In financial econometric literature, the GARCH model proposed by Bollerslev (1986) is a popular approach for measuring and forecasting financial volatility. The GARCH model can be written as

where r_t = log(P_t/P_t−1) is the log-return on day t, where P_t is the closing price on day t; μ is the conditional mean of r_t; h_t is the conditional variance of r_t; z_t is the return innovation, and ϵ_t is the standardized return innovation.

The EGARCH model was proposed by Nelson (1991), which has the capacity to capture the leverage effect, which has been found to be important for volatility forecasting. The EGARCH model is given by

where the coefficient γ captures the leverage effect when γ < 0.

Based on the Heterogeneous Market Hypothesis, the HAR model proposed by Corsi (2009) aims to capture the volatility dynamics in financial markets, which can be written as

where x_t is the realized measure of volatility on day t; xtw and xtm represent the average realized volatility for the past weekly and monthly periods, respectively. (corresponding to lags of 5 days, and 22 days).

Considering the fact that the RGARCH model cannot capture the leverage effect (asymmetric response of volatility to positive and negative shocks) adequately, Hansen and Huang (2016) extend the RGARCH to the REGARCH model, which can be written as

where u_t is the realized measure innovation, which is independent of the return innovation ϵ_t; d(ϵ_t) and υ(ϵ_t) are the leverage functions, which are used to capture the leverage effect, satisfying E_t−1[d(ϵ_t)] = E_t−1[v(ϵ_t)] = 0. If d₁ < 0, υ₁ < 0, it means that there exists leverage effect.

In the REGARCH model, Equations 11, 13 and 14 are referred to as the return equation, the variance (GARCH) equation and the measurement equation, respectively. The measurement equation relates the ex-post realized measure to the ex-ante conditional variance, with bias-correction coefficients ξ and φ used to correct the bias in the realized volatility measure caused by non-trading hours and market microstructure noise. Consistent with Takahashi et al. (2009), Koopman and Scharth (2012) and Wu et al. (2020), we assume that φ = 1 to facilitate model estimation and to improve out-of-sample performance.

The REGARCH model fall short in capturing the jumps (huge changes) in the asset prices. In light of this, this paper proposes the REGARCH-Jump model that extends the REGARCH model of Hansen and Huang (2016) to incorporate the dynamic jump intensity to model and forecast the volatility of crude oil futures markets. The specification of the REAGRCH-Jump model is as follows

where z_t and y_t denote the normal component and the jump component, respectively, which are assumed to be independent. The normal component z_t is assumed to be distributed as N(0, h_z,t), where h_z,t is the conditional variance of the normal component, which is governed by the REGARCH dynamic. The jump component y_t follows a compound Poisson process with jump intensity h_y,t and jump size Ytj, where Ytj is independently drawn from a normal distribution with mean θ and variance δ², that is, Ytj~i.i.d.N(θ,δ2). The mean and variance of the jump component, y_t, are given by θh_y,t and (θ2+δ2)hy,t, respectively. n_t is the number of jumps arriving between t − 1 and t, which follows a Poisson counting process with intensity h_y,t, that is, n_t ~ Poisson(h_y,t). The conditional probability of n_t can be written as

Therefore, the conditional expectation of the number of jumps arriving over the time interval (t − 1, t) equals the jump intensity, that is, E_t−1[n_t] = h_y,t. To describe the dynamics of jump intensity process h_y,t, we follow Maheu and McCurdy (2004) and assume that it follows the autoregressive conditional jump intensity model:

where ρ > 0, κ > 0, ψ > 0; ζ_t−1 denotes the jump intensity residual, which can be written as

It is clear that E(ζ_t−1) = 0. Thus, κ describes the persistence of the jump intensity process. If h_y,t is stationary, we have 0 < κ < 1. Then the unconditional jump intensity is given by E(hy,t)=ρ1-κ.

