To jump or not to jump: momentum of jumps in crude oil price volatility prediction

Abstract

A well-documented finding is that explicitly using jumps cannot efficiently enhance the predictability of crude oil price volatility. To address this issue, we find a phenomenon, “momentum of jumps” (MoJ), that the predictive ability of the jump component is persistent when forecasting the oil futures market volatility. Specifically, we propose a strategy that allows the predictive model to switch between a benchmark model without jumps and an alternative model with a jump component according to their recent past forecasting performance. The volatility data are based on the intraday prices of West Texas Intermediate. Our results indicate that this simple strategy significantly outperforms the individual models and a series of competing strategies such as forecast combinations and shrinkage methods. A mean–variance investor who targets a constant Sharpe ratio can realize the highest economic gains using the MoJ-based volatility forecasts. Our findings survive a wide variety of robustness tests, including different jump measures, alternative volatility measures, various financial markets, and extensive model specifications.

Introduction

The jump component is an essential and useful determinant in the prediction of the volatility dynamics of various asset prices, such as exchange rates, stock returns, and bond yields (see, e.g., Andersen et al. 2007; Corsi et al. 2010; Duong and Swanson 2015; Patton and Sheppard 2015; Clements and Liao 2017). However, an influential paper by Prokopczuk et al. (2016) argues that explicitly using jumps cannot efficiently enhance the out-of-sample forecast accuracy for the volatility of the crude oil futures market. Prokopczuk et al. (2016) provide a plausible explanation that unpredictable events such as political unrest and natural disasters in oil-exporting countries always trigger jumps in oil prices. Theoretically, jumps do not occur at each point in time. When there is no jump in the oil price, incorporating the jump component into the predictive model will probably lead to overfitting, in which the in-sample forecasting performance improves but the out-of-sample performance worsens. Hence, it is critical to investigate whether the jump component should be included in volatility models in real time. The main purpose of this study is to selectively use jumps (or jump model) and thereby discover useful forecasting information that is hidden in jumps. To this end, we propose a simple but successful strategy to improve the forecasting ability of the jump model. This is our main contribution to the literature on forecasting crude oil market volatility.

Our new strategy makes an optimal predictive model switch between the benchmark of the heterogeneous autoregressive realized variance (HAR-RV)¹ model pioneered by Corsi (2009) and the HAR-CJ jump model pioneered by Andersen et al. (2007), which is an extension that adds the jump component to the HAR-RV specification. The switching behavior is conditional on the relative predictive performances of these two models during a recent period of past time. Specifically, if the recent past predictive ability of the HAR-CJ model is superior to that of the HAR-RV model, we continue to select the HAR-CJ model to predict the oil futures market realized variance (RV) in the near future. Otherwise, the HAR-RV benchmark is used to generate volatility forecasts. The motivation is straightforward: We believe that the relatively strong forecasting power of the HAR-CJ model is persistent. That is, a jump model with a good past performance will have a good forecasting performance in the near future. We term this phenomenon ‘‘momentum of jumps’’ (MoJ), in the context of forecasting oil futures market volatility. To be precise, MoJ refers to the momentum (or persistence) of the predictive ability of a jump model.

Consistent with the empirical findings by Prokopczuk et al. (2016), our full-sample estimation results show that the overall impact of the jump component on the oil futures market RV is not pronounced. In other words, the benefits from the use of jumps appear to be negligible. This in-sample evidence also suggests that using the jump model (i.e., the HAR-CJ model) alone is unlikely to be feasible for out-of-sample forecasting exercises. Nevertheless, we observe that the HAR-CJ model outperforms the HAR-RV model during part of the out-of-sample period. More importantly, several relatively good periods are clustered and show a continuous pattern. Using a formal test proposed by Wang et al. (2018), we document the existence of the MoJ between the HAR-RV and HAR-CJ models for the oil futures market. The existence of the MoJ phenomenon suggests that if the jump model (i.e., the HAR-CJ model) could produce more accurate RV forecasts than the HAR-RV benchmark over a recent past period, the jump model would continuously produce more accurate RV forecasts in the near future. This lays the foundation for the success of our MoJ strategy in forecasting oil futures RV.

In our MoJ strategy, we use the mean squared error (MSE) loss function to assess the past predictive performance of the individual HAR-RV and HAR-CJ models during a look-back period. The MoJ strategy uses the model that shows a relatively good past forecasting performance to generate an RV forecast in the next forecast step. To rule out the concern that the success of the MoJ method is due to the forecast combination,² we further consider the simple mean combination as a competing model, which takes the equally weighted average of the individual HAR-RV and HAR-CJ forecasts. In terms of the out-of-sample evaluation, we rely on the statistic test of the model confidence set (MCS) originated by Hansen et al. (2011). We find convincing evidence that the MoJ strategy consistently exhibits a substantially stronger out-of-sample predictive ability than the competing models of the HAR-RV, HAR-CJ, and mean combination for not only the 1-day, 5-day, 10-day, and 22-day forecast horizons, but also for six widely used loss functions.

We further present a multitude of robustness tests and extensions. The results are summarized in seven streams. First, we consider the robustness regarding the jump detection test and jump model. Specifically, we additionally use two jump models, namely, the HAR-J model proposed by Andersen et al. (2007), which uses a simple measure of the jump component without a jump detection test, and the HAR-TCJ model proposed by Corsi et al. (2010), in which a threshold jump measure is used. In terms of the jump detection test, we consider three confidence levels: 1%, 0.5%, and 0.1%.

The second type of robustness test entails how to evaluate past forecasting performance in our MoJ strategy. We calculate the forecast error of past RV forecasts during 1-day, 5-day, 10-day, and 22-day look-back periods. Furthermore, the average MoJ forecasts are generated by the individual MoJ forecasts based on the various look-back periods. In addition, we use three different evaluation criteria to assess past forecasting performance.

Third, we consider the robustness of the model specification. The MoJ method is mainly based on the linear HAR models. Alternatively, we consider not only the nonlinear HAR models cast in logarithmic and standard deviation forms, but also the MIDAS model, which is regarded as a generalized version of the HAR framework.

Fourth, we consider alternative volatility estimators and forecast evaluation methods. The volatility estimators include the RV and realized kernel (RK). In addition, we use six different loss functions to evaluate forecast accuracy. Moreover, both the MCS test by Hansen et al. (2011) and the Diebold–Mariano (DM) test by Diebold and Mariano (1995) are used to calculate the significance level of forecast accuracy.

Fifth, we consider alternative estimation windows in out-of-sample forecasting exercises. On the one hand, we use both the rolling and expanding estimation windows. On the other hand, we consider different window sizes (i.e., various lengths of out-of-sample evaluation periods).

Sixth, we extend our competing models by considering other similar strategies, including alternative forecast combination approaches that also depend on individual models’ past forecasting performance and the three widely used shrinkage approaches of the ridge, lasso, and elastic net.

Finally, the MoJ strategy is extended to the stock market. That is, we use the MoJ strategy as well as the competing models to predict the stock market RV. Fortunately, we observe consistent results for all the above-mentioned robustness tests and extensions, which greatly alleviate the concern of data mining.

In a portfolio exercise, we explore the economic significance of the volatility forecasts of the MoJ strategy and the competing models. Specifically, we follow Bollerslev et al. (2018) and consider a specific case, in which a mean–variance investor who targets a constant Sharpe ratio allocates her wealth between a risky asset (i.e., oil futures) and a risk-free asset (i.e., risk-free bills). The corresponding results suggest that the four forecasting models used in this study deliver sizeable utility gains relative to the ones from a static model that uses the rolling sample average of past RVs. More importantly, the utility gains from our MoJ model relative to the ones from the static model are always highest. The mean–variance investor would be glad to pay at least 56 basis points to employ the MoJ model rather than the simple static model, which is, of course, economically significant.

