tree T, the input indicator space is recursively partitioned by the terminal nodes into non-overlapping regions . At the top level of the regression tree, the model uses a greedy algorithm to select the first partition in such a way that the partition variable and its partition point , defining the half-plane and , minimize the loss function:(20)where signifies the half-plane-specific data for RV, , is the half-plane-specific mean of RV. The region-specific squared error loss is minimized by selecting half-plane-specific means via inner minimization.
在回归树 T 中，输入指示空间 通过终端节点 递归地划分为 个非重叠区域 。在回归树的最顶层，模型使用贪婪算法选择第一个分区，使得分区变量 及其分区点 ，定义了半平面 和 ，最小化损失函数： (20) 其中 表示 RV 的半平面特定 数据， 是 RV 的半平面特定均值。通过内部最小化选择半平面特定均值，以最小化区域特定的平方误差损失。
The outer minimization finds the first optimal partitioning predictor and the first optimal partitioning point, resulting in a new regression tree with two terminal nodes, by searching through all possible combinations of and .
外部最小化找到第一个最优分区预测器和第一个最优分区点，通过搜索所有可能的 和 的组合，从而得到一个新的具有两个终端节点的回归树。
在回归树 T 中，输入指示空间 通过终端节点 递归地划分为 个非重叠区域 。在回归树的最顶层，模型使用贪婪算法选择第一个分区，使得分区变量 及其分区点 ，定义了半平面 和 ，最小化损失函数： (20) 其中 表示 RV 的半平面特定 数据， 是 RV 的半平面特定均值。通过内部最小化选择半平面特定均值，以最小化区域特定的平方误差损失。
The outer minimization finds the first optimal partitioning predictor and the first optimal partitioning point, resulting in a new regression tree with two terminal nodes, by searching through all possible combinations of and .
外部最小化找到第一个最优分区预测器和第一个最优分区点，通过搜索所有可能的 和 的组合，从而得到一个新的具有两个终端节点的回归树。
The above method is then repeated until the terminal tree's maximum number of terminal nodes or each terminal node's minimum number of observations, whichever comes first, is reached.
上述方法重复进行，直到达到终端树的最大终端节点数或每个终端节点的最小观察数，以先到者为准。
The final tree employs the region-specific averages of the variables to forecast RV after sending them to the best region (1 = indicator function):(21)
最终树使用变量的区域特定平均值，在将它们发送到最佳区域（1 = 指示函数）后预测 RV： (21)
上述方法重复进行，直到达到终端树的最大终端节点数或每个终端节点的最小观察数，以先到者为准。
The final tree employs the region-specific averages of the variables to forecast RV after sending them to the best region (1 = indicator function):(21)
最终树使用变量的区域特定平均值，在将它们发送到最佳区域（1 = 指示函数）后预测 RV： (21)
The “randomness” of the RF is reflected by two criteria: 1) the training sample obtained by resampling bootstrapping in the original sample has randomness, and 2) the grouping variables are random in nature.
射频的“随机性”通过两个标准体现：1）通过在原始样本中进行重采样自举得到的训练样本具有随机性；2）分组变量在本质上具有随机性。
To construct each regression tree, the present optimum grouping variable is obtained from a random subset of the candidate variables for all variables (rather than from the set of all variables).
为了构建每个回归树，当前最优分组变量是从所有候选变量的随机子集中获得的（而不是从所有变量的集合中获得）。
射频的“随机性”通过两个标准体现：1）通过在原始样本中进行重采样自举得到的训练样本具有随机性；2）分组变量在本质上具有随机性。
To construct each regression tree, the present optimum grouping variable is obtained from a random subset of the candidate variables for all variables (rather than from the set of all variables).
为了构建每个回归树，当前最优分组变量是从所有候选变量的随机子集中获得的（而不是从所有变量的集合中获得）。
3.3. Combination forecasting models
3.3. 组合预测模型
We introduce four popular ML models in the previous section, which allow us to obtain four individual predictions for volatility.