In the measurement Equation 21, h_t denotes the total unconditional variance of return, which can be written as

It has been well documented in the literature that the use of intraday high and low prices leads to more accurate estimate of volatility than daily returns (Molnár, 2011; Chou and Liu, 2010). Therefore, in the paper we utilize the intraday price range calculated from the intraday high and low prices as a proxy for the realized measure in the HAR, REGARCH specifications, and their extensions. The intraday price range of Parkinson (1980) is defined as

where H_t and L_t are the high and low prices observed at day t, respectively. The intraday range given by Equation 26 is not only a highly efficient volatility proxy, capturing information regarding the entire intraday trajectory of the price, but also robust to microstructure noise (Alizadeh et al., 2002; Brandt and Jones, 2006). In this paper, we set xt=RNGt2 the HAR, REGARCH specifications, and their extensions.

The proposed REGARCH-Jump model is intuitive and easy to implement. We can use the classical maximum likelihood method to estimate the parameters of the model. To be specific, the log-likelihood function of the REGARCH-Jump model can be written as

where Θ=(μ,ω,α,β,d1,d2,ξ,φ,σu,υ1,υ2,ρ,κ,ψ,θ,δ)′ is the vector of model parameters, ℓ(r; Θ) and ℓ(x|r; Θ) are the partial log-likelihood functions for the returns and realized measure, respectively. f_t−1(r_t) and f_t−1(x_t|r_t) denote the conditional probability density functions of r_t and x_t, respectively, which can be written as

where Pr_t−1(n_t = j) is the conditional probability of the number of jumps, which is given in Equation 22. It should be noted that the summation in Equation 30 must be truncated when implementing the maximum likelihood estimation. We truncate the summation at 50 jumps per day. In fact, since the tail probability of Poisson distribution at n_t ≥ 50 is sufficiently small and negligible, summing to n_t = 50 can ensures the accuracy of the estimation process. f_t−1(r_t ∣ n_t = j) in Equation 30 can be written as

In order to calculate the likelihoods (Equations 30–32), it is necessary to determine the normal innovation z_t and the number of jumps n_t, and further filter the conditional variance h_z,t+1 and the conditional jump intensity h_y,t+1. This process can be easily implemented by using analytic filtering. Applying Bayes' rule, the filtering density is given by

where the expressions on the right-hand side of the Equation 33 are given by Equations 22, 30 and 32. Pr_t(n_t = j) represents the ex-post inference on n_t, or the probability that j jumps have arrived between time t − 1 and t conditional on the information available at time t. The filtered number of jumps is then given by

The actual number of jumps, n_t, can be larger than one but is restricted to be less than fifty as mentioned above. According to Christoffersen et al. (2012), the filtering of the normal innovation z_t can be written as

Once ñ_t and z~t are known, we can infer the filtered h~z,t+1 and h~y,t+1 easily through Equations 20 and 23.

Finally, the parameters of the REGARCH-Jump model can be estimated via maximum likelihood method by solving

The loss function is commonly used to evaluate the forecasting accuracy of the competing models. In the paper, we employ four popular loss functions, including the mean absolute error (MAE), mean absolute percentage error (MAPE), mean squared error (MSE) and Quasi-likelihood (QLIKE). The four evaluation criteria are defined as

where T is the number of out-of-sample forecasts, h_t+1 and ĥ_t+1(m) denote the measured (true) volatility and forecasted volatility, respectively, and m stands for the GARCH, EGARCH, HAR, REGARCH or REGARCH-Jump models. It is worth noting that MSE and QLIKE are robust loss functions, which could provide a consistent ranking of the volatility models with a conditionally unbiased volatility proxy (Patton, 2011).

Since the true volatility is unobservable, the evaluation and comparison of volatility forecasting models requires the proxy of true volatility. In the paper, we employ the scaled realized volatility (RV) as a proxy of the true volatility. The scaled RV is defined as RVtscale=c×RVt, where RV_t is computed based 5-min intraday returns on day t and c=∑t=1Trt2/∑t=1TRVt. This adjustment is used to account for the non-trading hours.

