The role of global economic policy uncertainty in predicting crude oil futures volatility: Evidence from a two-factor GARCH-MIDAS model
aa r X i v : . [ q -f i n . S T ] J u l The role of global economic policy uncertainty in predicting crude oil futuresvolatility: Evidence from a two-factor GARCH-MIDAS model
Peng-Fei Dai a , Xiong Xiong a,b , Wei-Xing Zhou c,d,e, ∗ a College of Management and Economics, Tianjin University, Tianjin 300072, China b China Center for Social Computing and Analytics, Tianjin University, Tianjin 300072, China c School of Business, East China University of Science and Technology, Shanghai 200237, China d Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China e Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China
Abstract
This paper aims to examine whether the global economic policy uncertainty (GEPU) and uncertainty changes havedifferent impacts on crude oil futures volatility. We establish single-factor and two-factor models under the GARCH-MIDAS framework to investigate the predictive power of GEPU and GEPU changes excluding and including realizedvolatility. The findings show that the models with rolling-window specification perform better than those with fixed-span specification. For single-factor models, the GEPU index and its changes, as well as realized volatility, areconsistent effective factors in predicting the volatility of crude oil futures. Specially, GEPU changes have strongerpredictive power than the GEPU index. For two-factor models, GEPU is not an effective forecast factor for thevolatility of WTI crude oil futures or Brent crude oil futures. The two-factor model with GEPU changes containsmore information and exhibits stronger forecasting ability for crude oil futures market volatility than the single-factormodels. The GEPU changes are indeed the main source of long-term volatility of the crude oil futures.
Keywords:
Crude oil futures; Global economic policy uncertainty; Volatility forecasting; GARCH-MIDAS;Two-factor model
1. Introduction
The fast-growing commodity markets are attracting the attention of more and more investors and policy makers,since commodity futures broaden the instruments for financial market investment and play an important role in pre-venting systemic risk. On April 20, 2020, the West Texas Intermediate (WTI) crude oil futures closed at − $37 .
63 perbarrel. This event catches the eyes of the world and produces a profound influence on practitioners and policy makers(Ji et al., 2020). Therefore, commodity-related research also has practical significance.A large number of studies show that the commodity futures are valuable sources of diversification investmentfor investors and portfolio managers (Arouri et al., 2011; Fazelabdolabadi, 2019; Klein, 2017; Lucey et al., 2017).Geman and Kharoubi (2008) explore the diversification effect brought by crude oil futures contracts into a portfolioof stocks. Nguyen et al. (2020) study the hedging versus the financialization nature of commodity futures, and findthat gold can be seen as a hedge against unfavorable fluctuations in the stock market. Narayan et al. (2010) examinethe long-run relationship between oil and gold spot and futures markets. Hammoudeh et al. (2014) provide evidenceof low and positive correlations between commodity markets and stock markets and suggest that commodity futuresare a desirable asset class for portfolio diversification. Among all the commodities, the price dynamics of crude oilfutures and related energy futures play a crucial role in modern global economic and financial systems and our dailylife (Jones and Kaul, 1996; Sadorsky, 1999). ∗ Corresponding author.
Email address: [email protected] (Wei-Xing Zhou)
Preprint submitted to Energy Economics July 28, 2020 he research of crude oil futures can be divided into two strands: price evolution and fluctuation dynamics. Interms of price evolution, scholars have studied the correlation and influence mechanism of crude oil futures andcrude oil spot, other commodity futures. In addition, the impact factors of crude oil futures pricing and the influenceof crude oil futures market on other financial markets are discussed. Chang and Lee (2015) and Holmes and Otero(2019) investigate the correlation and the causality between crude oil futures and spot prices over time using differ-ent methods. Wang et al. (2020) and Liu et al. (2019) detect the correlation of the crude oil futures price with otherfutures price. Cheng et al. (2018) show that the interest rates, the most traditional financial instruments, have influ-ence on crude oil futures prices. Yan et al. (2018) and Ames et al. (2020) discuss other impact factors on crude oilfutures prices. By contrast, more scholars focus on the study of the volatility of crude oil futures market (Agnolucci,2009; Ergen and Rizvanoghlu, 2016; Hasanov et al., 2020; Jo¨et et al., 2017; Kang and Yoon, 2013; Liu et al., 2018;Zhang et al., 2019). Undoubtedly, exploring the sources of the crude oil futures market volatility is crucial to energyresearchers, financial practitioners and policy makers. Our contribution is to expand the literature on the determinantsof crude oil futures market volatility.The determinants of crude oil futures market volatility attract the attention of many scholars (Bakas and Triantafyllou,2019; Liu et al., 2018; Nguyen and Walther, 2020). For instance, Bakas and Triantafyllou (2019) study the predictivepower of macroeconomic uncertainty on the volatility of agricultural, energy and metals commodity markets. Intheir paper, the latent macroeconomic uncertainty is constructed by Jurado et al. (2015). Liu et al. (2018) investigatethe impact of news implied volatility and its sub-component on the volatility of commodity futures. The news im-plied volatility is introduced by Manela and Moreira (2017), which quantifies the information about uncertainty fromnewspaper articles. Bakas and Triantafyllou (2019) and Liu et al. (2018) also discuss the impact of economic policyuncertainty of the United States on some commodities. Fang et al. (2018) examine whether global economic policyuncertainty contains forecasting information for global gold futures market volatility. To our knowledge, rare literatureinvestigates in depth the influence of global economic policy uncertainty on crude oil futures market volatility.Nowadays, the ties between different economies are getting stronger and stronger, and the development of worldeconomy is highly integrated (Dai et al., 2019). At the same time, the internal factors and external environment thataffect economic development are changing over time. Consequently, the global economic policy uncertainty hasbecome a new normal, which is also time-varying. The study on uncertainty has attracted much attention (Bloom,2009; Castelnuovo and Tran, 2017; Moore, 2017; P´astor and Veronesi, 2012, 2013). For instance, P´astor and Veronesi(2012) and P´astor and Veronesi (2013) develop a general equilibrium model to study how policy uncertainty affectstock market. Baker et al. (2016) construct a seminal index as the proxy for economic policy uncertainty in the UnitedStates and 11 other major economies, which was initially put forward by Baker et al. (2013). Inspired by Baker et al.