Forecasting crude oil prices volatility with GARCH models

Universidade do Minho Escola de Economia e Gestão Monique Oliveira Moreira de Souza Forecasting Crude Oil Prices Volatility with GARCH Models January 2021 UMinho | 2021 Monique Oliveira Moreira de Souza Forecasting Crude Oil Prices Volatility with GARCH Models Monique Oliveira Moreira de Souza Forecasting Crude Oil Prices Volatility with GARCH Models A thesis presented for the degree of Master in Monetary, Banking and Financial Economics Research under the supervision of Professor Cristina Alexandra Oliveira Amado Universidade do Minho Escola de Economia e Gestão January 2021 ii DIREITOS DE AUTOR E CONDIÇÕES DE UTILIZAÇÃO DO TRABALHO POR TERCEIROS Este é um trabalho académico que pode ser utilizado por terceiros desde que respeitadas as regras e boas práticas internacionalmente aceites, no que concerne aos direitos de autor e direitos conexos. Assim, o presente trabalho pode ser utilizado nos termos previstos na licença abaixo indicada. Caso o utilizador necessite de permissão para poder fazer um uso do trabalho em condições não previstas no licenciamento indicado, deverá contactar o autor, através do RepositóriUM da Universidade do Minho. Licença concedida aos utilizadores deste trabalho Atribuição CC BY https://creativecommons.org/licenses/by/4.0/ iii STATEMENT OF INTEGRITY I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. iv AKNOWLEDMENTS This dissertation represents the final stage of a long journey that became especially challenging due to the pandemic crisis that still persists, fortunately I found all the support I needed. First, I would like to acknowledge my advisor, Professor Cristina Amado whose expertise and guidance were crucial not only for the results of this research work, but for my personal growth. Although we faced an atypical year, I learned a lot from all the honest and constructive criticisms during our online meetings. Finally, I would like to acknowledge my beloved family and friends, especially my husband Daniel, for all the words of encouragement and kind gestures during all the challenging moments I faced in this thesis. You are always there for me. v RESUMO As variações do preço do petróleo têm um grande impacto a nível financeiro em todo o mundo. Assim, o objetivo desta tese é realizar um estudo empírico de previsão da volatilidade dos preços do petróleo através da estimação de vários modelos univariados da classe GARCH para o período de Janeiro de 1986 até Dezembro de 2019. As previsões conduzidas fora da amostra para horizontes de um, cinco e vinte dias foram obtidas através do model confidence set (MCS) proposto por Hansen et al. (2011) que consiste numa sequência de testes que retornam um ranking com o desempenho de cada modelo e cuja grande vantagem é a possibilidade de seleção de mais do que um melhor modelo, o que proporciona uma solução mais realista. Os resultados empíricos indicam que os modelos que conseguem captar a persistência ou a longa memória na volatilidade e a assimetria na volatilidade do preço do petróleo, têm uma performance superior em todos os horizontes de previsão. Palavras-Chave: Mercado do Petróleo, Previsões de Volatilidade, GARCH, Model Confidence Set vi ABSTRACT Changes in crude oil prices have a major impact on finances worldwide. Thus, the proposal of this thesis is to conduct an empirical study to forecast the volatility of crude oil prices by estimating several univariate GARCH-class models for the period from January 1986 to December 2019. The out-of-sample forecasts horizons for one, five and twenty days were obtained through the model confidence set (MCS) procedure proposed by Hansen et al. (2011) which consists of a sequence of tests that return a ranking with the performance of each model and whose great advantage is the possibility of selecting more than one best model which provides a more realistic solution. The empirical results indicate that models that captures the long-memory or volatility persistence and asymmetry in crude oil prices volatility perform better in all forecasting horizons. Key Words: Crude Oil Market, Volatility forecasting, GARCH, Model Confidence Set vii "Prediction is very difficult, especially if it's about the future." Niels Bohr, Nobel laureate in Physics 1922 viii CONTENTS 1. Introduction .................................................................................................................................. 1 2. Brief Historical Overview of Crude Oil Price Returns .................................................. 4 3. Literature Review....................................................................................................................... 