For robustness, we adopt the model confidence set (MCS) test proposed by Hansen et al. (2011) to examine whether the difference in forecasting performance between the competing models is statistically significant. The MCS method tests a given set of competing models and identifies a set of optimal predictive models or MCS with a certain level of confidence. Specifically, the MCS test relies on an equivalence test δ_M and an elimination rule e_M. Let M⁰ be the initial set of all competing models. Set M = M⁰, and use the equivalence test δ_M to test the null hypothesis that the competing models have the same expected loss (equal forecasting performance), i.e.,

where d_uv,t = Loss_t(u)−Loss_t(v) is the loss difference between the models u and v. Hansen et al. (2011) propose the following statistics to test the null hypothesis H_0,M:

where d̄uv is the average loss difference, and var^(d̄uv) is the bootstrapped estimate of the variance of d̄uv, i.e., var(d̄uv). For a given significance level α, if the null hypothesis H_0,M is accepted, define M1-α*=M. Otherwise, we use the elimination rule eM=argmaxu∈Msupv∈Mtuv to exclude the model that has poor forecasting performance. This process is repeated until no model can be excluded. Finally, the set of surviving models M1-α*=M is referred to as the MCS, i.e. the set of optimal predictive models at a given confidence level of 1 − α.

Since the asymptotic distribution of the test statistics T_M is non-standard, this paper uses a block bootstrap of 10⁵ replications for approximate calculation. In addition, we set the significance level as α = 10%.

For our empirical analysis, we use data on the daily open, high, low and close prices for the Brent crude oil futures. We focus on Brent crude oil futures market due to the fact that Brent crude oil is currently considered to be the global oil pricing benchmark (Dowling et al., 2016; Zavadska et al., 2020). We employ the squared price range (RNGt2) as the realized volatility measure (i.e., x_t). The data is obtained from Wind Database of China, and the sample period is from January 4, 2005 to February 28, 2023.

Table 4 presents the descriptive statistics of the daily returns and price ranges of Brent crude oil futures. As can be seen from the table, the return distribution for the Brent crude oil futures is negatively skewed and leptokurtic, while the price range distribution is positively skewed and leptokurtic. The Jarque-Bera statistics indicate that both the return and price range distributions are non-Gaussian. The Ljung-Box Q-statistic for autocorrelation up to 20 lags shows that the price range series is highly autocorrelated, suggesting high persistence of Brent crude oil futures volatility.

Figure 1 presents the time series plots of the daily Brent crude oil futures returns and price ranges. It can be seen from the figure that the well-known behavior of volatility clustering is apparent. In addition, the Brent crude oil futures experience large fluctuations (jumps) during the periods of 2008–2009 global financial crisis (GFC), 2020 COVID-19 pandemic and 2022 Russo-Ukrainian conflict.

In this subsection, we estimate the five competitor models (GARCH, EGARCH, HAR, REGARCH and REGARCH-Jump) using the maximum likelihood method based on the in-sample data covering the period from January 4, 2005 to December 31, 2020. Table 5 reports the maximum likelihood estimates for the five models. As can be seen from the table, the daily coefficient of x_t in the HAR model is significantly positive, while the weekly coefficient is significantly negative. This indicates that daily information positively affects crude oil futures volatility, whereas weekly information has a negative impact. Additionally, the estimates of volatility persistence in the GARCH, EGARCH, REGARCH and REGARCH-Jump models are larger than 0.98, suggesting the stylized fact of high volatility persistence. Regarding the leverage parameters, γ, d₁, d₂, υ₁ and υ₂, they are all significantly different from zero. In particular, γ, d₁ and υ₁ are all negative, suggesting the presence of the leverage effect in the Brent crude oil futures market.