(2016), many scholars (Arbatli et al., 2017; Castelnuovo and Tran, 2017; Moore, 2017) propose many other indices fordifferent economies successively using different methods and study the influence of economic policy uncertainty onvarious financial markets. Since crude oil futures are highly correlated with the global economic environment, as wellas the national policy environment, it is crucial to investigate how the global uncertainty related to economic policyaffect crude oil futures markets volatility. Dai et al. (2020) construct an index for the aggregate global uncertaintyrelated to economic policy based on the principal component analysis. Hence, using this index as the proxy variablefor global economic policy uncertainty and the changes of the index as the proxy variable for the changes of globaleconomic policy uncertainty, we study the determinants of crude oil futures markets volatility.There are various methods to model and predict the volatility of the crude oil futures market, among whichthe GARCH-class models are the most widely used. Most empirical tests require data of the same frequency forthe volatility and its potential sources. To overcome this shortfall, Ghysels et al. (2004) and Ghysels et al. (2007)introduce and re-explore MIDAS regression models, which can deal with time series data sampled at different fre-quencies. Engle and Rangel (2008) propose the spline-GARCH model to combine the macroeconomic causes withlow-frequency volatility of equities. In their model, high-frequency return volatility is specified to be the productof a slow-moving component, represented by an exponential spline, and a unit GARCH. Engle et al. (2013) formu-late a new class of component models, i.e. GARCH-MIDAS models, which distinguish long-term movement fromshort-term movement. Wei et al. (2017) investigate the informative determinant in forecasting crude oil spot marketvolatility via employing the GARCH-MIDAS model. Asgharian et al. (2013) utilize the GARCH-MIDAS model toexamine the forecasting power of macroeconomic variables on short-term and long-term component of the varianceof equity returns. They detect a large group of macroeconomic variables including unexpected inflation, term pre-mium, per capita labour income growth, default premium, unemployment rate, short-term interest rate, and per capita2onsumption. Asgharian et al. (2013) augment the model by adding the level and variance of an economic variable tothe MIDAS model. Based on the model proposed by Asgharian et al. (2013), Fang et al. (2018) investigate whetherglobal economic policy uncertainty contains forecasting information for global gold futures market volatility. Thereis little literature focusing on the different effects between global economic policy uncertainty and its changes. Ourpaper contributes to the literature on modelling the influence of global economic policy uncertainty and its changeson crude oil futures volatility.The remainder of the paper is organized as follows. Section 2 describes the data. Section 3 presents the modelsand evaluation methods. In Section 4, the empirical results are reported. Section 5 concludes the paper.
2. Data description
The dominating global crude oil futures markets are the New York Mercantile Exchange (NYMEM) in the UnitedStates and the Intercontinental Exchange (ICE) in the United Kingdom. The two most important pricing benchmarksfor the global oil market are West Texas Intermediate (WTI) crude oil futures contracts traded on the NYMEM andBrent crude oil futures contracts traded on the ICE. In this work, we choose the two commodity futures to representthe crude oil futures market. And their “contract 1” are selected for subsequent analyses. We retrieve the daily pricesof the two crude oil futures from the web site of the U.S. Energy Information Administration and the prices are indollars per barrel. In order to match the data of the GEPU index, the samples are from 1 December 1998 to 31 October2019. Figure 1 illustrates the evolutionary price trajectories of the two commodity futures. The daily returns of crudeoil futures are calculated as follows r t = ln p t p t − ! , (1)where t is in units of trading days. Time P r i ce (a) Time P r i ce (b) Figure 1: The evolutionary trajectories of Brent crude oil futures price (a) and WTI crude oil futures price (b). The sample periodof the two crude oil futures is from 1 December 1998 to 31 October 2019.
Dai et al. (2020) construct a new global economic policy uncertainty index (GEPU) based on the principal com-ponent analysis, which performs comparatively and slightly better in some situations as the GDP-weighted GEPU ofDavis (2016). The monthly GEPU index between December 1998 and October 2019 is calculated for our subsequentanalysis. In order to carry out the calculation, we select 21 EPU indices representing various economies’ economicpolicy uncertainty . The 21 economies are Australia, Brazil, Canada, Chile, China, Colombia, France, Germany,Greece, India, Ireland, Italy, Japan, South Korea, Mexico, the Netherlands, Russia, Spain, Sweden, the United King-dom, and the United States. In addition to the GEPU, we pay attention to the corresponding uncertainty change which The EPU indices of the 21 economies are publicly available at .
3s named GEPU change. The GEPU changes are calculated as follows ∆ GEPU m = ln GEPU m GEPU m − ! , (2)where m is in units of months. Figure 2 illustrates the time series of the monthly GEPU inedx and its changes.Comparing Fig. 2 and Fig. 1, we see that the GEPU index rose rapidly around the global financial crisis in 2008-2009,while both the Brent and WTI crude oil futures prices plummeted during the period. After the global financial crisis,the crude oil futures prices recovered and stabilized, whereas the GEPU index maintained at a relatively high level. Time G E P U i nd e x Time -1-0.500.51 G E P U c h a ng e Figure 2: The time evolution of the monthly GEPU index and its changes. The sample period of the GEPU index is from December1998 to October 2019, while the sample of GEPU change covers the period from January 1999 to October 2019.