8 4. Methodology...............................................................................................................................14 The Class of GARCH Models .........................................................................................14 Forecasting Model Evaluation .....................................................................................18 5. Empirical Aplication ................................................................................................................20 Preliminary Data Analysis ............................................................................................20 Estimation Results ...........................................................................................................24 Forecasting Crude Oil Volatility ..................................................................................29 6. Conclusions .................................................................................................................................33 7. References ...................................................................................................................................35 6 In the early 1980s, the opposite happened as a result of the energy crisis of the 1970s, a lower consumer demand followed which caused an ‘oil glut’ and a consequent decrease in crude oil prices. In 1990, the first Gulf War occurred when Iraq invaded Kuwait, causing the crude oil prices to skyrocket from $34 per barrel to nearly $77. After a U.S.-led military coalition managed to remove Saddam Hussein's Iraqi forces from Kuwait in early 1991, the prices dropped again. In 2007, after the subprime mortgages depreciation quickly turned into an international banking crisis in 2008, the world emerged in a severe financial crisis, consequently oil prices had a sharp decline, as the general drop in asset prices increased the demand for energy around the world decreased as well as the demand for crude oil. Through the 2010s there was again an ‘oil glut’ due a combination of many factors. During 2014–15 OPEC members consistently exceeded their production ceiling and China, the world’s biggest importer of crude oil, experienced a sharp slowdown in economic growth. At the same time, in response to record oil prices, shale oil production in the United States almost doubled from 2008 levels, due to significant technology improvements. In 2016, all these factors caused a collapse in oil prices because, although the supply glut increased it failed to meet the expected global growth. Still, according to BP’s 2019 Statistical Review of World Energy, in 2018 OPEC accounted for nearly 44% of global oil production and 81.5% of the world´s proven oil reserves. At the present OPEC counts with eight more nations, in addition to the five founding countries: Libya, United Arab Emirates, Algeria, Nigeria, Gabon, Angola, Equatorial Guinea, and Congo, totalizing 13 members. Recently, on April 20th, 2020, the crude market made history for the worst reasons, as WTI ended the day at minus $37 per barrel, entering on negative territory for the first time. Although the time series analysis for this study is between 1986 and 2019 is inevitable to mention that the effect of COVID-19 pandemic on crude oil prices represent the biggest contraction ever on its price’s history. 7 Another peculiarity about crude oil, besides its high influence in energy markets and great acceptance among financial assets, is its tangible dimension, that contribute to extremely high levels of price volatility, such as future supplies and reserves, political upheavals, adverse weather conditions, issues with the refineries or pipelines and world crises. There are three factors that define crude oil grades and directly affect barrel prices: the American Petroleum Institute (API) gravity which represents an index that measures crude oil density and normally ranges from 15 to 45 degrees, where a lower density means a higher API which indicates a light crude and the opposite a heavy crude; the sulfur content which is characterized as sweet when has fewer impurities, but is harder to find, or sour when the sulfur level is higher than 0.5%, however is less expensive due to lower quality; and finally, the proximity to tidewater and refineries. Thus, it is easy to understand why WTI is also known as “Texas Light Sweet”, since it has an API gravity around 39.6 degrees per barrel rating it as a light crude, contains about 0.24% sulfur, rating it as a sweet crude which is not only used as a benchmark in oil pricing but also as the underlying commodity of NYMEX oil future contracts. As WTI is a light crude oil, its properties and production site make it ideal for being refined in the United States, mostly in the Midwest and Gulf Coast regions; for those reasons, the town of Cushing, in Oklahoma has been the price settlement point for WTI on the NYMEX for over three decades and became the major oil supply hub for crude oil in North America. WTI historical spot prices FOB 2 in U.S. dollars per barrel since January 2nd, 1986 can be downloaded free of charges from U.S. Energy Information Administration (EIA) website. The graph from Figure 2 represents the annual average price for WTI. 2 Free on Board is a transportation term that indicates that the price for goods includes delivery at the seller's expense to a specified point and no further. 8 Figure 2. Evolution chart of WTI prices between 1986 - 2019 WTI prices have increased dramatically, rising from just 15.05 dollars per barrel in 1986, to a peak of 99.67 dollars per barrel in 2008. 