We organize the paper as follows. ’Related literature and our contribution” section reviews the related literature and highlights the paper’s contribution. “RV and HAR models” section details the methodology of RV and HAR models. “Data and in‑sample results” section presents the data and in-sample results, while  “Out‑of‑sample analyses” section provides the out-of-sample analyses. “Robustness checks” section provides a wide variety of robustness checks. “Extension and application” section presents extensions and an economic application of the MoJ strategy. “Conclusion” section concludes this paper.

Related literature and our contribution

In this section, we review the related literature on (a) jump behavior in the crude oil market, (b) jumps and crude oil volatility forecasting, and (c) the momentum of predictability (MoP). Moreover, we separately discuss the innovative work of this paper in the three aspects.

Jump behavior in the crude oil market

Crude oil prices are characterized by jump behavior. Gronwald (2012) argues that a large quantity of total oil price volatility is triggered by jumps. Wilmot and Mason (2013) document that jumps help to improve a model’s ability to explain crude oil prices. Bouri (2019) finds that the jumps in the sovereign risks of major oil-exporting countries are significantly driven by oil price volatility jumps. Bouri and Gupta (2020) present that crude oil price jumps and macroeconomic news surprises are likely to occur synchronously, indicating the sensitivity of crude oil prices to macroeconomic news. Bouri et al. (2021) provide evidence that the spillover effect of jumps in crude oil and other financial markets is notable. In contrast, this paper provides new insights into the jump behavior of crude oil prices. That is, we find a novel phenomenon, MoJ, in the forecasting of oil futures market volatility. The predictive ability of jumps is confirmed to be persistent.

Jumps and crude oil volatility forecasting

Andersen et al. (2007) is perhaps the first study that uses jump information to forecast the RV of financial assets. Following this seminal work, a growing number of studies rely on jump information to improve the predictability of crude oil volatility (see, e.g., Liu et al. 2018; Ma et al. 2019; Dutta et al. 2021). In contrast, the work by Prokopczuk et al. (2016) most closely relates to this paper. Prokopczuk et al. (2016) explore the role of jumps in forecasting energy market volatility and find that explicitly modeling jumps does not significantly improve the forecast accuracy for the volatility of the oil futures market. However, their study is silent on how to improve the accuracy of oil price volatility forecasts. Our paper contributes to their study by providing a solution to the problem of improving forecast accuracy. Specifically, we propose the MoJ strategy, which selectively uses the jump model and thereby successfully captures the useful forecasting component contained in jumps.

MoP

Our paper also contributes to the literature on MoP (see Wang et al. 2018; Zhang et al. 2019a). Wang et al. (2018) find the MoP that a univariate predictive regression with one macroeconomic variable, which generates more accurate return forecasts than the benchmark of historical average during several past months, can continue to successfully predict stock market returns in the near future. Zhang et al. (2019a) document the existence of the MoP between low- and high-frequency forecasting models in the case of forecasting stock market volatility, thus establishing a new mixed-frequency model. In this sense, the MoJ proposed by our paper is not completely new, as it has been empirically confirmed by the related studies of Wang et al. (2018) and Zhang et al. (2019a). However, the MoP findings of the two related studies, in other words, motivate and support us to investigate the MoJ. In contrast, our paper provides a new study that focuses on the momentum of the forecasting performance of the jump model in predicting oil futures market volatility. This is necessary and meaningful because the role of jumps in forecasting oil futures market volatility is found to be limited. To address this issue, we rely on the fundamental of the MoP and thereby present an efficient model, i.e., the MoJ strategy.

More broadly, the MoJ strategy, as well as the MoP, is related to the conditional combination approaches that are based on past predictive performance (see, e.g., Stock and Watson 2004; Yang 2004; Giacomini and White 2006). In contrast, the contribution of this paper is not a technical innovation but a novel idea of clustered jumps (i.e., MoJ). Moreover, we document that our MoJ strategy can outperform a popular forecast combination approach that is conditional on past predictive performance.

RV and HAR models

RV and jump

The quadratic variation (QV) of the asset price return process can be given by

$$QV_{t} = \int_{t - 1}^{t} {\sigma^{2} (z)dz + \sum\limits_{t - 1 < z \le t} {\nu^{2} (z)} } ,$$

(1)

where in the continuous-time jump diffusion process, $\sigma (z)$ represents a stochastic volatility process and $\nu (z)$ measures the size of discrete jumps. On the right hand of Eq. (1), the first term is the so-called integrated variance (IV), which is regarded as the continuous sample path component of QV, while the second one represents the jump (discontinuous) component of QV.

Andersen and Bollerslev (1998), Barndorff-Nielsen and Shephard (2002), and Andersen et al. (2003) emphasize that the RV estimator uniformly converges to QV in probability as the sampling frequency increases. The RV measure can be calculated as the summation of squared intraday oil price returns,

$$RV_{t} = \sum\limits_{j = 1}^{N} {r_{t,j}^{2} } ,$$

(2)

where $RV_{t}$ refers to the realized variance measure on trading day t, $N = 1/\Delta$, $\Delta$ is the sampling interval for intraday returns, and $r_{t,j}$ denotes the jth intraday oil futures market return during day t. Since RV converges to QV, we have

$$RV_{t} \to \int_{t - 1}^{t} {\sigma^{2} (z)dz} + \sum\limits_{t - 1 < z \le t} {\nu^{2} (z)} ,$$

(3)

for $N \to \infty$ or $\Delta \to 0$. Barndorff-Nielsen and Shephard (2004) further propose an estimator dubbed realized bipower variation (BPV), which takes the form of

$$BPV_{t} = \kappa_{1}^{ - 2} \sum\nolimits_{j = 2}^{N} {\left| {r_{t,j} } \right|} \left| {r_{t,j - 1} } \right|,$$

(4)

where $\kappa_{1} = \sqrt {{2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi }}$. As the sampling frequency increases ($N \to \infty$ or $\Delta \to 0$), BPV is a consistent estimator of IV. That is, we have

$$BPV_{t} \to \int_{t - 1}^{t} {\sigma^{2} (z)dz} ,$$

(5)

for $N \to \infty$. Combining Eqs. (3) and (5), we can consistently estimate the jump (discontinuous) component of QV as

$$RV_{t} - BPV_{t} \to \sum\limits_{t - 1 < z \le t} {\nu^{2} (z)} ,$$

(6)

for $N \to \infty$. To ensure that each jump estimate is nonnegative, Andersen et al. (2007) truncate jump measures at zero, which is also suggested by Barndorff-Nielsen and Shephard (2004). Consequently, the daily jump measure on day t is given by

$$J_{t} = \max \{ RV_{t} - BPV_{t} ,0\} .$$

(7)

We follow Andersen et al. (2007) and employ the ratio statistic to detect significant jumps. The jump detection test based on the ratio statistic is given by

$$Z_{t} = \Delta^{ - 1/2} \frac{{(RV_{t} - BPV_{t} )/RV_{t} }}{{\sqrt {(\kappa_{1}^{ - 4} + 2\kappa_{1}^{ - 2} - 5)\max \{ 1,TQ_{t} /BPV_{t}^{2} \} } }},$$