Table 1 presents the summary statistics of the four time series, where the data frequency, mean, minimum, max-imum, standard error, skewness and kurtosis are reported. The GEPU index and its changes are monthly, whosesampling frequencies are lower than the daily crude oil futures returns. All the means of the time series are closeto zero except for the GEPU index whose mean is 152.81. The distributions of the GEPU index and its changes arepositively skewed and leptokurtic, while the two crude oil futures returns’ distributions are negatively skewed andleptokurtic. In addition, We verify whether each time series is stationary using the augmented Dickey-Fuller (ADF)test. We find that the the GEPU index and its changes, as well as the two crude oil futures returns time series, arestationary. The result of the stationary test for the GEPU index is in line with that for the GDP-weighted GEPU(Fang et al., 2019). Thus, all of the time series can be modelled directly.
Table 1Summary statistics of the crude oil futures returns, the GEPU index, and the GEPU changes.
Variable Obs. Freq. Mean Min. Max. Std. dev. Skew. Kurt. ADF r (Brent) 5258 daily 3.27E-04 − .
14 0.14 0.02 − .
11 6.02 − . ∗∗∗ r (WTI) 5258 daily 2.99E-04 − .
17 0.16 0.02 − .
14 7.04 − . ∗∗∗ GEPU
251 monthly 152.81 50.55 314.70 52.03 1.14 4.33 − . ∗∗∗ ∆ GEPU
250 monthly 2.90E-03 − .
66 0.65 0.18 0.46 4.70 − . ∗∗∗ Note: ADF represents the statistics calculated from the unit root test, and ∗∗∗ , ∗∗ , ∗ denote the null hypothesis is rejected at 1%, 5%, 10% statisticalsignificance level respectively. . Empirical methodology We utilize the GARCH-MIDAS model to investigate the effects of economic policy uncertainty on the daily pricevolatility of energy futures markets. Engle et al. (2013) propose the GARCH-MIDAS model to study the contributionof macroeconomic variables to stock volatility. They decompose the volatility of low-frequency time series into twocomponents: short-term volatility and long-term volatility. The long-run volatility is determined by low-frequencymacroeconomic factors, while the short-run volatility depends on the dynamics of the high-frequency time seriesitself. Following this thread, we introduce the GEPU index and its changes into the GARCH-MIDAS model so asto explore the effects of economic policy uncertainty on the long-term volatility of crude oil futures. Besides themacroeconomic variable, the contribution of the realized volatility is also considered. To carry out better the research,two models are specified, the single-factor model and two-factor model.
The GARCH-MIDAS model could be formally expressed as follows. The return on day i in period t (which maybe a week, a month, a quarter or longer) follows the following process: r i , t = E i − , t (cid:0) r i , t (cid:1) + √ τ t g i , t ε i , t , ∀ i = , . . . , N t (3)and E i − , t (cid:0) r i , t (cid:1) = µ, ε i , t | Φ i − , t ∼ N (0 , , (4)where N t is the number of trading days in each period, Φ i − , t is the information set up to day i −
1, and ε i , t is theinnovation term. We set µ as a constant since the mean daily return of crude oil futures is quite small. Eq. (3) impliesthe volatility of the return is decomposed into two parts: one is a short-term volatility component represented by g i , t ,and the other is a long-term volatility component represented by τ t .The dynamics of the short-term volatility component g i , t is assumed to be a daily GARCH (1 ,
1) process: g i , t = (1 − α − β ) + α ( r i − , t − µ ) τ t + β g i − , t , (5)where α > β > α + β <
1, while the long-term volatility component τ t is specified as smoothed realizedvolatility in the spirit of the MIDAS regression: τ t = m + θ K X k = φ k ( ω , ω ) RV t − k , (6)where RV t = N t X i = r i , t (7)is the realized volatility in period t , K is the number of periods over which we smooth the realized volatility, m is theintercept, and θ is the slope denoting the impact of realized volatility on long-term volatility. Following Engle et al.(2013), the weighting scheme in Eq. (6) is assigned by a two-parameter Beta polynomial: φ k ( ω , ω ) = k ω − ( K − k ) ω − P Kj = j ω − ( K − j ) ω − . (8)Eqs. (3-8) constitute the single-factor model pertaining to realized volatility under the GARCH-MIDAS framework(Model I) . The realized volatility figured out from Eq. (7) is fixed in period t , where we set the period as a monthbecause the GEPU index and its changes are monthly data.We further consider the single-factor model with the realized volatility in rolling-window specification. In thiscase, Eq. (3), Eq. (5) and Eq. (8) are almost unchanged, and only expressions of Eq. (6) and Eq. (7) are modified. The5olling-window realized volatility is expressed as RV (rw) i = N ′ X j = r i − j , (9)where r i − j denotes the backward rolling daily returns across various months, and N ′ is the number of trading days inone month. For simplicity, we pose N ′ =