3. LITERATURE REVIEW The revolution in modeling and forecasting volatility began in 1982, and had as starting point the seminal paper of Engle whom instead of assuming a constant oneperiod forecast variance, as in traditional econometric models, presented a new class of stochastic processes to model the effects of serial correlation and conditional heteroskedasticity that became known as the autoregressive conditional heteroskedasticity (ARCH) model (Engle, 1982). The model describes important stylized facts, very common in financial returns, such as fat-tailed distributions which have a much higher probability of extreme profits or losses than predicted by the normal distribution; volatility clustering characterized by large price changes being followed by large price changes and small price changes being followed by small price changes; long-range dependence, also called long memory or long-range persistence which is observed when the autocorrelation function of absolute or squared returns decays slowly across lags; and finally, the asymmetry of gains and losses which denotes that the distribution of losses has a heavier tail and, therefore, a greater influence on future volatilities than the distribution of gains. 9 A few years later, the generalized autoregressive conditional heteroskedasticity (GARCH) models was proposed by Bollerslev (1986) with a more flexible lag structure, capable of representing current variances in terms of past variances, allowed GARCH models to quickly gain greater acceptance due to its high performance in modelling and forecasting time-dependent volatility. Palm’s (1996) extensive literature about the developments of the first 15 years in modeling time-varying volatility using ARCH and GARCH processes on financial applications demonstrated that empirical applications achieved significant improvements, especially on statistical properties regarding financial time series as they were not accounted for any existing models until then. Even nowadays, the importance of ARCH and GARCH models as tools to model time series data is indisputable. However, since it is intrinsically symmetric, when forecasting skewed time series, the asymmetric GARCH model often leads to biased results. Consequently, several extensions and alternative specifications for the standard GARCH model have been presented in an attempt to appropriately take into account the volatility persistence and assist in modeling conditional volatility, but so far a complete model covering a variety of stylized facts is yet to be proposed. Bollerslev and Ole Mikkelsen (1996) and Granger and Ding (1996) research works about non-stationary behavior of volatility using long-memory persistence became very popular as they focused on the volatility of high-frequency data, especially in financial time series and crude oil prices is no exception, as persistence shows that shocks to conditional variance die at a hyperbolic rate which is slower than the exponential rate of the decay of shocks in GARCH models as accounted by Baillie (1996), whereby in this dissertation I shall use a sample period longer than 30 years, as it makes sense to apply a modelling procedure for long-term volatility movements over long observation periods. Unfortunately, when volatility persistence is high forecasts are harmed because when fitting a single-regime GARCH models on crude oil price changes, potential structural changes may be ignored and the model persistence is not reduced causing often inaccurate predictions. To solve this issue, regime-switching GARCH models were introduced into the GARCH-type models, in order to model nonlinear behavior 10 and address potential structural changes contributing to the improvement of asset prices volatility forecasts. Hamilton (1988, 1989) proposed the Markov Regime-Switching GARCH (MRSGARCH) model to allow the regimes in the Markov chain to have different volatility structures, capturing effectively potential state transition and nonlinearity which were relevant to Zhang and Wang (2015) study about the reasons behind WTI strong volatility from 2003 to 2012. The first surprising issue was the difference between the market-trading price, that is influenced by the speculation in crude oil markets and the fundamental crude oil price, that is determined by crude oil supply and demand which fluctuated less than the market-trading price. Thus, Zhang and Wang (2015) defined the deviation between crude oil markettrading price and its fundamental price as crude oil price bubbles and measured them with multivariate regression model and then employed the MRS model of Hamilton (1989) to measure the evolution of crude oil price bubbles over time. The empirical results indicated that during these price bubbles process, prices evolved between two regime-switching states: the stable state and the upheaval state. The price bubbles regime-switching process also followed a time homogeneous first-order Markov chain that made possible to statistically differentiate these two states during the price bubbles processes concluding that, in most cases, the stable states were often observed after upheaval states, being the stable state more persistent than the upheaval state, because during most of the time there were not huge surprising events during the sample period which highlight the importance of future policy actions in order to prevent instabilities in crude oil markets. Li et al. (2013) also proposed a regime-switching model named Mixture Memory GARCH (MM-GARCH) which is a mixture of the standard GARCH and the Fractionally Integrated GARCH (FIGARCH) model proposed by Baillie et al. (1996) and became often used to describe the long-range dependence on conditional variances of financial time series which was also supported by the empirical study by Zhang et 11 al. (2019) for in-sample data, where they concluded