(8)

where $TQ_{t}$ denotes the realized tripower quarticity measure. Statistically, TQ is expressed as

$$TQ_{t} = \Delta^{ - 1} \kappa_{{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}}^{ - 3} \sum\nolimits_{j = 3}^{N} {\left| {r_{t,j} } \right|}^{{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}} \left| {r_{t,j - 1} } \right|^{{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}} \left| {r_{t,j - 2} } \right|^{{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}} ,$$

(9)

where $\kappa_{{{4 \mathord{\left/ {\vphantom {4 3}} \right. \kern-\nulldelimiterspace} 3}}}^{{}} = 2^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}}} {{\Gamma ({7 \mathord{\left/ {\vphantom {7 6}} \right. \kern-\nulldelimiterspace} 6})} \mathord{\left/ {\vphantom {{\Gamma ({7 \mathord{\left/ {\vphantom {7 6}} \right. \kern-\nulldelimiterspace} 6})} {\Gamma (0.5)}}} \right. \kern-\nulldelimiterspace} {\Gamma (0.5)}}$. As the test statistic in Eq. (8) closely follows a standard normal distribution, the significant jump (SJ) is naturally expressed as

$$SJ_{t} = I(Z_{t} > \Theta_{\alpha } )(RV_{t} - BPV_{t} ),$$

(10)

where $I (\cdot)$ refers to an indicator function, which equals one if a significant jump happens and zero otherwise, and $\Theta_{\alpha }$ denotes the threshold value that is calculated by the cumulative standard normal distribution at the confidence level of 1 − α. We follow Andersen et al. (2007) and rely on the significance level of 0.5%. To ensure that the sum of the continuous and discontinuous components equals the whole RV, the continuous component is then identified as

$$C_{t} = I(Z_{t} \le \Theta_{\alpha } )RV_{t} + I(Z_{t} > \Theta_{\alpha } )BPV_{t} .$$

(11)

HAR models

In terms of our forecasting strategies, the benchmark method is naturally the HAR-RV model, which is pioneered by Corsi (2009). The HAR-RV model is probably the most popular volatility model. This is because the model captures some stylized facts of asset return volatility such as long memory and multi-scaling behavior. Furthermore, the HAR-RV model is tractable, as it merely includes three variables without any hyperparameter tuning. Therefore, its model specification is straightforward, and can be shown as

$$RV_{t + 1:t + h} = \varphi_{0} + \varphi_{d} RV_{t} + \varphi_{w} RV_{t - 4:t} + \varphi_{m} RV_{t - 21:t} + \omega_{t + 1:t + h} ,$$

(12)

where $RV_{{t + {1}:t + h}} = ({1 \mathord{\left/ {\vphantom {1 h}} \right. \kern-\nulldelimiterspace} h})(RV_{t + 1} + \cdots + RV_{t + h} )$. In particular, $RV_{t}$, $RV_{t - 4:t}$, and $RV_{t - 21:t}$ denote the daily, weekly, and monthly RVs, respectively, all of which are available up to day t.

More importantly, our MoJ strategy requires a jump model. We choose the HAR-CJ model, which is pioneered by Andersen et al. (2007) and has been widely used by a host of literature on predicting asset return volatility (see, e.g., Sévi 2014; Prokopczuk et al. 2016; Wang et al. 2016; Buncic and Gisler 2017; Zhang et al. 2019a, 2021). Mathematically, the HAR-CJ model takes the following form:

$$RV_{t + 1:t + h} = \varphi_{0} + \varphi_{cd} C_{t} + \varphi_{cw} C_{t - 4:t} + \varphi_{cm} C_{t - 21:t} + \varphi_{sd} SJ_{t} + \varphi_{sw} SJ_{t - 4:t} + \varphi_{sm} SJ_{{t - 2{1}:t}} + \omega_{t + 1:t + h} {.}$$

(13)

For robustness, we also consider alternative jump measures and jump models in Sect. 6.1.

Data and in-sample results

Data

Following Sévi (2014), Haugom et al. (2014), and Zhang et al. (2022), we choose a well-known oil price benchmark, West Texas Intermediate (WTI). The intraday price data of the WTI futures are obtained from Tick Data. The whole sample period is between January 3, 2012 and May 11, 2018.

The 5-min RV is commonly used by a substantial body of literature on predicting oil futures market RV (see, e.g., Haugom et al. 2014; Sévi 2014; Ma et al. 2019; Yang et al. 2019; Zhang et al. 2019c; Niu et al. 2021). Overall, Liu et al. (2015) argue that it is extremely difficult to outperform the 5-min RV by using any other volatility measures from a wide range of estimators and financial assets. Thus, we rely on the 5-min interval as the sampling frequency to calculate the oil futures market RV.

In-sample results

The in-sample estimation results of the individual HAR-RV and HAR-CJ models are reported in Table 1. One striking observation follows the table immediately. The HAR predictors (namely, $RV_{t}$, $RV_{t - 4:t}$, and $RV_{t - 21:t}$) always yield significant coefficients, while the regression coefficients of the lagged daily and weekly SJs are always insignificant. Although the R² of HAR-CJ is greater than that of HAR-RV, the improvement in the R² is limited. Overall, the full-sample estimation results suggest that the jump components do not contain a powerful explanatory ability for future oil futures RV. This also implies that a straightforward approach of using the jump model (i.e., the HAR-CJ) alone is unlikely to be feasible. This evidence echoes the findings of Prokopczuk et al. (2016), who document that the jump components are not useful for the in-sample predictability of crude oil price volatility.

Table 1 Full-sample estimation results of the HAR-RV and HAR-CJ models

Out-of-sample analyses

Forecasting methodology of individual HAR regression models

The in-sample estimation analysis only provides the predictive information of the regression models (namely, the HAR-RV and HAR-CJ), while it is silent on our MoJ strategy and the mean combination approach. In real time, financial investors and practitioners pay more attention to the out-of-sample forecasting test as it is more relevant to examining genuine predictive ability. Moreover, an in-sample analysis is probably influenced by econometric issues such as the Stambaugh bias (see Busetti and Marcucci 2013), small-sample size distortion, and over-fitting, whereas an out-of-sample test is less likely to be influenced. Therefore, it is more crucial to assess the out-of-sample forecasting ability of the volatility models used.

In this study, we generate the out-of-sample RV forecasts for the individual HAR-RV and HAR-CJ models by employing a rolling estimation window. Specifically, we decompose the whole sample period into an in-sample training period and an out-of-sample forecasting period. The former contains the initial 819 observations, while the latter contains the remaining 800 observations. When we obtain each out-of-sample RV forecast, we roll the estimation window forward by not only discarding the first used observation, but by also including one new observation. Finally, the MoJ and mean combination forecasts are produced by the individual HAR-RV- and HAR-CJ forecasts.