22 in our models. Thereupon, the long-term volatility process is redefinedaccordingly as follows: τ (rw) i = m (rw) + θ (rw) K X k = φ k ( ω , ω ) RV (rw) i − k . (10)Finally, we adjust the low-frequency variable in Eq. (3) and Eq. (5) into daily variable. The adjusted Eq. (3) and Eq. (5),together with Eq. (9) and Eq. (10) form the single-factor model with rolling-window realized volatility (Model II).Next, we turn to the models that incorporate the GEPU index and its changes directly. We consider the fixed-span specification where the value of long-term volatility is the same on any day in a month and the rolling-windowspecification which has time-varying long-term volatility in a month. For the fixed-span model with the GEPU indexor its changes, the long-term volatility term τ is expressed as follows: τ t = m l + θ l K X k = φ k ( ω , ω ) GEPU t − k , (11)and τ t = m ∆ + θ ∆ K X k = φ k ( ω , ω ) ∆ GEPU t − k , (12)where ∆ GEPU denotes the GEPU changes. The single-factor model with the GEPU index (Model III) is consisted ofEq. (3), Eq. (5) and Eq. (11). Eq. (12) with Eq. (3) and Eq. (5) form the single-factor model with the GEPU changes(Model IV). Turning to the rolling-window setting, the long-term volatility term τ related to the GEPU index or itschanges is specified as: τ (rw) i = m (rw) l + θ (rw) l K X k = φ k ( ω , ω ) GEPU (rw) i − k , (13)and τ (rw) i = m (rw) ∆ + θ (rw) ∆ K X k = φ k ( ω , ω ) ∆ GEPU (rw) i − k . (14)The rolling-window specifications are calculated by the trading day, hence the long-term volatility is not an unchangedvalue in any month. We adjust Eq. (3) and Eq. (5) into low-frequency expressions, which together with Eq. (13) orEq (14) form the single-factor model with the GEPU index (Model V) or its changes (Model VI) respectively. Allweighting schemes φ k ( ω , ω ) appeared in the specifications above have the same definition as presented in Eq. (8). Apart from the influence of realized volatility itself, how economic policy uncertainty affects the volatility ofcrude oil futures is worth studying. In order to find out the answer, we integrate the GEPU index or its changes withthe realized volatility and get a new presentation of long-term volatility with fixed span: τ t = m + θ RV K X k = φ k ( ω , ω ) RV t − k + θ MV K X k = φ k ( ω , ω ) M t − k . (15)6ccordingly, the rolling-window long-term volatility case is specified as: τ (rw) i = m (rw) + θ (rw) RV K X k = φ k ( ω , ω ) RV (rw) i − k + θ (rw) MV K X k = φ k ( ω , ω ) M (rw) i − k . (16)The variable M in Eq. (15) and Eq. (16) represents the GEPU index or its changes. As for M (rw) , firstly we makethe macroeconomic variable M to be the daily index through copying the corresponding monthly value to each day inthat month. Then we carry out the rolling-window calculation in Eq. (16). We adopt the same lag order for realizedvolatility and macroeconomic variables in both fixed-span and rolling-window versions. The weighting coefficientshave the same definition as in Eq. (8). Eq. (15) with Eq. (3) and Eq. 5 constitute the two-factor model with fixed span(GEPU index: Model VII and GEPU changes: Model VIII), and Eq. (16) with adjusted Eq. (3) and Eq. (5) constitutethe two-factor model with rolling window (GEPU index: Model IX and GEPU changes: Model X).Finally, the total conditional variance in Eq. (3) is shown as follows: σ i , t = τ t · g i , t . (17) We calibrate the models using full sample data and investigate the explanatory ability of the models. In orderto implement model evaluation, we start with in-sample calibrations. We estimate the models using a calibrationwindow and then use the estimated parameters to make out-of-sample variance prediction. We choose a thirteen-year calibration window for both WTI oil and Brent oil, then data lagged three years before the calibration windoware needed to compute the historical realized volatility and economic policy uncertainty. To evaluate the varianceprediction of a specific model, we use two popular loss functions, root mean squared error (RMSE) and root meanabsolute error (RMAE), defined as follows:
RMS E = vut S S X s = (cid:16) σ s + − E s (cid:16) σ s + (cid:17)(cid:17) (18)and RMAE = vut S S X s = (cid:12)(cid:12)(cid:12)(cid:12) σ s + − E s (cid:16) σ s + (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) , (19)where σ s + is the actual daily total variance on day s + E s (cid:16) σ s + (cid:17) is the predicated daily total variance for day s + S is the length of prediction interval.In order to further verify the quality of the two models, we will conduct robustness test with computing two moreloss functions, root mean squared deviation (RMSD) and root mean absolute deviation (RMAD), defined as follows: RMS D = vut S S X s = ( σ s + − E t ( σ s + )) (20)and RMAD = vut S S X s = | σ s + − E s ( σ s + ) | . (21)Finally, for the sake of comparing the predictive accuracy of two competing models, the DM test proposed byDiebold and Mariano (2002) is adopted: DM = ¯ D √ var ( D s ) ∼ N (0 ,
1) (22)7 s = E , s − E , s (23)where E A , s and E B , s are the forecast errors of two competing models A and B respectively, ¯ D is the mean of the timeseries D s , and var ( D s ) is the variance of D s .
4. Empirical results
In this section, we present the calibration results of all the single-factor models in Section 4.1 and two-factormodels in Section 4.2 from the full sample and investigate the explanatory ability of the models to the long-termvolatility of crude oil futures. Next, in Section 4.3, we consider the in-sample estimation and make evaluation for thepredictive performance of the models using the out-of-sample prediction errors. Concerning model calibration, we set ω =
1, following Engle et al. (2013) and Asgharian et al. (2013).