that regime-switching models perform better than the single-regime models. More recently, Zhang et al. (2019) employed a great number of linear and nonlinear GARCH-type models to forecast the volatility of daily and weekly WTI and Brent crude oil prices, from 1986 until June 2017 the single-regime GARCH, GJR-GARCH and EGARCH models were applied. The empirical results showed that, although the MRS-GARCH models provided superior results than the single-regime GARCH and MM-GARCH models in in-sample results, when confronted with out-of-sample experiments the single-regime GARCH models are not inferior than the regime-switching GARCH models, concluding that the regime-switching models do not provide an indisputable improvement in the accuracy of forecasts for crude oil price volatility. As a matter of fact, Zhang et al. (2019) recommended that volatility forecasting models should not simply elect multiple regimes for crude oil prices because the simplest models as the singleregime models in most cases outperforms regime-switching models. Kang et al. (2009) investigated the efficacy of volatility long-memory models and its performance to forecast Brent, Dubai and WTI crude oil daily prices, from 1992 until 2006 combining the Component GARCH (CGARCH) model proposed by Ding and Granger (1996) and Engle and Lee (1999) which has one component to capture short-term and another component to capture the long-term effects of volatility, together with FIGARCH model proposed by Baillie (1996). The out-of-sample forecasts achieved better results and proved that persistence or long-memory models are more useful than the linear GARCH and the Integrated GARCH (IGARCH) models proposed by Engle and Bollerslev (1986). However, it is important to highlight that the long-memory volatility CGARCH and FIGARCH models cannot capture the asymmetric leverage effect. Nevertheless, when Kang et. al (2009) paper was extended a further analysis conducted by Wei et. al (2010) employed a large number of linear and nonlinear GARCH-type models and six loss functions as a forecasting criteria and the Superior Predictive Ability (SPA) test of Hansen (2005) to ensure more robust results, but unlike Kang et. al (2009), none of the nine GARCH-type models used (including the FIGARCH) demonstrated to be superior to the others, across any loss functions for 12 both Brent and WTI prices. However, the nonlinear GARCH-type models could capture long-memory and asymmetric volatility with greater accurate forecasts than the linear ones, especially over long-time horizons, warning that when a volatility model is chosen it should not only take into account existing research, but also the correlation between the sample data, the loss functions and ultimately, the forecast aim. Recently, Hasanov et al. (2020) paper examined a greater number of volatility models, their forecasting performances and the relevance of some statistical complexities regarding four types of petroleum futures contracts traded on NYMEX. In this work nine univariate models were used, such as the random walk, historical mean, moving average, exponentially smoothing, linear regression, autoregressive, GARCH, threshold GARCH, GARCH-in-mean, the vector autoregression and bivariate GARCH to forecast WTI daily volatility of heating oil, unleaded gasoline and natural gas futures contracts, from February 1988 to December 2018 3 . They also polled all single-regime models, with two-state Markov-Switching models proposed by Haas et al., (2004) under three different distributions (Gaussian, Student’s t, and skew Student’s t) and used the MCS test of Hansen et al. (2011) to analyze the forecasting accuracy and account for structural breaks. The results showed that the Markov-Switching models are particularly useful for heating oil and unleaded gas markets but for crude oil and natural gas the results remained unchanged across many forecasting horizons. In addition, the authors included statistical models on the out-of-sample volatility forecast analysis that widely confirmed the results obtained on Sadorsky (2006), as GARCH-type models appeared to be superior in one-period forecasts results on crude oil futures case. Unlike previous forecasting studies, related with the energy market volatility using GARCH-type models which ignored the optimality of the mean equation in the 3 The only exception is the unleaded gasoline data which was discontinued after December of 2006 13 context of a single moving window approach Hasanov et al. (2020) considered the relevance of the mean equation optimality in addition to the significance of conditional distributions, using a large number of the moving windows to find out whether the introduction of statistical complexities such as regime switches and alternative distribution functions lead to any forecasting accuracy gains. In most cases, results revealed that models with optimal mean were greater than models that ignore optimality in the mean which highlights the importance of optimality in the mean of GARCH-type models. In sum, until the early 2010s there were few articles analyzing the key features of crude oil prices volatility because once the standard GARCH model cannot capture persistence or long memory, there is not an agreement about the best way of modelling and forecasting crude oil prices volatility and neither about an specific GARCH-type model, that consistently