Forecasting methodology based on the MoJ

Wang et al. (2018) present a similar phenomenon, termed MoP, in which the stock return predictability of univariate regressions is persistent. Specifically, a superior past predicting performance of a univariate regression model that uses a single economic variable is commonly followed by a superior future predicting performance. Furthermore, Zhang et al. (2019a) document that the MoP also exists between GARCH-class and HAR-RV-type models. Along the same lines, we propose the MoJ. To be precise, the MoJ refers to the MoP between the volatility forecasting models with and without jumps. In our case, we have two strands of RV forecasts, separately given by the HAR-RV and HAR-CJ models. We then continue to employ the volatility model whose past predictive performance is relatively good. Following Zhang et al. (2019a) and Wang et al. (2018), we assess whether the past predictive performance of the HAR-CJ model is superior to that of the HAR-RV model as follows.

$$pp_{t + 1:t + h} (k) = I\left( {\sum\limits_{i = t - h - k + 1}^{t - h} {(RV_{i + 1:i + h} - \widehat{RV}_{i + 1:i + h}^{CJ} )^{2} } - \sum\limits_{i = t - h - k + 1}^{t - h} {(RV_{i + 1:i + h} - \widehat{RV}_{i + 1:i + h}^{RV} )^{2} } < 0} \right),$$

(14)

where k denotes the length of the look-back evaluation period,³$I( \cdot )$ denotes an indicator function, $RV_{t + 1:t + h}$ is the true RV on days $t + 1:t + h$, and $\widehat{RV}_{i + 1:i + h}^{CJ}$ and $\widehat{RV}_{i + 1:i + h}^{RV}$ are the HAR-CJ and HAR-RV forecasts, respectively, for $RV_{i + 1:i + h}$. Based on the relative past performance, as defined by $pp_{t + 1:t + h} (k)$, we can readily obtain the corresponding MoJ forecast as follows.

$$\widehat{RV}_{t + 1:t + h}^{MoJ} (k) = \left\{ \begin{gathered} \widehat{RV}_{t + 1:t + h}^{CJ} ,{\text{ if }}pp_{t + 1:t + h} (k) = 1 \hfill \\ \widehat{RV}_{t + 1:t + h}^{RV} ,{\text{ if }}pp_{t + 1:t + h} (k) = 0. \hfill \\ \end{gathered} \right.$$

(15)

Simple mean forecast combination

Our MoJ strategy switches between the benchmark and jump models by observing their relative past forecasting performances. This model selection approach can be treated as a particular combination approach, in which the weight of each model is a binary variable that equals either 0 or 1. For comparison, we also use the equal-weighted combination forecasts, ${1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}(\widehat{RV}_{t + 1:t + h}^{CJ} + \widehat{RV}_{t + 1:t + h}^{RV} )$. Here, we do not consider any more complex weighting schemes when combining forecasts. This is because the famous “forecast combination puzzle” suggests that the simple mean cannot be systematically outperformed by many other sophisticated combination methods in out-of-sample prediction exercises (see, e.g., Stock and Watson 2004; Rapach et al. 2010).

Evaluation framework

We employ six commonly used loss functions to provide a quantitative assessment of the out-of-sample predictive performance for different volatility forecasting strategies: the Quasi-Likelihood (QLIKE), mean squared error (MSE), mean absolute error (MAE), mean squared percentage error (MSPE), mean absolute percentage error (MAPE), and mean squared logarithmic error (MSE-LOG) loss functions, which are statistically expressed as

$$QLIKE:L(\widehat{RV}_{t + 1:t + h} ,RV_{t + 1:t + h} ){\text{ = log(}}\widehat{RV}_{t + 1:t + h} ) + \frac{{RV_{t + 1:t + h} }}{{\widehat{RV}_{t + 1:t + h} }},$$

(16)

$$MSE: \, L(\widehat{RV}_{t + 1:t + h} ,RV_{t + 1:t + h} ) = \left( {\widehat{RV}_{t + 1:t + h} - RV_{t + 1:t + h} } \right)^{2} ,$$

(17)

$$MAE: \, L(\widehat{RV}_{t + 1:t + h} ,RV_{t + 1:t + h} ) = \left| {\widehat{RV}_{t + 1:t + h} - RV_{t + 1:t + h} } \right|,$$

(18)

$$MSPE: \, L(\widehat{RV}_{t + 1:t + h} ,RV_{t + 1:t + h} ) = \left( {1 - \frac{{\widehat{RV}_{t + 1:t + h} }}{{RV_{t + 1:t + h} }}} \right)^{2} ,$$

(19)

$$MAPE: \, L(\widehat{RV}_{t + 1:t + h} ,RV_{t + 1:t + h} ) = \left| {1 - \frac{{\widehat{RV}_{t + 1:t + h} }}{{RV_{t + 1:t + h} }}} \right|,$$

(20)

and

$$MSE{ - }LOG: \, L(\widehat{RV}_{t + 1:t + h} ,RV_{t + 1:t + h} ) = \left( {\log (\widehat{RV}_{t + 1:t + h} ) - \log (RV_{t + 1:t + h} )} \right)^{2} ,$$

(21)

respectively, where $RV_{t + 1:t + h}$ is the true RV for days $t + 1:t + h$, and $\widehat{RV}_{t + 1:t + h}$ denotes the RV forecast given by one of the predictive strategies. Patton (2011) recommends the use of the QLIKE and MSE loss functions because the two are robust to the presence of noise in the volatility proxy. Nonetheless, we employ more loss functions to show a comprehensive test.

To ascertain the confidence level of the different models’ out-of-sample forecast accuracies, we follow extensive literature on predicting RV (see, e.g., Patton and Sheppard 2009; Liu et al. 2015; Gong and Lin 2018; Zhang et al. 2019c, 2020; Calzolari et al. 2021; Dai et al. 2022) and employ the MCS econometric method pioneered by Hansen et al. (2011). An MCS refers to a subset of all the used models into which the best model falls with a specific confidence level. Generally, a model that delivers a larger MCS p value is more likely to show the best forecasting performance. Following Hansen et al. (2011) and Zhang et al. (2019c), we choose the confidence (significance) level of 90% (10%). In other words, a model whose MCS p value is greater than 0.1 falls into the MCS. Finally, it should be noted that the MCS p values we report below are all calculated based on the range statistic; however, the results are similar when we rely on the semi-quadratic statistic.

Forecasting performance

Table 2 presents the MCS test results. We summarize the table with one key observation. Our MoJ strategy always produces the highest MCS p value (i.e., 1). In contrast, the HAR-RV, HAR-CJ, and mean combination models generate substantially lower MCS p values, most of which are lower than 0.1, indicating that the corresponding models cannot enter the MCS at the 10% significance level. Overall, the reported MCS p values indicate that our MoJ strategy exhibits significantly better forecasting performance than the competing models of the individual HAR-RV and HAR-CJ models as well as the mean combination. Furthermore, we observe that the relatively powerful predictive ability of the MoJ strategy consistently exists not only across various loss functions but also across various forecast horizons.

Table 2 Out-of-sample forecasting performance based on the MCS test

Testing the MoJ

Wang et al. (2018) and Zhang et al. (2019a) both highlight that the success of their MoP strategies relies on the presence of the MoP. Therefore, we need to investigate whether the superiority of our MoJ strategy is supported by the existence of the MoJ. More precisely, the MoJ refers to the momentum of the predictive ability of the forecasting strategy with jumps relative to the one without jumps. That is, we should examine whether a better past predictive performance of the HAR-CJ models relative to that of the HAR-RV model can generally result in a better future performance. Statistically, the future predictive performance of the HAR-CJ relative to that of the HAR-RV over days $t + 1:t + h$ is defined as

$$fp_{t + 1:t + h} = I\left( {(RV_{t + 1:t + h} - \widehat{RV}_{t + 1:t + h}^{CJ} )^{2} - (RV_{t + 1:t + h} - \widehat{RV}_{t + 1:t + h}^{RV} )^{2} < 0} \right).$$

(22)

In a statistical sense, a cross-sectional dependence of $pp_{t + 1:t + h} (k)$ and $fp_{t + 1:t + h}$ implies the existence of the MoJ. Following the related studies of Zhang et al. (2019a) and Wang et al. (2018), we rely on the chi-square statistic proposed by Pesaran and Timmermann (2009) to test the null hypothesis that $pp_{t + 1:t + h} (k)$ and $fp_{t + 1:t + h}$ are not cross-sectional dependent in the presence of serial dependencies for each series itself against the alternative hypothesis that the two time-series variables are cross-sectional dependent. In this sense, if the null hypothesis of no dependence between $pp_{t + 1:t + h} (k)$ and $fp_{t + 1:t + h}$ is rejected, we statistically prove that the MoJ exists.