In order to test the explanatory power of the models precisely, we estimate the parameters of the models using fullsample data. Panel A of Table 2 provides the parameter estimates of the single-factor models with fixed-span RV androlling-window RV (rw) . In Panel A of Table 2, all the α ’s and β ’s values of the two crude oil futures are significantlydifferent from 0 at 1% level and the sum of α and β for each commodity futures is less than and close to 1, whichimplies the short-term volatility of returns in crude oil futures market has clustering features. The parameter θ reflectshow realized volatility affects long-term volatility of crude oil futures return. All the estimates of θ shown in PanelA of Table 2 are significantly positive at the 1% level, which means the realized volatility has a positive influence onthe long-term volatility of crude oil futures return. Comparing the differences between θ in Model I and Model II, wefind that θ in Model I is less than that in Model II for both commodities. The realized volatility with rolling-windowpattern has greater impact on long-term volatility of crude oil futures return. The value of BIC in Model II is smallerthan that in Model I, which is the evidence that realized volatility using rolling-window expression in Eq. (9) is abetter explanatory factor.Fig. 3 illustrates the annualized long-term volatility and total volatility of the crude oil futures returns which arecalculated from the single-factor models with fixed-span RV and rolling-window RV (rw) . As can be seen from thefigure, the long-term volatility curve from Model II is smoother than that from Model I. The evolutionary trend of thelong-term volatility is consistent with the corresponding total volatility. Certainly, the difference between them is alsovisible by eye-balling.Next, we discuss the individual influence of economic policy uncertainty on crude oil futures volatility. The single-factor models (Model III to Model VI) satisfy our requirement to examine the effect of economic policy uncertainty.Similar to realized volatility, the GEPU index and its changes are set as two versions, fixed-span specification androlling-window specification. The rolling-window specification about the macroeconomic variable (the GEPU indexand its changes) is defined as follows: MV (rw) i = N ′ N ′ X j = MV i − j , (24)where MV i is a daily variable and its value equals to the corresponding monthly value, MV (rw) i is the mean of a monthearlier before the i -th day. It’s worth mentioning that we could carry out preciser analysis if we have the real dailydata of economic policy uncertainty.Model III and Model IV are single-factor models with fixed-span specification, while Model V and Model VI aresingle-factor models with rolling-window specification. The whole sample data is selected to estimate the parametersof the four models, which are also presented in Table 2. Panel B of Table 2 reports the parameter estimates of single-factor models with fixed-span specification, while Panel C of Table 2 reports the parameter estimates of single-factormodels with rolling-window specification.In Panel B and Panel C of Table 2, the values of α and β are all significantly different from 0 at the 1% level andall the β > .
9, which means the estimated short-term volatility from the single-factor model with economic policyuncertainty exhibits strong volatility clustering. Concerning the single-factor model with the GEPU index, the values8 able 2Parameter calibration of single-factor models
Commodity MIDAS Regressor µ α β θ ω m LLF BIC
Panel A: Model I and Model II
Brent oil Fixed RV ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗ . ∗∗∗ − RV (rw) ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗ . ∗∗∗ − RV ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ .
949 0 . ∗∗∗ − RV (rw) ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ .
744 0 . ∗∗∗ − Panel B: Model III and Model IV (fixed-span version)
Brent oil
GEPU . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗ − . − ∆ GEPU . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗ − GEPU ∗ . ∗∗∗ . ∗∗∗ . ∗∗ . ∗ − ∆ GEPU . ∗∗∗ . ∗∗∗ . ∗∗∗ .
011 5.069E-04 ∗∗∗ − Panel C: Model V and Model VI (rolling-window version)
Brent oil
GEPU (rw) . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗ − . − ∆ GEPU (rw) . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗ − GEPU (rw) ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗ − ∆ GEPU (rw) . ∗∗∗ . ∗∗∗ . ∗∗∗ .
148 5.037E-04 ∗∗∗ − Note: This table reports the parameter estimates of all the six single-factor models (Model I – Model VI). The samples for Brent oil and WTI oilare both from 1 December 1998 to 31 October 2019. LLF is the log-likelihood function and BIC indicates the Bayesian information criterion. Thenumbers in the parentheses are the standard deviation. Superscripts ∗∗∗ , ∗∗ , and ∗ denote respectively the significance levels at 1%, 5%, and 10%.Panel A reports the parameter estimates of Model I and Model II. Fixed RV denotes the realized volatility with fixed span, which is calculatedfrom Eq. (7). Rolling-window RV (rw) represents the realized volatility with rolling window, which is calculated from Eq. (9). Panel B reports theparameter estimates of Model III and Model IV. Panel C reports the parameter estimates of Model V and Model VI. The the GEPU index and itschanges here use the rolling-window version, whose monthly value is copied to the value of each day of the corresponding month.
002 2005 2008 2011 2014 2017 2020
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Figure 3: Total volatility and its long-term component of crude oil futures return from single-factor model with realized volatility.The left column is realized volatility with fixed span (Model I) and the right column is realized volatility with rolling window(Model II). The top row is for Brent oil, while the bottom row is for WTI oil. The plots show standard deviations on an annualizedscale. Total volatility and its long-term component of crude oil futures return from single-factor models. θ for the two commodity futures are significantly positive at the 5% level both in the fixed-span version and rolling-window version, which implies that the long-term volatility of crude oil futures return responds to global uncertaintyof economic policy positively. The long-term volatility is heavy when the economic policy uncertainty is high withoutconsidering the realized volatility. Similarly, the long-term volatility of both commodities futures respond to economicpolicy uncertainty changes towards the same direction. The GEPU changes have consistent contributions to the crudeoil futures volatility since the θ values of the two commodities are significantly positive at the 1% level both in thefixed-span version