Moreover, the jump intensity parameters (ρ, κ, ψ) are all positive and statistically significant, suggesting the presence of time-varying jump intensity and that the REGARCH-jump model is correctly specified. In particular, the parameter κ is estimated to be 0.9578, which provides evidence of high persistence of the conditional jump intensity process. The estimated mean jump size θ in the REGARCH-Jump model is significantly negative, while the estimated jump volatility δ is significantly positive. The unconditional mean of dynamic jump intensity is E(hy,t)=ρ1-κ=0.1493, implying that jumps arrive at a frequency of 37.6 jumps per year. In addition, the contribution of return jumps to the total return variance is (θ2+δ2)hy,t(θ2+δ2)hy,t+hz,t=12.73%, which is close to the results reported in Christoffersen et al. (2012) (12%~15%), Andersen et al. (2007) (14.6%) and Pan et al. (2020) (11%), suggesting that our estimation results of jump intensity is reasonable.

Finally, we can observe that the REGARCH-Jump model improves the empirical return fit relative to all other models (GARCH, EGARCH and REGARCH) in terms of the partial log-likelihood for the returns ℓ(r) and the full log-likelihood ℓ(r, x).

Figure 2 presents the time series plots of the jump intensity and the number of jumps. It is clear that the jumps have a time-varying feature, and the number of jumps related to market uncertainty events have increased during the periods of 2008–2009 global financial crisis, 2015 global oil price decline and 2020 COVID-19 pandemic. Note also that, the number of jumps in most cases is less than one jump per day.

In the in-sample analysis, we document that the jump intensity and the number of jumps has time-varying characteristic. Moreover, we show that incorporating the extreme-value information and jump dynamics into the REGARCH model could improve the model's ability to fit the Brent crude oil futures returns. In this subsection, we investigate the importance of accounting for the extreme-value information and jump dynamics for forecasting the crude oil futures volatility relying on our proposed REGARCH-Jump model. Importantly, we compare the out-of-sample forecasting performance of the REGARCH-Jump model with that of the GARCH, EGARCH, HAR and REGARCH models.

The out-of-sample forecast exercise is performed relying on a rolling window scheme. The out-of-sample period is from January 4, 2021 to February 28, 2023. Figure 3 presents the out-of-sample volatility forecasts for the competing models. Table 6 reports the out-of-sample forecast evaluation results based on the four loss functions. As can be seen from the table, the REGARCH model offers smaller loss values (MAE, MAPE, MSE, QLIKE) than the GARCH and EGARCH models. In particular, the REGARCH-Jump model yields the smallest loss values in all cases. Our findings indicate that incorporating the extreme-value information and jump dynamics into volatility model is important for improving the out-of-sample volatility forecasts.

Further, Table 7 presents the MCS test results for the out-of-sample forecasts of competing models. It is clear that the REGARCH-Jump model is always included in the MCS, and always has the highest MCS p-value (p = 1), confirming the superior performance of the REGARCH-Jump model over all other models in forecasting the crude oil futures volatility.

For robustness, we perform the out-of-sample forecast exercise for different out-of-sample windows. To be specific, we consider two alternative out-of-sample windows: 200 and 400. The out-of-sample forecast evaluation results for the alternative out-of-sample windows are reported in Table 8.

In line with the results reported in Tables 6, 7, the REGARCH-Jump model that accounts for the extreme-value information and jump dynamics yields the most accurate volatility forecasts, dominating all other models. This result confirms that the superior performance of the REGARCH-Jump model in forecasting the crude oil futures volatility is robust to alternative out-of-sample forecast windows.

To illustrate the economic value of the improved volatility forecasts from the REGARCH-Jump model, we perform a VaR analysis in this subsection. Accurate measurement of financial market risk is of great significance to the investors, policy makers and regulators who are trying to manage the risk of portfolio as well as to maintain the functioning and the stability of financial markets. The standard tool for measuring market risk is VaR, which is intuitive, simple and easy to compute. It is used by financial institutions and financial regulators worldwide for market risk mointoring and management.