proved to be superior, as discussed in studies by Kang et al. (2009), Wei et al. (2010), Zhang et al. (2019) and Hasanov et al. (2020). In addition, since the combination of GARCH models with loss functions during forecasts estimation may lead to heterogeneous results, authors also recommend caution when combining volatility models with loss functions to evaluate forecasting performances. Bhowmik and Wang (2020) found that in the last 10 years there have been significant improvements in the research work and, although GARCH-type models have increased and presented excellent models, each model has its own strengths and weaknesses. Therefore, in order to achieve better results, other statistical techniques must be combined. However, the authors observed that few researchers used multivariate GARCH statistical techniques to analyze market volatility and returns and asymmetric GARCH models such as EGARCH, GJR-GARCH and TGARCH were often used to capture the effect of negative news on volatility of stock returns. Finally, the authors also concluded that when accounting for the model optimality, structural breaks and the use of asymmetric heavy-tailed distribution functions in the estimations, lead to significant forecasting accuracy gains. 14 4. METHODOLOGY THE CLASS OF GARCH MODELS ARCH-type models were the pioneers in modeling conditional variance processes, since they were the first to allow the incorporation of volatility clustering and mean reversion which became instantly a very flexible and powerful tool to describe the time-varying nature of conditional volatility that characterizes financial assets. Therefore, nowadays the most popular volatility model is the generalized ARCH of Bollerslev (1986), since it is more parsimonious than the ARCH model and its simplicity of implementation contributed to its great acceptance and since GARCH also avoids overfitting thanks to the ability to account for the main stylized facts common in financial returns, GARCH models became widely accepted in quantitative finance to measure and forecast volatility. However, despite its popularity, GARCH models treat shocks symmetrically which led to the proposal of several asymmetric extensions in the financial econometrics literature and thus the GARCH-family models specifications emerged to jointly estimate the conditional mean and conditional variance equations. This thesis will focus on the following seven univariate models to model and forecast crude oil returns volatility: the standard GARCH, EGARCH, GJR-GARCH, TGARCH, CGARCH, APARCH, NGARCH models. The GARCH model of Bollerslev (1986) uses both an autoregressive and moving average structure to the conditional variance. Since the return series may be linearly dependent, an Autoregressive Moving Average (ARMA) model may be needed for the conditional mean and the GARCH model to fit the conditional variance. In this dissertation the continuously compounded return series 𝑟𝑡 shall have the following ARMA(m,n)-GARCH(p,q) specification: 𝑟𝑡= 𝐸(𝑟𝑡|𝛺𝑡−1) + 𝜇𝑡 𝜇𝑡=𝜀𝑡 √ℎ𝑡 (1) 15 where 𝛺𝑡−1 is the information set up to time 𝑡−1, 𝜀𝑡 is the independently and identically distributed (i.i.d.) standardized innovations conditionally with a standard normal distribution or the Students’t t-distribution. The conditional mean can be written as follows: 𝑟𝑡= 𝐶+ ∑𝜙𝑖𝑟𝑡−1 𝑚 𝑖=1 +∑𝜃𝑗𝜀𝑡−𝑗 𝑛 𝑗=1 , (2) where 𝑟𝑡 is the daily returns, C is a constant term, 𝜙𝑖 are the parameters of the autoregressive component of order 𝑚, 𝜃𝑗 are the parameters of the moving average component of order 𝑛, and both orders 𝑚 and 𝑛 are non-negative integers. ℎ𝑡= 𝜔+ ∑𝛽𝑖ℎ𝑡−1 𝑝 𝑖=1 +∑𝛼𝑗𝜀𝑡−1 2 𝑞 𝑗=1 , (3) where 𝜔 is the long-rung volatility with condition 𝜔 > 0, 𝛽𝑖≥ 0;𝑖 =1,…,𝑝 and 𝛼𝑗≥ 0;𝑗 =1,…𝑞 to ensure the non-negativity of the variance process. The sum of 𝛽𝑖+𝛼𝑗 quantifies the persistence of shocks to volatility, if equal to the unity then every shock has an infinite persistence. When 𝛽𝑖+𝛼𝑗< 1, then GARCH(p,q) model is covariance stationary. A more specific model is the ARMA(1,1)-GARCH(1,1) model that is defined as: 𝑟𝑡= 𝐶+ 𝜙𝑟𝑡−1 +𝜃𝜀𝑡−1 (4) ℎ𝑡= 𝜔+ 𝛽ℎ𝑡−1 +𝛼𝜀𝑡−1 2 (5) A more flexible GARCH extension is the exponential GARCH (EGARCH) model proposed by Nelson (1991) who argued that the non-negative restrictions on parameters α and β in the standard GARCH are very restrictive, thus the EGARCH does not have any parameters restrictions. EGARCH also accommodates asymmetry where negative shocks can have a larger impact on volatility than positive shocks and by modeling the logarithm of the variance, will always be positive regardless of the sign of the parameters. EGARCH is defined as: log(ℎ𝑡)= 𝜔+ 𝛼𝑧𝑡−1 + 𝛾 (|𝑍𝑡−1|−Ε|𝑍𝑡−1|)+ 𝛽log(ℎ𝑡−1), (6) 22 Table 1. Descriptive Statistics – 01/02/1986-12/31/2013 Prices Returns Observations 7064 7063 Mean 40.8489 0.0002 Median 25.45 0.0007 Minimum 10.25 -0.4064 Maximum 145.31 0.1915 Standard Deviation 30.0394 0.0254 Skewness 1.1029 -0.7627 Kurtosis -0.0753 14.9717 The plot of the autocorrelation of a time series by lag is called the autocorrelation function (ACF) and is a useful qualitative tool to assess the presence of linear dependence in the returns at individual lags. The ACF for log-returns plotted in Figure 5 up to 250 lags, shows that several both lower and higher order autocorrelation coefficients are significantly different from zero. Figure 5. Autocorrelation function for log-returns. The ACF for absolute returns and squared returns plotted in Figures 6 and 7, respectively, shows that the autocorrelation coefficients are statistically significant within 5% up to the first half, that is approximately until the 125th lag, are significantly different from zero which indicates strong dependence in the higher order movements in the returns, providing evidence for the presence of timevarying volatility in the oil returns. 