and rolling-window version. The long-term volatility of crude oil futures returns is heavy when theglobal economic policy uncertainty changes strongly without considering the realized volatility. Even the GEPU isat low level, obvious change will lead to heavy volatility. This is a quite interesting provisional result which impliesour follow-up research meaningful. Comparing the results of the fixed-span version and the rolling-window versionof each commodity in Table 2, we find that the BIC of the fixed-span version is not smaller than that of the rolling-window version. Therefore, the rolling-window version of the single-factor model is preciser than the fixed-spanversion, no matter the factor is the GEPU index or its changes.Fig. 4 illustrates the estimated annualized total volatility and its long-term component of crude oil futures derivedfrom the rolling-window version of the single-factor model with the GEPU index and its changes (Model V and ModelVI) . The left column is for the GEPU index and the right column is for the GEPU changes. We find that the long-term volatility estimated from the single-factor model with GEPU changes (Model VI) is closer to its correspondingtotal volatility. In each plot, the pair of total volatility and long-term volatility evolve in a similar trend during thewhole sample period. Thus, the short-term volatility is more sensitive to extreme events which usually cause drasticfluctuations of the crude oil futures market. Obvious separation also exists between the two trajectories of the totalvolatility and long-term volatility, induced by the short-term volatility. Extreme events also affect the development oflong-term volatility, thus the long-term volatility follows cyclical pattern presented in the figure.The implication about the value of θ could be visually reviewed in Fig. 4. For example, the GEPU index andits changes rise around the global financial crisis of 2008, and the estimated contemporaneous long-term volatilityfrom Model V and Model VI ascend simultaneously. By comparing the two columns, we discover that the long-termvolatility related to the GEPU changes has higher resolutions than that related to the GEPU index. Looking back tothe results of the single-factor models with realized volatility, we can see the long-term volatility related to the GEPUchanges also has higher resolutions. Considering the period from 2011 to 2014, for instance, long-term volatilitycurve estimated from the single-factor models with realized volatility (Model I and Model II) is quite gentle comparedwith that from the single-factor models with GEPU changes (Model V and model VI) for both Brent oil and WTI oil.Hence, economic policy uncertainty changes act as a more effective factor in the single-factor model to explain thevolatility of crude oil futures market. Unlike the single-factor models, the two-factor models combine macroeconomic variable (GEPU index or GEPUchanges) with realized volatility of commodity futures returns. We utilize the two-factor models in Section 3.2 to studythe extra explanatory power of macroeconomic variable to long-term volatility of crude oil futures returns eliminatingthe influence of realized volatility. In line with the single-factor models, we select the full sample data to deduce theestimates of all parameters. In calibrating the two-factor models with GEPU changes, we eliminate the daily returnsof the first month of the crude oil futures. The parameter estimates in the two-factor models with fixed span arelisted in Panel A of Table 3 and Panel B of Table 3 reports the parameter estimates in two-factor models with rollingwindow. As far as the rolling-window version of two-factor model is concerned, we calculate the realized volatilityand macroeconomic variable with rolling-window specification.In Table 3, the β values for all the commodities are significantly positive at the 1% level and their values are closeto the corresponding estimates of the single-factor models in Table 2. The two-factor models describe the volatilityclustering of crude oil futures returns’ short-term volatility, which is consistent with the conclusion from traditionalGARCH-class models (Agnolucci, 2009; Ergen and Rizvanoghlu, 2016; Lv and Shan, 2013). Comparing the lastcolumns in Panel A and Panel B of Table 3, we note that, for each commodity futures, BIC of two-factor model withfixed-span GEPU index is larger than that with rolling-window GEPU index for each commodity futures and BIC of To save space, we do not present the results from the fixed-span version here.
002 2005 2008 2011 2014 2017 2020
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Figure 4: Total volatility and its long-term component of crude oil futures estimated from Model V and Model VI. The left columnis for the GEPU index (Model V) and the right column is for the GEPU changes (Model VI). The two rows display in turn theresults for Brent oil and WTI oil. The plots show standard deviations on an annualized scale. able 3Parameter calibration of two-factor models Commodity MV µ α β θ RV θ MV ω m LLF BIC
Panel A: Model VII and Model VIII (fixed-span version)
Brent oil
GEPU ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ − .
002 5 . ∗∗ ∗∗∗ − ∆ GEPU ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ ∗∗∗ − GEPU ∗ . ∗∗∗ . ∗∗∗ . − . ∗ .
142 6.588E-04 ∗∗∗ − ∆ GEPU . ∗∗∗ . ∗∗∗ .
011 0 . ∗∗ . ∗∗ ∗∗∗ − Panel B: Model IX and Model X (rolling-window version)
Brent oil
GEPU (rw) ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ − .
001 7 . ∗∗ ∗∗∗ − ∆ GEPU (rw) ∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ ∗∗∗ − GEPU (rw) ∗ . ∗∗∗ . ∗∗∗ . − . ∗ .
397 6.262E-04 ∗∗∗ − ∆ GEPU (rw) ∗ . ∗∗∗ . ∗∗∗ . ∗∗ . ∗∗ . ∗∗ ∗∗∗ − Note: This table reports the parameter estimates of the four two-factor models (Model VII – Model IX). The samples for Brent oil and WTI oilare both from 1 December 1998 to 31 October 2019. LLF is the log-likelihood function and BIC indicates Bayesian information criterion. Thenumbers in the parentheses are the standard deviation. The superscripts ∗∗∗ , ∗∗ , and ∗ denote respectively the significance level at 1%, 5%, and 10%.Panel A reports the parameter estimates of Model VII and Model VIII and Panel B reports the parameter estimates of Model IX and Model X. TheGEPU index and its changes here use the rolling-window version, whose monthly value is copied to the value of each day of the correspondingmonth. GEPU or GEPU (rw) mean that the GEPU index has an opposite effect on the volatilityof crude oil futures when considering realized volatility. In Table 3, the coefficients of
GEPU and
GEPU (rw) , θ MV ,is − .
002 and − .
001 for Brent oil. These values are not significantly different from 0. For WTI oil, the coefficientsof
GEPU and
GEPU (rw) are − .
023 and − .