We follow Wang et al. (2018) and report the p-values for the Pesaran and Timmermann (2009) statistics in Table 3. As expected, all the p values for the different lengths of the look-back periods and forecast horizons are less than 0.001. That is, the null hypothesis of independence between $pp_{t + 1:t + h} (k)$ and $fp_{t + 1:t + h}$ is rejected below the 0.1% significance level. This evidence suggests that the MoJ does exist between the HAR-CJ model with and the HAR-RV model without jump information. In other words, we statistically document that a better past performance of the HAR-CJ model is always associated with a better future forecasting performance. Of course, the existence of the MoJ phenomenon is the fundamental driving force of our MoJ method.

Table 3 Testing results of the momentum of jumps

To further provide a visual impression of the model switching between the HAR-CJ and HAR-RV models, we plot the dynamics of the model selection between the two models in Fig. 1 for various forecast horizons, which is based on the case of k = 5. We summarize this graphical device with two major observations. First, we observe that our MoJ strategy sometimes selects the HAR-RV model and sometimes selects the HAR-CJ model. This implies that the jump component cannot always provide useful information for forecasting the oil futures market RV during the entire out-of-sample period; however, it contains useful forecasting information during part of the out-of-sample period. That is, the HAR-RV and HAR-CJ models cannot outperform each other completely. This evidences the potential success of our MoJ strategy (which uses both the HAR-RV and HAR-CJ models) in selecting the relatively good model. Second and more importantly, the model selection between the HAR-RV and HAR-CJ is highly persistent. That is, we observe the momentum of model selection. To be precise, the MoJ strategy persistently selects one model between the HAR-RV and HAR-CJ for a relatively long period. Therefore, the model that shows a relatively good past forecasting performance tends to yield a relatively good future performance. This appealing selection pattern contributes to the success of the MoJ strategy.

Fig. 1

Model selection between the HAR-RV and HAR-CJ during the out-of-sample forecasting period. The HAR-RV or HAR-CJ model is selected by the MoJ strategy when its past forecasting performance is relatively good. A selected (discarded) model at each forecast step is shown as a blue (white) bar

Robustness checks

The primary forecasting performance reported in Table 2 shows that our out-of-sample results are robust to a multitude of loss functions and forecast horizons. Furthermore, we provide ten robustness tests in this section. These robustness tests alleviate the concern about data mining, thus validating our results.

Alternative jump models

We use the prevailing HAR-CJ model to incorporate the jump component. However, there are many other jump models used to predict financial market RV. To alleviate the concern about the arbitrary use of the jump model, we additionally consider two popular forecasting models that also use the jump component. The first new jump model is termed HAR-J, which is also originated by Andersen et al. (2007). The HAR-J model specification is given by

$$RV_{t + 1:t + h} = \varphi_{0} + \varphi_{d} RV_{t} + \varphi_{w} RV_{t - 4:t} + \varphi_{m} RV_{t - 21:t} + \varphi_{J} J_{t} + \omega_{t + 1:t + h} ,$$

(23)

The second new jump model is termed HAR-TCJ, which is pioneered by Corsi et al. (2010). To detect jumps, Corsi et al. (2010) depend not only on a new test statistic, termed C-Tz, but also on the threshold bipower variation (TBPV) to calculate the threshold jump measure as $TJ_{t} = I(C{ - }Tz_{t} > \Theta_{\alpha } )(RV_{t} - TBPV_{t} )$. The continuous counterpart is calculated as $TC_{t} = RV_{t} - TJ_{t}$. Consequently, the HAR-TCJ model takes the regression form of

$$RV_{t + 1:t + h} = \varphi_{0} + \varphi_{tcd} TC_{t} + \varphi_{tcw} TC_{t - 4:t} + \varphi_{tcm} TC_{t - 21:t} + \varphi_{TJ} TJ_{t} + \omega_{t + 1:t + h} .$$

(24)

Thus far, we have the HAR-RV benchmark model and two new jump models of the HAR-J and HAR-TCJ. We can generate two new MoJ and mean combination forecasts by separately using the two new jump models. Tables 4 and 5 provide the forecasting results when we use the HAR-J and HAR-TCJ jump models, respectively. In the HAR-J case, we find that the MoJ model falls into the MCS at the 10% significance level for all the 24 cases (6 different loss functions and 4 different forecast horizons). Furthermore, our MoJ model generates the greatest MCS p value (i.e., 1) for 22 out of the 24 cases. The HAR-J model generates the greatest MCS p value for only 2 cases and survives in the MCS test for several cases. However, the HAR-J model as well as the other two competing models fails to remain in the MCS for most of the 24 cases. The results suggest that the MoJ method has significantly stronger predictive power than the competing methods for most of the cases (i.e., most loss functions and forecast horizons). We observe similar results when using the HAR-TCJ as the jump model. Thus, our forecasting results are robust to different jump models.

Table 4 MCS out-of-sample forecasting test using the HAR-J jump model

Table 5 MCS out-of-sample forecasting test using the HAR-TCJ jump model

In this subsection, we use two alternative jump measures to explore the robustness of the MoJ strategy. Additionally, sampling frequency is an important factor in detecting jumps (see, e.g., Lyócsa et al. 2020; Maneesoonthorn et al. 2020). However, we leave it for future research due to data constraints.

Alternative look-back periods

Our MoJ strategy relies on recent past forecasting performance that is assessed based on a look-back period whose length is defined as k. The previously reported forecasting results of the MoJ model are based on a weekly (k = 5) look-back period. In this subsection, we follow Wang et al. (2018) and use a few reasonable look-back periods to generate an average MoJ forecast. More specifically, we consider daily (1-day), weekly (5-day), biweekly (10-day), and monthly (22-day) look-back periods, and thereby generate $\widehat{RV}_{t + 1:t + h}^{MoJ} (k)$ for k = 1, 5, 10, 22. The average MoJ forecast is then given by

$$\widehat{RV}_{t + 1:t + h}^{MoJ - AVG} { = }\frac{1}{4}\sum\limits_{{k \in \{ 1,5,10,22\} }} {\widehat{RV}_{t + 1:t + h}^{MoJ} (k)} .$$

(25)

Table 6 presents the corresponding MCS results when the average MoJ strategy is used. Expectedly, the MoJ strategy continues to deliver the highest MCS p values for all the 24 cases and, of course, consistently falls into the MCS at the 10% significance level. In contrast, the competing models of the HAR-CJ and mean combination survive in the MCS for only 1 out of 24 cases (i.e., the 1-day forecast horizon and MSE loss function). The robust results suggest that our MoJ strategy consistently outperforms the competing models. Finally, it should be noted that the MoJ forecasting results based on the individual look-back periods (i.e., k = 1, 10, 22) are tabulated in our Additional file 1. The forecasting results are robust to alternative look-back periods.

