Forecasting crude oil price intervals and return volatility via autoregressive conditional interval models

Abstract

Crude oil prices are of vital importance for market participants and governments to make energy policies and decisions. In this paper, we apply a newly proposed autoregressive conditional interval (ACI) model to forecast crude oil prices. Compared with the existing point-based forecasting models, the interval-based ACI model can capture the dynamics of oil prices in both level and range of variation in a unified framework. Rich information contained in interval-valued observations can be simultaneously utilized, thus enhancing parameter estimation efficiency and model forecasting accuracy. In forecasting the monthly West Texas Intermediate (WTI) crude oil prices, we document that the ACI models outperform the popular point-based time series models. In particular, ACI models deliver better forecasts than univariate ARMA models and the vector error correction model (VECM). The gain of ACI models is found in out-of-sample monthly price interval forecasts as well as forecasts for point-valued highs, lows, and ranges. Compared with GARCH and conditional autoregressive range (CARR) models, ACI models are also superior in volatility (conditional variance) forecasts of oil prices. A trading strategy that makes use of the monthly high and low forecasts is further developed. This trading strategy generally yields more profitable trading returns under the ACI models than the point-based VECM.

Keywords:

• ACI model
• interval-valued crude oil prices
• range
• trading strategy
• volatility forecast

Notes

1 We use futures prices rather than spot prices. There are mainly two reasons. First, the futures market is a forum to disseminate crude oil information. It delivers market price signals that are essential for risk monitoring. Second, the WTI futures on NYMEX is one of the most actively traded contracts all over the world.

2 Monthly average futures price is the average daily futures prices over the trading days within each month.

3 The results are not reported here for space but are available upon request.

4 To determine the lag orders in an ACI model for oil prices, we can refer to the implied point-valued equation (3.8)(3.8) $$\{\begin{array}{c}\Delta M_t={\alpha }_0+{\gamma }_{1}E{C}_{t-1}-{\beta }_1\Delta M_{t-1}+{\beta }_2\Delta M_{t-2}+\sum _{j=1}^2{\theta }_j{u}_{M,t-j}+{\gamma }_2{M}_{\Delta \mathbf{SPE},t-2}+u_{M,t},\hfill \\ \Delta R_t={\beta }_0+{\gamma }_{1}E{C}_{t-1}+{\beta }_1\Delta R_{t-1}+{\beta }_2\Delta R_{t-2}+\sum _{j=1}^2{\theta }_j{u}_{R,t-j}+{\gamma }_2{R}_{\Delta \mathbf{SPE},t-2}+u_{R,t},\hfill \end{array}$$(3.8) . An ARMA(2,2) is adequate for both midpoint and range. Similarly, second-lagged values of the differenced speculation variable are significant to predict the crude oil midpoint.

5 $\mathbf{E}{\mathbf{C}}_{t-1}=[\frac{1}{2}E{C}_{t-1},\frac{3}{2}E{C}_{t-1}]$ ensures that $E{C}_{t-1}$ has the same coefficient in the range equation (3.4)(3.4) $$\Delta R_t={\beta }_0+{\gamma }_{1}E{C}_{t-1}+\sum _{j=1}^2{\beta }_j\Delta R_{t-j}+\sum _{j=1}^2{\theta }_j{u}_{R,t-j}+{\gamma }_2{R}_{\Delta \mathbf{SPE},t-2}+u_{R,t},$$(3.4) and the midpoint equation (3.5)(3.5) $$\Delta M_t={\alpha }_0+{\gamma }_{1}E{C}_{t-1}+\sum _{j=1}^2{\beta }_j\Delta M_{t-j}+\sum _{j=1}^2{\theta }_j{u}_{M,t-j}+{\gamma }_2{M}_{\Delta \mathbf{SPE},t-2}+u_{M,t},$$(3.5) .

6 The modified ACI model can be extended to a more general ACI(p, q) model of order (p, q): $$\Delta {\mathbf{F}}_t={\alpha }_0+{\beta }_0{I}_0+{\gamma }_1\mathbf{E}{\mathbf{C}}_{t-1}+\sum _{j=1}^p{\beta }_j\Delta {\mathbf{F}}_{t-j}+\sum _{j=1}^p{\beta }_j^{*}\Delta {\mathbf{F}}_{t-j}^{*}+\sum _{j=1}^q{\gamma }_j{\mathbf{u}}_{t-j}+\sum _{j=1}^q{\gamma }_j^{*}{\mathbf{u}}_{t-j}^{*}+{\mathbf{u}}_t,$$ where ${\mathbf{u}}_{t-j}^{*}=[-u_{H,t-j},-u_{L,t-j}].$ Among other things, this can be used to address the important empirical stylized facts in interval data, such as the fact that the midpoints at times t – j and t are negatively correlated while the ranges are positively correlated.

7 Suppose a point particle undergoes a one-dimensional continuous random walk with a diffusion constant D. Parkinson (1980) mentioned that the difference l between the maximum and minimum positions is a good estimator for the diffusion constant.

8 In the futures market, there is a practice of daily settlement or marking to market. Investors may take a risk arising from large margin calls. Therefore, we also compute daily gains or losses of long positions before every sell action is triggered during the evaluation period. It is found that the trading obtained from an ACI model usually has a smaller margin account risk than that from the point-valued time series model for m = 2, 3, and 4, respectively. This indicates again that the use of interval information can improve profitability of crude oil modeling. The relevant empirical results are available from the authors upon request.

9 When the trading rule for m = 2 is considered, the VEC model with and without speculation deliver different trading results during the period from 2004 to 2008. As it is well known, crude oil prices had increased markedly in those years. In May 2008, WTI crude oil futures prices reached as high as U.S. $125.46 per barrel. As a result, a number of slightly different buy-sell actions can lead to a great difference in the trading profits. This explains why the two types of VEC models differ greatly in the average returns for m = 2.

Additional information

Funding

We thank three anonymous referees and the editor Esfandiar Maasoumi for their comments and suggestions. All remaining errors are solely ours. Han thanks support from China NNSF Grant (Nos. 71403231, 71671183), Hong and Wang thank support from China NNSF Grant (No. 71988101) entitled as Econometric Modeling and Economic Policy Studies, and Sun thanks support from China NNSF Grant (Nos. 71703156, 72073126, 72091212).