23 Figure 6. Autocorrelation function for absolute log-returns. Figure 7. Autocorrelation function for squared log-returns. In order to formally detect if the returns have serial autocorrelation, the Ljung–Box test proposed by Ljung and Box (1978) was applied to test whether the returns are jointly autocorrelated at multiple lags. The results of Table 2 show that: the null hypothesis of no serial autocorrelation until the 10th order is rejected as the p-value indicates that there is significant autocorrelation in the returns at the 5% level of significance. Therefore, the hypothesis that the return series is white noise is rejected and conclude there is strong evidence of autocorrelation in the data. 24 Table 2. Ljung-Box test for autocorrelation Lags 𝛸2 p-value Reject H0 1 14.179 8.336 e-04 True 5 36.501 7.541e-07 True 10 52.602 8.826e-08 True Finally, the ARCH-Lagrange Multiplier (LM) test introduced by Engle (1982) is applied to test for conditional heteroskedasticity in the returns. Table 3. Engle’s ARCH-LM test for conditional heteroskedasticity Lags 𝛸2 p-value Reject H0 1 76.753 < 2.2e-16 True 5 307.19 < 2.2e-16 True 10 363.02 < 2.2e-16 True The results of Engle’s ARCH-LM test shown in Table 3 assess the significance of ARCH effects for WTI returns. The p-value indicate that the null hypothesis of no ARCH effects is rejected at the 5% level of significance which shows the presence of ARCH effects in WTI returns, therefore, the GARCH model is necessary to model in order to capture the presence of conditional heteroskedasticity in the returns. ESTIMATION RESULTS In this study, volatility forecasts are computed using an estimation window of twenty-eight years which consists of 7,063 daily returns observations to cover both economic expansions and contractions. The length of the forecast periods will be 1, 5 and 20-day-ahead. As already discussed in Chapter 4, the modelling of the conditional variance shall be the focus of this work. However, before conducting the empirical results of variance forecasting, the in-sample data was examined to determine which types of models perform best in-sample. If the full sample were applied it would result in very rigid models with flat variance term structures. 25 The approach for modeling the conditional mean and the conditional variance was to search over several alternatives of 𝐴𝑅𝑀𝐴(𝑚,𝑛)−𝐺𝐴𝑅𝐶𝐻(𝑝,𝑞) models where 𝑚 and 𝑛 ranged between 0 to 3 while 𝑝 and 𝑞 parameters were either 1 or 2, since in most cases that is sufficient to capture the conditional dependencies. When comparing among different specifications of ARMA-GARCH-type models in order to select the optimal model, I used the statistical measures of model information criteria, such as the Akaike Information Criterion (AIC) proposed by Akaike (1974), the Bayesian Information Criterion (BIC) proposed by Schwarz (1997), the Hannan-Quinn Information Criterion (HQIC) proposed by Hannan and Quinn (1979). The AIC, BIC and HQIC can be computed respectively as: 𝐴𝐼𝐶 = −2ln(𝐿)+2𝑘, (22) 𝐵𝐼𝐶 = −2ln(𝐿)+ln(𝑛)𝑘, (23) 𝐻𝑄𝐼𝐶 = 2ln[ln[𝑛]]𝚔−2ln(𝐿), (24) where 𝑛 is the number of observations, 𝑘 is the number of estimated parameters and shows the degree to which the number of model parameters is penalized and L is the value of the likelihood function evaluated at the parameter estimates. When comparing among ARMA-GARCH-type models, the aim is to select the model with the smallest value information criteria for each specification however, BIC and HQIC are stricter in penalizing loss of degree of freedom than AIC since the higher the number of parameters 𝑘 the more the model is penalized. Tables 4 and 5 present the in-sample estimation results for the different volatility models assuming for the innovations the standard normal distribution and the Student’s t-distribution, respectively. First, the results reported in Tables 5 and 6 show that for the ARMA-GARCH, ARMAEGARCH, ARMA-GJR-GARCH, ARMA-TGARCH, ARMA-NGARCH and ARMA-APARCH models, 𝛽 is close to 1 which indicates a high degree of volatility persistence (𝛼+𝛽) which is also an greater indication of the presence of long memory in volatility. 26 Second, the asymmetric leverage coefficients γ reported in Table 4 and 5 are all significantly different from 0 in the ARMA-EGARCH, ARMA-TGARCH and ARMAAPARCH models while GJR-GARCH leverage coefficient γ is only different from 0 in Table 6 which indicates that WTI market also exhibits a leverage effect. Third, the values of the power coefficient δ shown in Tables 4 and 5 for the ARMANGARCH and ARMA-APARCH models feature a relatively high and statistically significant leverage as both models present positive coefficients which emphasizes that negative returns have a bigger impact on the conditional variance than positive returns. Fourth, Tables 4 and 5 report that the estimated sum 𝛼+𝛽 assumes the lowest value for the ARMA-CGARCH model which indicates that the short-run volatility component is weaker while the long-run volatility component (ρ) is estimated as 0.9986−0.9935, indicating that the permanent component of the conditional variance evidences a high degree of persistence. Finally, the ARMA(3,3)-CGARCH(1,1) model performs best for the standard normal distribution and the ARMA(3,3)-EGARCH(1,1) model performs best for the Student’s t-distribution regarding the LL, AIC, BIC and HQIC criteria as shown in Tables 4 and 5. 