Table 6 MCS out-of-sample forecasting test based on various look-back periods

Nonlinear HAR models

A commonly considered issue for model specification is whether to use linear or nonlinear HAR models. With this in mind, we further employ nonlinear HAR models that are cast in logarithmic and standard deviation forms (see also Andersen et al. 2007; Corsi et al. 2010; Prokopczuk et al. 2016). Mathematically, the logarithmic HAR-RV and HAR-CJ models are given by

$$\ln (RV_{t + 1:t + h} ) = \varphi_{0} + \varphi_{d} \ln (RV_{t} ) + \varphi_{w} \ln (RV_{t - 4:t} ) + \varphi_{m} \ln (RV_{t - 21:t} ) + \omega_{t + 1:t + h}$$

(26)

and

$$\begin{aligned} \ln (RV_{t + 1:t + h} ) = \,& \varphi_{0} + \varphi_{cd} \ln (C_{t} ) + \varphi_{cw} \ln (C_{t - 4:t} ) + \varphi_{cm} \ln (C_{t - 21:t} ) \\ & + \varphi_{sd} \ln (SJ_{t} + 1) + \varphi_{sw} \ln (SJ_{t - 4:t} + 1) + \varphi_{sm} \ln (SJ_{{t - 2{1}:t}} + 1) + \omega_{t + 1:t + h} , \\ \end{aligned}$$

(27)

respectively. The square-root counterparts are expressed as

$$RV_{t + 1:t + h}^{{{{1} \mathord{\left/ {\vphantom {{1} {2}}} \right. \kern-\nulldelimiterspace} {2}}}} = \varphi_{0} + \varphi_{d} RV_{t}^{{{{1} \mathord{\left/ {\vphantom {{1} {2}}} \right. \kern-\nulldelimiterspace} {2}}}} + \varphi_{w} RV_{t - 4:t}^{{{{1} \mathord{\left/ {\vphantom {{1} {2}}} \right. \kern-\nulldelimiterspace} {2}}}} + \varphi_{m} RV_{t - 21:t}^{{{{1} \mathord{\left/ {\vphantom {{1} {2}}} \right. \kern-\nulldelimiterspace} {2}}}} + \omega_{t + 1:t + h}$$

(28)

and

$$\begin{aligned} RV_{t + 1:t + h}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} =\, & \varphi_{0} + \varphi_{cd} C_{t}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \varphi_{cw} C_{t - 4:t}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \varphi_{cm} C_{t - 21:t}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \\ & + \varphi_{sd} SJ_{t}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \varphi_{sw} SJ_{t - 4:t}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \varphi_{sm} SJ_{t - 21:t}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} + \omega_{t + 1:t + h} , \\ \end{aligned}$$

(29)

respectively.

Tables 7 and 8 present the corresponding MCS results when we employ the logarithmic and square-root HAR models, respectively. In the logarithmic version, our MoJ strategy generates the highest p values for 22 out of 24 cases and falls into the MCS at the 10% significance level across all the 24 cases. Moreover, we observe a better forecasting performance of the MoJ strategy than the competing models in the square-root version. Overall, the MoJ strategy still shows a significantly stronger predictive ability than the competing models when nonlinear HAR models are employed. Our forecasting results are thus robust to the use of linear and nonlinear HAR models.

Table 7 MCS out-of-sample test using the logarithmic HAR-RV and HAR-CJ models

Table 8 MCS out-of-sample test using the square-root HAR-RV and HAR-CJ models

MIDAS model

A growing number of studies rely on the MIDAS regression model to forecast financial market volatility (see, e.g., Ghysels et al. 2006, 2007; Forsberg and Ghysels 2007; Santos and Ziegelmann 2014; Ma et al. 2019). The HAR model imposes constant weights on lagged RVs, while the MIDAS model allows a more flexible weighting scheme. In this sense, the HAR model appears to be a special case of the MIDAS model. Therefore, we further examine the forecasting ability of the MoJ strategy based on the MIDAS model. Specifically, we use our MoJ strategy to switch between the MIDAS-RV and MIDAS-CJ models.

The MIDAS-RV model can be shown as

$$RV_{t + 1:t + h} = \mu + \beta \sum\limits_{k = 0}^{{k^{\max } - 1}} {b(k,\theta_{1} ,\theta_{2} )RV_{t - k} } + \omega_{t + 1:t + h} ,$$

(30)

where $k^{\max }$ refers to the maximal lag length of the included RVs and the weighting function, $b(k,\theta_{1} ,\theta_{2} )$, provides the lag coefficients of lagged RVs. Consistent with the previously used HAR models, we use $k^{\max } { = 22}$.⁴ The weighting function, $b(k,\theta_{1} ,\theta_{2} )$, is given by

$$b(k,\theta_{1} ,\theta_{2} ) = \frac{{g\left( {\frac{k}{{k^{\max } }},\theta_{1} ,\theta_{2} } \right)}}{{\sum\nolimits_{j = 1}^{{k^{\max } }} {g\left( {\frac{j}{{k^{\max } }},\theta_{1} ,\theta_{2} } \right)} }},$$

(31)

where $g(x,y,z) = x^{y - 1} {{(1 - x)^{z - 1} } \mathord{\left/ {\vphantom {{(1 - x)^{z - 1} } {f(y,z)}}} \right. \kern-\nulldelimiterspace} {f(y,z)}}$ and $f(y,z) = {{\Gamma (y)\Gamma (z)} \mathord{\left/ {\vphantom {{\Gamma (y)\Gamma (z)} {\Gamma (y + z)}}} \right. \kern-\nulldelimiterspace} {\Gamma (y + z)}}$. It should be noted that the weighting scheme always delivers positive weights, which ensures that the RV forecasts are positive. We refer the reader to Ghysels et al. (2007) for more details regarding the weighting scheme.

Consistent with Santos and Ziegelmann (2014) and Ma et al. (2019), the MIDAS-CJ model can be shown as

$$RV_{t + 1:t + h} = \mu + \beta_{1} \sum\limits_{k = 0}^{{k^{\max } - 1}} {b(k,\theta_{1}^{C} ,\theta_{2}^{C} )C_{t - k} } + \beta_{2} \sum\limits_{k = 0}^{{k^{\max } - 1}} {b(k,\theta_{1}^{SJ} ,\theta_{2}^{SJ} )SJ_{t - k} } + \omega_{t + 1:t + h} .$$

(32)

We provide the forecasting results in Table 9 when using the MIDAS-RV and MIDAS-CJ models to replace the HAR-RV and HAR-CJ models, respectively. The MoJ model consistently falls into the MCS at the 10% significance level for all the cases. Conversely, the competing models of the MIDAS-RV, MIDAS-CJ, and mean combination approach hardly fall into the MCS. This indicates that the MoJ model exhibits substantially better predictive ability than the competing models. Thus, the forecasting results remain robust to the alternative use of the MIDAS or HAR frameworks.

Table 9 MCS out-of-sample test based on the MIDAS-RV and MIDAS-CJ models

Alternative volatility estimators

It is commonly known that the actual measure of asset price volatility is unobservable. With this in mind, we additionally employ another widely used volatility measure, termed realized kernel (RK), which is originally proposed by Barndorff-Nielsen et al. (2008). RK has the appealing property that it is not affected by market microstructure noise. Statistically, the RK is calculated as

$$RK_{t} = \sum\limits_{p = - P}^{P} {k(\frac{p}{P + 1})} \gamma_{p} ,$$

(33)

where

$$\gamma_{p} = \sum\limits_{j = \left| p \right| + 1}^{N} {r_{t,j} r_{t,j - \left| p \right|} }$$

(34)

and k(x) refers to the Parzen kernel function. For more details, please refer to Barndorff-Nielsen et al. (2009).

Table 10 presents the corresponding out-of-sample results when we use the RK to predict oil futures market volatility in all the four models used. The MoJ strategy continues to exhibit substantially stronger forecasting power than the competing models. Specifically, the MoJ model generates the highest MCS p values (i.e., 1) for 23 out of 24 cases and falls into the MCS for all the 24 cases. In contrast, the three competing models enter the MCS for no more than 3 cases. The RK evidence suggests that our out-of-sample results are robust to the use of alternative volatility estimators.