019 and they are significantly different from 0 at the 10% level, but thecorresponding θ RV values in the same model are not significantly different from 0. The calibrated results of the two-factor models with the GEPU index show that Model VII and Model IX are not suitable for estimating the long-termvolatility of both crude oil futures, Brent oil and WTI oil.Table 3 also shows that the coefficients θ MV of the GEPU changes in the two-factor models for the two commodityfutures are all significantly positive at least at the 5% level. Meanwhile, the corresponding coefficients θ RV of realizedvolatility in the same two-factor model with rolling window are significantly positive at least at the 5% level as well(see Panel B of Table 3). The results indicate that the two-factor model with GEPU changes works well for the crudeoil futures market. In the two-factor model, the GEPU changes have a positive effect on the crude oil futures marketvolatility when the realized volatility is considered together. Moreover, we draw a conclusion that the two-factormodel with GEPU changes performs better than the single-factor model with GEPU changes by comparing the valuesof BIC in Panel C of Table 2 and in Panel B of Table 3.In order to interpret the estimated results of the two-factor models intuitively, we plot the curves of total volatilityand its long-term component of the two crude oil futures in Fig. 5. Visual inspection of the figure reveals that thetwo-factor model with GEPU changes provides better fits. Consistent with previous analyses, the estimated long-termvolatility curve of WTI oil from the two-factor model with GEPU index does not characterize its real evolution trendprecisely. For Brent oil, the estimated long-term volatility curve from the two-factor model with GEPU index between2011 and 2015 is quite gentle while the curve from the two-factor model with GEPU changes has high resolutions.Focusing on the right columns in Fig. 3, Fig. 4 and Fig. 5, we observe that the long-term volatility curves in Fig. 5is the closest to the corresponding total volatility curve and they have higher resolutions. Therefore, the two-factormodel with GEPU changes is more appropriate to describe the crude oil futures volatility. In this section, we appraise the forecasting ability of all the models. To evaluate the variance prediction loss ofa specific model, we select two popular loss functions mentioned in Section 3.3,
RMS E and
RMAE , as the relevantmeasure. As the rolling-window version of the models preforms better than the corresponding fixed-span version, wejust provide the evaluation results of the rolling-window version. The sample intervals of Brent oil and WTI oil arethe same which covers from 1 December 1998 to 31 October 2019.In order to ensure a five-year sample period for out-of-sample prediction evaluation, we take a thirteen-year cali-bration window for WTI oil and Brent oil. Before the calibration window, there are additional three-year-lagged dataneeded to calculate the historical realized volatility. That is, we choose the data from period between 1 December1998 and 31 December 2014 for the in-sample calibration and the data from period between 1 January 2015 and31 October 2019 for the out-of-sample prediction. Table 4 not only reports the results of in-sample calibration andout-of-sample evaluation for WTI oil and Brent oil, but also lists the results of the full sample estimation in this table.The loss function values of the single-factor models with rolling window are presented in Panel A of Table 4.Comparing the values of
RMS E and
RMAE between every two single-factor models, we find that Model VI causesless loss than other two models (Model II and Model V) when the full sample of Brent oil and WTI oil is concerned.The facts reveal that the GEPU changes is the more competitive factor to improve the interpretation capability ofthe single-factor model, compared with GEPU index and realized volatility. As far as the predictive ability of thesingle-factor model is concerned, the GEPU changes do not outperform other two factors, since the out-of-sample-values of
RMS E and
RMAE of Model VI are not always the smallest among the three models for the two commodityfutures. However, the GEPU changes are indeed the volatility forecasting factor with better performance than theGEPU index. Panel B of Table 4 reports the loss function values of the two-factor models with rolling window. Wedo not find clear and unified numerical magnitude relationship between the out-of-sample-values of
RMAE of Model14
002 2005 2008 2011 2014 2017 2020
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Time A nnu a li ze d vo l a tilit y Total volatilityLong-term volatility
Figure 5: Total volatility and its long-term component of crude oil futures estimated from Model IX and Model X. The left columnis for the GEPU index and the right column is for the GEPU changes. The two rows display in turn the results for Brent oil andWTI oil. The volatility in each plot is annualized. able 4Results of model evaluation Commodity Model Full Sample In-Sample Out-of-Sample
RMS E RMAE RMS E RMAE RMS E RMAEPanel A: Single-factor models
Brent oil II 1.003E-03 2.198E-02 9.772E-04 2.149E-02 1.087E-03 2.346E-02V 1.000E-03 2.249E-02 9.733E-04 2.226E-02 1.096E-03 2.394E-02VI 9.953E-04 2.195E-02 9.876E-04 2.095E-02 1.089E-03 2.270E-02WTI oil II 1.295E-03 2.395E-02 1.323E-03 2.397E-02 1.091E-03 2.458E-02V 1.292E-03 2.421E-02 1.323E-03 2.431E-02 1.196E-03 2.496E-02VI 1.282E-03 2.383E-02 1.309E-03 2.367E-02 1.191E-03 2.464E-02
Panel B: Two-factor models
Brent oil IX 1.003E-03 2.198E-02 9.779E-04 2.145E-02 1.088E-03 2.319E-02X 9.972E-04 2.187E-02 1.083E-03 2.296E-02 8.713E-04 2.040E-02WTI oil IX 1.294E-03 2.394E-02 1.325E-03 2.375E-02 1.192E-03 2.431E-02X 1.283E-03 2.380E-02 1.310E-03 2.361E-02 1.187E-03 2.449E-02
Note: This table reports the results of full sample estimation, in-sample calibration and out-of-sample evaluation of the rolling-window models forWTI oil and Brent oil. The full samples of Brent oil and WTI oil are both from December 1998 to October 2019 and the out-of-sample predictioncover the period from November 2014 to October 2019. Panel A reports the calculation results of loss functions (
RMS E and
RMAE ) for thesingle-factor models with rolling window (Model II, Model V and Model VI). Panel B reports the calculation results of loss functions (
RMS E and
RMAE ) for the two-factor models with rolling window (Model IX and Model X).