27 Table 4: Estimation results of volatility models for standard normal distribution ARMA (3,3) GARCH (1,1) ARMA (3,3) EGARCH (1,1) ARMA (3,3) GJR-GARCH (1,1) ARMA (3,3) CGARCH (1,1) ARMA (3,3) TGARCH (1,1) ARMA (3,3) NGARCH (1,1) ARMA (3,3) APARCH (1,1) μ 0.0304 (0.0216) 0.0285 (0.0232) 0.0306 (0.0219) 0.0396 (0.0217) 0.0259 (0.0191) 0.02794 (0.0338) -0.0504 (0.007) 𝜙1 -0.1207 (0.0192) 0.052 (0.0187) -0.1207 (0.0194) 0.0458 (0.0108) -0.1145 (0.0048) -0.1141 (0.0268) 0.1269 (0.0000) 𝜙2 -0.1526 (0.0265) -0.3549 (0.0066) -0.1525 (0.0262) -0.3557 (0.013) -0.2129 (0.0082) -0.1746 (0.0895) -0.0204 (0.0012) 𝜙3 0.7667 (0.0097) 0.7995 (0.0139) 0.7668 (0.0097) 0.793 (0.0085) 0.7128 (0.0157) 0.7339 (0.0911) 0.8929 (0.0004) 𝛳1 0.1025 (0.0164) -0.0754 (0.009) 0.1025 (0.0165) -0.0647 (0.0085) 0.0948 (0.0043) 0.0964 (0.0267) -0.1096 (0.0002) 𝛳2 0.1276 (0.0219) 0.3266 (0.0092) 0.1275 (0.0219) 0.3297 (0.0101) 0.1772 (0.0051) 0.1457 (0.0826) 0.0129 (0.0001) 𝛳3 -0.8062 (0.0047) -0.8277 (0.0000) -0.8063 (0.0047) -0.8184 (0.0000) -0.7574 (0.0151) -0.7755 (0.0872) -0.9057 (0.0057) ω 0.0569 (0.019) 0.0259 (0.0151) 0.057 (0.0191) 0.0319 (0.0068) 0.0228 (0.0119) 0.0335 (0.0116) 0.0305 (0.06) α 0.0925 (0.0164) -0.0074 (0.0119) 0.0927 (0.0222) 0.0859 (0.0203) 0.1019 (0.0213) 0.1039 (0.0159) 0.1009 (0.0554) β 0.9035 (0.0158) 0.9889 (0.0075) 0.9035 (0.0161) 0.6560 (0.0669) 0.9144 (0.0193) 0.9088 (0.0137) 0.9123 (0.004) δ 1.3998 (0.1389) 1.3333 (0.4353) η 0.0674 (0.0075) ρ 0.9986 (0.0009) γ 0.192 (0.1327) -0.0003 (0.0188) 0.0621 (0.0706) 0.0183 (0.0853) α + β 0.996 (0.0322) 0.9815 (0.0194) 0.9962 (0.0383) 0.7419 (0.0872) 1.0163 (0.0406) 1.0127 (0.0296) 1.0129 (0.0874) LL -15476 -15465 -15476 -15450 -15470 -15464 -15466 AIC 4.3850 4.3822 4.3853 4.3784 4.3837 4.3821 4.3830 BIC 4.3947 4.3929 4.3960 4.3901 4.3943 4.3938 4.3956 HQIC 4.3883 4.3859 4.3890 4.3824 4.3873 4.3862 4.3873 Notes: Robust standard errors are in parenthesis. 28 Table 5: Estimation results of volatility models for Student’s t-distribution ARMA (3,3) GARCH (1,1) ARMA (3,3) EGARCH (1,1) ARMA (3,3) GJR-GARCH (1,1) ARMA (3,3) CGARCH (1,1) ARMA (3,3) TGARCH (1,1) ARMA (3,3) NGARCH (1,1) ARMA (3,3) APARCH (1,1) μ 0.0569 (0.0214) 0.0463 (0.0222) 0.0528 (0.0220) 0.0602 (0.0217) 0.0478 (0.0429) 0.0274 (0.0352) 0.0195 (0.0389) 𝜙1 -0.0062* (0.0076)* 1.5319 (0.0006) -0.0168 (0.0077) -0.2379* (0.1723)* -0.4038 (0.0355) -0.2659 (0.0823) -0.4811 (0.0100) 𝜙2 -0.3976 (0.0127) -0.2813 (0.0022) -0.4056 (0.0126) -0.2811* (0.1734)* -0.4492 (0.0088) -0.3181 (0.0408) -0.5265 (0.0081) 𝜙3 0.7377 (0.0076) -0.3679 (0.0019) 0.7275 (0.0077) 0.6314 (0.1966) 0.4515 (0.0117) 0.5838 (0.0196) 0.3731 (0.0384) 𝛳1 -0.0025* (0.0035)* -1.547 (0.0008) 0.0082 (0.0031) 0.2297* (0.1665)* 0.3922 (0.0312) 0.2562 (0.0785) 0.4696 (0.0096) 𝛳2 0.3783 (0.0113) 0.3077 (0.0000) 0.3865 (0.0111) 0.2675* (0.1672)* 0.4310 (0.0076) 0.3011 (0.0404) 0.5086 (0.0079) 𝛳3 -0.7545 (0.0000) 0.3523 (0.0000) -0.7441 (0.0000) -0.6609 (0.1914) -0.4853 (0.0115) -0.6163 (0.0187) -0.4059 (0.0357) ω 0.0593 (0.0202) 0.0152 (0.0014) 0.0588 (0.0206) 0.047 (0.0072) 0.0233 (0.1616) 0.0288 (0.0069) 0.0269 (0.0072) α 0.0679 (0.0151) -0.0229 (0.0087) 0.0573 (0.0159) 0.0493 (0.0181) 0.0763 (0.2879) 0.0799 (0.0087) 0.0761 (0.0094) β 0.9228 (0.0169) 0.9896 (0.0001) 0.9238 (0.0171) 0.6586 (0.0737) 0.9322 (0.2874) 0.9280 (0.0078) 0.9311 (0.0089) 𝛿 1.2042 (0.1163) 1.1348 (0.1149) η 0.0603 (0.0122) ρ 0.9935 (0.0026) γ 0.1424 (0.0054) 0.0183 (0.0136) 0.1756 (0.6189) 0.1574 (0.0651) α + β 0.9907 (0.032) 0.9667 (0.0089) 0.9811 (0.033) 0.7079 (0.0918) 1.0085 (0.5753) 1.0079 (0.0165) 1.0072 (0.0183) LL -15246 -15222 -15245 -15243 -15229 -15232 -15227 AIC 4.3204 4.3137 4.3202 4.3200 4.3156 4.3169 4.3158 BIC 4.3311 4.3253 4.3318 4.3326 4.3273 4.3295 4.3294 HQIC 4.3241 4.3177 4.3242 4.3244 4.3197 4.3212 4.3205 Notes: Robust standard errors are in parenthesis. * denote non-significance at the 5% level. 29 FORECASTING CRUDE OIL VOLATILITY Although the ARMA-CGARCH model and the ARMA-EGARCH model are a good fit for in-sample data, this does not necessarily imply that these models can accurately forecast volatility for the out-of-sample data. Thus, this section employs the MCS procedure proposed by Hansen et al. (2011) to evaluate the performance of volatility forecasts computed by the different ARMAGARCH-type models estimated in section 5.2 with the standard normal (N) and standardized Student’s-t (STD) distributions for the out-of-sample period from January 2nd, 2014 to December 31st, 2019 which consists of 1505 observations. The MCS sequence of tests is firstly performed to reduce the initial number of models while at the same time generates a model confidence set which includes the superior set of models (SSM) with Equal Predictive Ability (EPA) with the prespecified confidence level (α) of 10% for this study which set a confidence level of 90% for MCS (𝑀 0.9 ∗) procedure. This flexibility helps to discriminate models with respect to desired characteristics which for this