Table 10 MCS out-of-sample forecasting test based the volatility measure of realized kernel

Other robustness tests

In this subsection, we provide additional robustness tests from five different aspects. First, we consider various significance levels for the jump detection test. The previously reported forecasting results are based on the 0.5% significance level, which is suggested by Andersen et al. (2007). For the consideration of robustness, we follow Corsi et al. (2010) and Prokopczuk et al. (2016) and additionally use the 1% and 0.1% significance levels for the jump detection test.

Second, Rossi and Inoue (2012) and Inoue et al. (2017) both present that out-of-sample forecasting performance is often influenced by the choice of forecasting window size. Therefore, we further employ two window sizes. Specifically, the first 1019 and 619 observations are used as the initial training samples, while the rest of the observations are in the out-of-sample period.

Third, while the rolling estimation window can mitigate the impact of structural breaks (see, e.g., Clark and McCracken 2009), the rolling scheme also discards initial observations when the window rolls forward. The discarded observations perhaps contain useful information for forecasting future RV. For this consideration, we alternatively employ the recursive (expanding) estimation window to obtain out-of-sample RV forecasts.

Fourth, as shown in Eq. (14), the past predictive performance is evaluated based on the MSE form. For the consideration of robustness, we separately use the QLIKE and MAE forms to evaluate past forecasting performance.

Fifth, in addition to the MCS test, we consider another popular test, the Diebold–Mariano (DM) test (Diebold and Mariano 1995).⁵ Based on the DM test, we compare each forecasting model with the HAR-RV benchmark to investigate whether the MoJ model shows the highest forecasting gains.

For the sake of brevity, all the results for the five different types of robustness tests are tabulated in the Additional file 1. In short, we find robust results that the MoJ strategy consistently surpasses the competing models.

Extension and application

Stock market evidence

Stock market volatility forecasting is equally important and popular in the academic literature (see, e.g., Wang et al. 2016; Clements and Liao 2017; Zhang et al. 2020). Therefore, the question arises as to whether our MoJ strategy is useful for forecasting stock market volatility. Thus, we extend the MoJ strategy to the stock market. Specifically, we use the MoJ strategy as well as the three competing models to produce the RV forecasts of the S&P 500 Index. The entire sample period spans January 2, 2009 to December 30, 2016, which includes 2001 observations. The first 1201 observations are used as the initial training sample, while the rest 800 observations are in the out-of-sample period. A rolling estimation window is employed to produce the stock market RV forecasts.

Table 11 presents the corresponding MCS results for forecasting stock market volatility. We find similar results for the crude oil futures and stock markets. The MoJ model is in the MCS at the 10% significance level for all the 24 cases. Furthermore, the MoJ model yields the highest MCS value and significantly beats the competing model in 22 out of 24 cases. This evidence suggests that the MoJ strategy is also feasible and useful for forecasting stock market volatility.

Table 11 MCS out-of-sample forecasting test for stock market

Portfolio performance

We examine the economic value of the RV forecasts of the MoJ strategy and the competing models in an asset allocation exercise. Following Bollerslev et al. (2018), we assume that a mean–variance investor will allocate her wealth between a risky asset (i.e., WTI futures) and a risk-free asset (i.e., risk-free bills) with a constant Sharpe ratio. Compared to the related approaches (see, e.g., Fleming et al. 2001, 2003; Campbell and Thompson 2008; Rapach et al. 2010; Zhang et al. 2019c), which rely on both the return and volatility forecasts, the portfolio exercise proposed by Bollerslev et al. (2018) depends exclusively on the volatility forecast. This is appealing since forecasting returns is notoriously difficult (see, e.g., Campbell and Thompson 2008; Welch and Goyal 2008; Rapach et al. 2010).

In the portfolio exercise of Bollerslev et al. (2018), the investor invests a fraction, $w_{t}$, of her current (i.e., time t) portfolio in WTI futures with a return of $r_{t + 1}$ and the rest in risk-free bills with a return of $r_{t}^{f}$. Correspondingly, her future portfolio return becomes $r_{t + 1}^{p} = w_{t} r_{t + 1}^{{}} + (1 - w_{t} )r_{t}^{f} = w_{t} r_{t + 1}^{e} + r_{t}^{f}$, where $r_{t + 1}^{e} = r_{t + 1}^{{}} - r_{t}^{f}$. Excluding the constant terms, which depend only on time-t variables, we can approximate the expected utility as

$$U(w_{t} ) = w_{t} E_{t} (r_{t + 1}^{e} ) - \frac{\gamma }{2}w_{t}^{2} Var(r_{t + 1}^{e} ),$$

(35)

where $\gamma$ denotes the investor’s risk aversion coefficient and $Var(r_{t + 1}^{e} ) = E_{t} (RV_{t + 1} )$. To focus exclusively on volatility forecasting, Bollerslev et al. (2018) assume that the conditional Sharpe ratio, which is written as $SR \equiv {{E_{t} (r_{t + 1}^{e} )} \mathord{\left/ {\vphantom {{E_{t} (r_{t + 1}^{e} )} {\sqrt {E_{t} (RV_{t + 1} )} }}} \right. \kern-\nulldelimiterspace} {\sqrt {E_{t} (RV_{t + 1} )} }}$, is constant. Consequently, the expected utility can be rewritten as

$$U(w_{t} ) = w_{t} SR\sqrt {E_{t} (RV_{t + 1} )} - \frac{\gamma }{2}w_{t}^{2} E_{t} (RV_{t + 1} ),$$

(36)

which simply relies on the portfolio weight, $w_{t}$, and the expected RV, $E_{t} (RV_{t + 1} )$. Maximizing the expected utility given by Eq. (36), we can obtain the optimal portfolio weight for oil futures as follows.

$$w_{t}^{*} = \frac{{{{SR} \mathord{\left/ {\vphantom {{SR} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }}}{{\sqrt {E_{t} (RV_{t + 1} )} }}.$$

(37)

Given Eq. (37), we can derive that the conditional standard deviation of the portfolio’s risky part is $\sqrt {Var(w_{t}^{*} r_{t + 1}^{e} )} = {{SR} \mathord{\left/ {\vphantom {{SR} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }$. This indicates that the investor targets an optimal volatility of ${{SR} \mathord{\left/ {\vphantom {{SR} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }$. When the forecast of $\sqrt {E_{t} (RV_{t + 1} )}$ is greater than the “risk target” of ${{SR} \mathord{\left/ {\vphantom {{SR} \gamma }} \right. \kern-\nulldelimiterspace} \gamma }$ (that is, $w_{t}^{*} < 1$), the investor only allocates part of her wealth to the risky asset of oil futures. On the contrary, when the predicted volatility risk of $\sqrt {E_{t} (RV_{t + 1} )}$ is smaller than this risk target (that is, $w_{t}^{*} > 1$), the investor must rely on leverage to achieve her target.