IX and Model X for the two commodity futures. In contrast, the full sample values and out-of-sample-values of
RMS E of Model X is smaller than that of Model IX for both commodity futures. This is strong evidence that thetwo-factor model with GEPU changes (Model X) is more effective to predict the crude oil futures volatility than thetwo-factor model with GEPU index (Model IX). Analysing the out-of-sample prediction error in Table 4, we suggestthat the two-factor model with GEPU changes contains more information and has stronger prediction power than thesingle-factor models.The results of robustness test for the model evaluation are given in Table 5. As we can see from this table, thevalues of
RMS D and
RMAD of Model X are both smaller than that of Model IX, which implies that the two-factormodel with GEPU changes is exactly more suitable to predict the crude oil futures volatility.
Table 5Robustness test of two-factor model evaluation
Commodity Model Full Sample In-Sample Out-of-Sample
RMS D RMAD RMS D RMAD RMS D RMAD
Brent oil IX 1.469E-02 0.1074 1.524E-02 0.1095 1.403E-02 0.1057X 1.463E-02 0.1070 1.521E-02 0.1092 1.392E-02 0.1051WTI oil IX 1.590E-02 0.1110 1.574E-02 0.1101 1.669E-02 0.1151X 1.588E-02 0.1109 1.570E-02 0.1098 1.668E-02 0.1150
Note: This table shows the calculation results of loss functions (
RMS D and
RMAD ) for the two-factor models with rolling window (Model IX andModel X). The full samples of Brent oil and WTI oil are both from December 1998 to October 2019 and the out-of-sample prediction cover theperiod from November 2014 to October 2019. × ” to represent these situation in Table 6. According to this table, in term of the single-factor model, the predictiveeffect of the GEPU changes obviously outperforms that of the GEPU index since the corresponding t -statistic is − .
85 for Brent oil and − .
42 for WTI oil. Nevertheless, the realized volatility in the rolling-window version is thebest predictive factor when considering the single-factor model. The forecasting capacity of the models is improvedsignificantly (at the 1% level) after adding the realized volatility to the single-factor model with GEPU index or itschanges. On the other hand, the predictive power of the single-factor model with realized volatility is enhancedwhen integrating the GEPU index or its changes in the model. Summarizing the results in Table 6, we can draw theconclusion that Model X has the best performance in predicting crude oil futures volatility.
Table 6DM test for the out-of-sample performance of different models in rolling-window version
Brent oil WTI oilModel II V VI IX X II V VI IX XII × − − + + ∗∗ × − − + + ∗∗ × (-0.54) (-0.58) (1.28) (1.98) × (-0.27) (-0.87) (1.43) (2.02)V + × + ∗∗∗ + ∗∗∗ × + × + ∗∗∗ + ∗∗∗ × (0.54) × (5.85) (3.15) × (0.27) × (3.42) (2.80) × VI + − ∗∗∗ × × + ∗∗∗ + − ∗∗∗ × × + ∗∗∗ (0.58) (-5.85) × × (3.06) (0.8 7) (-3.42) × × (3.63)IX − − ∗∗∗ × × + − − ∗∗∗ × × + (-1.28) (-3.15) × × (1.25) (-1.43) (-2.80) × × (0.70)X − ∗∗ × − ∗∗∗ − × − ∗∗ × − ∗∗∗ − × (-1.98) × (-3.06) (-1.25) × (-2.02) × (-3.63) (-0.70) × Note: This table shows the DM test results between every two models among all the two-factor models in rolling-window version (Model II,Model V, Model VI, Model IX and Model X). The out-of-sample of Brent oil and WTI oil for DM test are from 1 November 2014 to 31 October2019. “ × ” represents that the model corresponding to the row are not compared with models corresponding to the column. “ − ” denotes the modelcorresponding to the row having higher accuracy compared with the model corresponding to the column, “ + ” denotes the model correspondingto the row that owns lower accuracy compared with the model corresponding to the column. The numbers in the parentheses are the t -statistic.Subscripts ∗∗∗ , ∗∗ , and ∗ denote being significance at the 1%, 5%, and 10% levels respectively.
5. Conclusions
In this work, we establish two types of models under the GARCH-MIDAS framework: single-factor models andtwo-factor models. Firstly, we employ the single-factor models to investigate respectively the impact of realizedvolatility, GEPU index and its changes on the crude oil futures volatility. Our empirical results show that the threefactors produce significantly positive impacts on both commodity futures. From the perspective of single-factormodels, the GEPU index and its changes are indeed the determinants of the crude oil futures volatility. A comparisonof the competing models show that the GEPU changes outperform the GEPU index in predicting crude oil futuresvolatility. In addition, empirical results manifest that the single-factor models with rolling-window specificationperform better than that with fixed-span specification.Then, we utilize the two-factor models to estimate respectively the impacts of GEPU index and its changes on thecrude oil futures volatility when eliminating the effect of realized volatility. The calibration results of the two-factormodel with GEPU changes reveal that GEPU changes can be a volatility forecasting factor for the crude oil futuresmarket even when we include the realized volatility as an existing predictive factor. Moreover, our empirical resultssuggest that the two-factor model with GEPU changes is more suitable to describe the crude oil futures volatility. Theincrease of GEPU changes will result in the increase of the long-term volatility of crude oil futures. In addition, similar17o the single-factor model, the two-factor model with rolling-window specification exhibits better performance. Thefindings of model evaluation indicate that the two-factor model with GEPU changes contains more information andhas stronger predictive power for the crude oil futures volatility than the single-factor model.Our study indicates that the changes in global uncertainty of economic policy have significantly positive impactson the long-term volatility of crude oil futures. We advise financial practitioners and policy makers to follow theguidance, taking global uncertainty changes into account when they want to predict the volatility of crude oil futures.This will help improve investment strategies and policy makers’ decisions and probably lower or prevent the systemicrisk of commodity markets.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grants Nos. 71532009, U1811462and 71790594), Fundamental Research Funds for the Central Universities, Tianjin Development Program for Innova-tion and Entrepreneurship, and Program of Shanghai Academic Research Leader.
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