我们在前一节介绍了四种流行的机器学习模型，这些模型使我们能够获得四个关于波动性的独立预测。
However, the stability of out-of-sample predictions cannot be guaranteed due to model uncertainty, and different ML models might be effective for different scenarios (Li and Tang, 2021). To solve this issue, we build on the earlier literature (Liang et al., 2020; Zhang et al., 2019) and apply the five combination approaches introduced by Rapach et al. (2010) to make forecasting more robust and further exploit the predictive potential of higher-order moments. Statistically, the combination volatility forecast is given as:(22)where denotes the forecasting of China's oil market volatility on day , represents the individual prediction yielded by the ML model, represents the combined weight of the individual prediction created at t, and N is the number of all the individual ML models used. The five combination weighting techniques are described in detail in the sections that follow.
然而，由于模型的不确定性，无法保证样本外预测的稳定性，不同的机器学习模型可能适用于不同的场景（Li 和 Tang，2021）。为了解决这个问题，我们借鉴了早期文献（Liang 等人，2020；Zhang 等人，2019）并应用了 Rapach 等人（2010）提出的五种组合方法，以提高预测的稳健性并进一步挖掘高阶矩的预测潜力。从统计学的角度来看，组合波动率预测如下： (22) 其中 表示对第 天中国石油市场波动的预测， 代表由 机器学习模型产生的单个预测， 代表在 t 时刻创建的 单个预测的加权组合，N 是所有单个机器学习模型的总数。接下来的章节将详细描述五种组合加权技术。
我们在前一节介绍了四种流行的机器学习模型，这些模型使我们能够获得四个关于波动性的独立预测。
However, the stability of out-of-sample predictions cannot be guaranteed due to model uncertainty, and different ML models might be effective for different scenarios (Li and Tang, 2021). To solve this issue, we build on the earlier literature (Liang et al., 2020; Zhang et al., 2019) and apply the five combination approaches introduced by Rapach et al. (2010) to make forecasting more robust and further exploit the predictive potential of higher-order moments. Statistically, the combination volatility forecast is given as:(22)where denotes the forecasting of China's oil market volatility on day , represents the individual prediction yielded by the ML model, represents the combined weight of the individual prediction created at t, and N is the number of all the individual ML models used. The five combination weighting techniques are described in detail in the sections that follow.
然而，由于模型的不确定性，无法保证样本外预测的稳定性，不同的机器学习模型可能适用于不同的场景（Li 和 Tang，2021）。为了解决这个问题，我们借鉴了早期文献（Liang 等人，2020；Zhang 等人，2019）并应用了 Rapach 等人（2010）提出的五种组合方法，以提高预测的稳健性并进一步挖掘高阶矩的预测潜力。从统计学的角度来看，组合波动率预测如下： (22) 其中 表示对第 天中国石油市场波动的预测， 代表由 机器学习模型产生的单个预测， 代表在 t 时刻创建的 单个预测的加权组合，N 是所有单个机器学习模型的总数。接下来的章节将详细描述五种组合加权技术。
- •Mean combination: Each individual forecast is the same (i.e., ).
均值组合：每个个体预测都相同（即 ）。 - •Median combination: The median of the N individual predictions is used in this combination strategy.
中位数组合：本组合策略使用 N 个个体预测的中位数。 - •Trimmed mean combination: The maximum and minimum individual predictions are assigned to , and the remaining individual forecasts are .
修剪平均值组合：将最大和最小个体预测值分配给 ，其余个体预测值分配给 。 - •Discount mean square prediction error (DMSPE) combination: The combined weight of the individual forecast on day is computed as , where , is the actual on day , the first training sample has m total observations, and θ stands for a discount factor. We consider two values of (1 and 0.9) and employ DMSPE(1) and DMSPE(0.9) in this study.
折扣均方预测误差（DMSPE）组合：第 天的单个预测的加权组合计算为 ，其中 ， 是第 天的实际 ，第一个训练样本有 m 个总观测值，θ代表折扣因子。我们考虑了 的两个值（1 和 0.9），并在本研究中使用了 DMSPE(1)和 DMSPE(0.9)。
Furthermore, an additional effective forecasting combination model, the polynomially weighted average with multiple rates (ML-Poly), is considered. This model involves the learner making online sequential predictions a series of rounds with the assistance of K experts.
此外，还考虑了一种额外的有效预测组合模型，即多项式加权平均多速率（ML-Poly）。该模型涉及学习者借助 K 位专家进行一系列在线序列预测。
In each round
此外，还考虑了一种额外的有效预测组合模型，即多项式加权平均多速率（ML-Poly）。该模型涉及学习者借助 K 位专家进行一系列在线序列预测。
In each round