Substituting Eq. (37) into (36), we can realize an expected utility from the optimally targeted portfolio as follows.

$$U(w_{t}^{*} ) = \frac{{SR^{2} }}{2\gamma }.$$

(38)

However, in practice, $E_{t} (RV_{t + 1} )$ is not available. Using the RV forecast of $\widehat{RV}_{t + 1}$ for day t + 1, we can realize an expected utility of

$$U(\widehat{RV}_{t + 1} ) = \frac{{SR^{2} }}{\gamma }\left( {\frac{{\sqrt {RV_{t + 1} } }}{{\sqrt {\widehat{RV}_{t + 1} } }} - \frac{1}{2}\frac{{RV_{t + 1} }}{{\widehat{RV}_{t + 1} }}} \right).$$

(39)

We empirically report the average utility during the out-of-sample forecasting period. Accordingly, the reported average utility is calculated as

$$\overline{U}(\widehat{RV}) = \frac{1}{q}\sum\limits_{t = R}^{R + P - 1} {\frac{{SR^{2} }}{\gamma }\left( {\frac{{\sqrt {RV_{{t{ + 1}}} } }}{{\sqrt {\widehat{RV}_{{t{ + 1}}} } }} - \frac{1}{2}\frac{{RV_{{t{ + 1}}} }}{{\widehat{RV}_{{t{ + 1}}} }}} \right)} ,$$

(40)

where R and P denote the lengths of in- and out-of-sample periods, respectively. Following Bollerslev et al. (2018), we set the risk aversion coefficient and annualized Sharpe ratio to be γ = 2 and SR = 0.4, respectively.⁶ Consequently, $U(w_{t}^{*} ) = 4\%$, implying that the investor is happy to pay 4% of her wealth to obtain the $w_{t}^{*}$ portfolio of the risky asset rather than to exclusively invest in risk-free bills.

Table 12 reports the portfolio performance evaluated based on the average realized utility. Particularly, we follow Bollerslev et al. (2018) and additionally use a static model as the benchmark in portfolio performance. Under the assumption that volatility risk is constant, the static model simply takes the rolling sample average of in-sample RVs as the RV forecast.⁷ Two important findings emerge. First, all the four forecasting models deliver substantially higher realized utilities than the static model. The utility gains, which are computed as the difference between the realized utilities of our previously used forecasting models and that of the static model, are mostly above 50 basis points. The realized utility can be regarded as the portfolio’s profit (or return) adjusted by volatility risk. Therefore, this evidence means that the investor is happy to forego 50 basis points to have access to the four econometric models rather than to simply use the static model. Second and more importantly, the utility gain from the MoJ model is the largest of the gains from all the four models used. This means that the investor is happy to pay more fees to use the MoJ model than to use the other three competing models. In other words, our MoJ model can deliver the largest economic gains for the assumed investor in a real portfolio exercise.

Table 12 Portfolio performance

A comparison with alternative strategies

In this subsection, we extend our competing models by considering two strands of similar forecasting strategies. First, the MoJ approach also works like the discount mean squared prediction error (DMSPE) combination method. Both the MoJ and DMSPE strategies depend on the past predictive performances of individual models. The difference is that the MoJ model imposes a binary weight of 0 or 1 on the individual HAR-RV and HAR-CJ models, while the DMSPE method produces a continuous weight between 0 and 1 for the two models based on their past forecasting performances. Several recent studies explicitly show that the DMSPE method can improve out-of-sample forecast accuracy (see, e.g., Rapach et al. 2010; Wang et al. 2019; Dai et al. 2021). Statistically, the DMSPE weight for individual forecast i on days $t + 1:t + h$ is given by

$$\omega_{i,t + 1:t + h} = {{\phi_{i,t + 1:t + h}^{ - 1} } \mathord{\left/ {\vphantom {{\phi_{i,t + 1:t + h}^{ - 1} } {\sum\limits_{{\ell \in \{ {\text{RV}},{\text{ CJ}}\} }} {\phi_{\ell ,t + 1:t + h}^{ - 1} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{{\ell \in \{ {\text{RV}},{\text{ CJ}}\} }} {\phi_{\ell ,t + 1:t + h}^{ - 1} } }},{\text{ for}}\quad \, i = {\text{ RV}},{\text{CJ}},$$

(41)

where

$$\phi_{i,t + 1:t + h} = \sum\nolimits_{s = R}^{t - h} {\delta^{t - h - s} \left( {RV_{s + 1:s + h} - \widehat{RV}_{s + 1:s + h}^{i} } \right)^{2} } ,$$

(42)

R denotes the length of the initial training period, and $\delta$ refers to the discount factor. We follow Rapach et al. (2010) and rely on two values of $\delta$, that is, 1 and 0.9. We then obtain two corresponding approaches, dubbed DMSPE(1) and DMSPE(0.9).

Second, the MoJ strategy is similar to shrinkage approaches, which push the coefficients of jump components toward 0 when the recent past performance of the HAR-CJ is worse than that of the HAR-RV model. Therefore, we compare the forecast accuracy of the MoJ with those of alternative shrinkage methods, including the ridge, lasso, and elastic net.⁸ These shrinkage methods have been demonstrated to perform well in forecasting asset price returns and volatilities (see, e.g., Li et al. 2015; Li and Tsiakas 2017; Zhang et al. 2019c).

The corresponding comparison results are shown in Table 13. Our MoJ strategy consistently produces the highest MCS p-values (i.e., 1) for all loss functions and forecast horizons. In contrast, the alternative competing models of DMSPE(1), DMSPE(0.9), ridge, lasso, and elastic net produce substantially lower MCS p values, most of which are smaller than 0.1, implying that the corresponding models cannot enter the MCS at the 10% significance level. We thus conclude that the MoJ strategy also outperforms the two DMSPE and three shrinkage methods in the out-of-sample forecasting test.

Table 13 A comparison of alternative forecasting models

Conclusion

The jump component is not informative for forecasting oil futures market volatility (Prokopczuk et al. 2016). To improve the efficiency of using jump information, we propose the MoJ strategy, which switches between the HAR-RV model without jumps and the HAR-CJ model incorporating jump information based on their relative past forecasting performances. The MoJ approach depends on the momentum of the jump model’s predictive ability. More precisely, the MoJ implies that a good past predictive performance of the jump model (i.e., the HAR-CJ model) typically delivers a good future predictive performance.

Empirically, the in-sample estimation results suggest that the jump component does not contain a powerful explanatory ability for future oil futures RV. This evidence implies that a straightforward approach of using the jump model alone is unlikely to be feasible. In addition, based on six prevailing loss functions, the MCS out-of-sample forecasting test provides convincing evidence that the MoJ strategy outperforms the HAR-RV, HAR-CJ, and the mean combination of the two HAR models. Furthermore, we document the existence of the MoJ in forecasting the oil futures market RV; that is, the stronger predictive power of the jump model is persistent. This lays the foundation for the success of the MoJ strategy.

The results of the superiority of our MoJ model are found to be robust to various forecast horizons (ranging from 1-day to 22-day horizons), alternative jump models, various look-back periods, alternative volatility estimators, the use of the HAR or MIDAS framework, the use of linear and nonlinear HAR models, different forecasting windows, and many other robustness perspectives. In addition, we extend the MoJ model to the prediction of stock market RV and obtain consistent forecasting results. Finally, in a portfolio exercise, we explore the economic significance of the RV forecasts of the MoJ strategy and the competing models. A mean–variance investor who targets a constant Sharpe ratio can realize sizeable utility gains relying on the MoJ-based RV forecasts to allocate her portfolio.

Our empirical findings have some useful implications for the participants in the crude oil market. For example, volatility forecasting is commonly used in the applications of asset allocation and risk management. While directly using jumps is likely to be useless when forecasting crude oil price volatility, the participants still need to indirectly consider jump information. Our MoJ approach is a successful example that can enhance the predictability of oil market volatility and thereby improve the performance of asset allocation and risk management. There are, of course, many other ways to address jump information. This is an interesting field for future research. Machine learning is probably a better choice for discovering more useful information in jumps.