study, are their forecasting performances for 1, 5, and 20-day-ahead which corresponds to 1 day, 1 week, and 1 month trading periods in business days. The higher the number of eliminated models, the higher the heterogeneity of the competing forecasts. Tables 6, 7 and 8 presents the MCS superior set of models (SSM) ranked according to their p-values for 1, 5, and 20-day forecasts horizons in terms of four different loss criteria that consequently serve different practical uses: the AE1 and AE2 denotes the absolute error and are known for devaluing the outliers, SE1 denotes the squared error and measures the difference between the estimator and what is estimated according to the expected value of the squared error loss, therefore, is characterized by heavily weighting outliers and finally, the QLIKE denotes the QuasiLikelihood and corresponds to the percentage absolute errors or the loss implied by a Gaussian likelihood, hence is characterized by allowing overdispersion. 30 Table 6: MCS superior models for 1-day-ahead volatility forecasting AE1 AE2 QLIKE SE1 EGARCH(STD) 1.000 TGARCH(N) 1.000 NGARCH(STD) 1.000 APARCH(N) 1.000 APARCH(N) 1.000 GJR-GARCH(N) 0.999 GJR-GARCH(STD) 0.999 APARCH(STD) 0.997 NGARCH(STD) 1.000 APARCH(N) 0.793 APARCH(STD) 0.998 NGARCH(STD) 0.991 GJR-GARCH(STD) 1.000 GARCH(N) 0.625 GARCH(STD) 0.998 GJR-GARCH(STD) 0.991 APARCH(STD) 0.996 NGARCH(N) 0.309 APARCH(N) 0.996 TGARCH(STD) 0.991 TGARCH(STD) 0.994 EGARCH(N) 0.990 GJR-GARCH(N) 0.972 CGARCH(STD) 0.994 TGARCH(N) 0.989 GARCH(STD) 0.972 GARCH(STD) 0.957 CGARCH(N) 0.989 CGARCH(STD) 0.969 GJR-GARCH(N) 0.747 GARCH(N) 0.912 TGARCH(N) 0.917 CGARCH(N) 0.747 NGARCH(N) 0.887 EGARCH(STD) 0.916 GARCH(N) 0.663 EGARCH(STD) 0.803 EGARCH(N) 0.869 NGARCH(N) 0.641 GJR-GARCH(N) 0.719 NGARCH(N) 0.803 EGARCH(N) 0.444 CGARCH(STD) 0.529 CGARCH(N) 0.803 TGARCH(N) 0.246 TGARCH(STD) 0.509 GARCH(N) 0.744 Notes: The table shows the corresponding p-values computed according to TR statistics under four loss functions. The confidence level for MCS is 90%. Table 7: MCS superior models for 5-day-ahead volatility forecasting AE1 AE2 QLIKE SE1 EGARCH(STD) 1.000 EGARCH(N) 1.000 EGARCH(N) 1.000 APARCH(N) 1.000 EGARCH(N) 0.999 NGARCH(N) 1.000 EGARCH(STD) 1.000 NGARCH(STD) 0.999 GJR-GARCH(N) 0.999 EGARCH(N) 1.000 GJR-GARCH(STD) 0.999 GJR-GARCH(STD) 0.999 APARCH(STD) 0.991 APARCH(STD) 0.997 APARCH(STD) 0.997 NGARCH(STD) 0.981 GARCH(STD) 0.997 APARCH(N) 0.997 GJR-GARCH(STD) 0.979 CGARCH(STD) 0.996 NGARCH(STD) 0.994 TGARCH(STD) 0.974 TGARCH(STD) 0.994 TGARCH(N) 0.992 TGARCH(N) 0.970 APARCH(N) 0.993 TGARCH(STD) 0.984 GARCH(STD) 0.969 CGARCH(N) 0.982 GARCH(N) 0.917 GJR-GARCH(N) 0.955 GARCH(N) 0.175 EGARCH(STD) 0.899 CGARCH(STD) 0.942 GJR-GARCH(N) 0.165 GARCH(STD) 0.893 NGARCH(N) 0.907 NGARCH(N) 0.146 CGARCH(N) 0.854 GARCH(N) 0.876 CGARCH(STD) 0.807 CGARCH(N) 0.759 Notes: The table shows the corresponding p-values computed according to TR statistics under four loss functions. The confidence level for MCS is 90%. 31 The estimated SSM for all forecast’s horizons considered the 14 estimated models which suggests that WTI may not be affected by some stylized facts such as the leverage effect or complex nonlinear conditional volatility dynamics. The homogeneous composition of the final SSM, report an empirical evidence that this series appear to be accurately described by the standard normal and the Student’s t distributions of the single-regime ARMA-GARCH-type models. The four loss functions also treat outliers very differently which result in some dissimilarities, however, they agree on which models should be preferred except for AE2. In general, the results of the ARMA-EGARCH, ARMA-GJR-GARCH, ARMA-NGARCH and ARMA-APARCH models perform better in smaller forecasting horizons such as the one and five-day-ahead, while the ARMA-EGARCH, ARMA-GJR-GARCH, ARMANGARCH and ARMA-GARCH models perform better in forecasts for the 20-dayahead. Table 8: MCS superior models for 20-day-ahead volatility forecasting AE1 AE2 QLIKE SE1 GJR-GARCH(STD) 1.000 EGARCH(N) 1.000 CGARCH(STD) 1.000 APARCH(N) 1.000 GARCH(STD) 1.000 GARCH(N) 0.864 EGARCH(N) 1.000 NGARCH(N) 1.000 NGARCH(STD) 1.000 CGARCH(N) 0.188 NGARCH(N) 0.999 GARCH(N) 1.000 APARCH(STD) 1.000 GJR-GARCH(STD) 0.998 EGARCH(N) 0.999 EGARCH(STD) 1.000 GJR-GARCH(N) 0.995 GARCH(STD) 0.999 TGARCH(STD) 1.000 NGARCH(STD) 0.992 GJR-GARCH(STD) 0.999 EGARCH(N) 0.999 APARCH(N) 0.992 EGARCH(STD) 0.995 CGARCH(STD) 0.999 CGARCH(N) 0.992 APARCH(STD) 0.987 APARCH(N) 0.999 APARCH(STD) 0.992 CGARCH(N) 0.987 GARCH(N) 0.985 GARCH(N) 0.986 NGARCH(STD) 0.973 CGARCH(N) 0.963 EGARCH(STD) 0.956 TGARCH(STD) 0.965 NGARCH(N) 0.896 TGARCH(STD) 0.935 GJR-GARCH(N) 0.951 GJR-GARCH(N) 0.695 TGARCH(N) 0.718 CGARCH(STD) 0.912 TGARCH(N) 0.145 GARCH(STD) 0.633 TGARCH(N) 0.611 Notes: The table shows the corresponding p-values computed according to TR statistics under four loss functions. The confidence level for MCS is 90%. 38 Zhang, Y.-J. et al. (2019) “Volatility forecasting of crude oil market: can the regime switching GARCH model beat the single-regime GARCH models?,” International Review of Economics & Finance, 59, pp. 302–317. doi: 10.1016/j.iref.2018.09.006. Zhang, Y.-J. and Wang, J. (2015) “Exploring the WTI crude oil price bubble process using the markov regime switching model,” Physica A: Statistical Mechanics and its Applications, 421, pp. 377–387. doi: 10.1016/j.physa.2014.11.051.