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Time-series-based forecasting is essential to determine how past events affect future events. This paper compares the performance accuracy of different time-series models for oil prices. Three types of univariate models are discussed: the exponential smoothing (ES), Holt-Winters (HW) and autoregressive intergrade moving average (ARIMA) models. To determine the best model, six different strategies were applied as selection criteria to quantify these models’ prediction accuracies. This comparison should help policy makers and industry marketing strategists select the best forecasting method in oil market. The three models were compared by applying them to the time series of regular oil prices for West Texas Intermediate (WTI) crude. The comparison indicated that the HW model performed better than the ES model for a prediction with a confidence interval of 95%. However, the ARIMA (2, 1, 2) model yielded the best results, leading us to conclude that this sophisticated and robust model outperformed other simple yet flexible models in oil market.

Forecasting is the process of developing hypotheses about future events [

Reliable estimates are required to judge the accuracy of a forecast. Therefore, assessing the quality of a model in terms of historical data is unacceptable. In contrast, the precision of a forecast should be analysed in terms of how the model handles new data that were not previously used to determine the quality of the model [

Two types of time-series-based forecasting models exist: univariate and multivariate. Univariate forecasting involves using historical data to predict the value of a continuous variable that serves as the response or output variable [

The last two decades have seen significant developments in ES, which has become one of the most noteworthy forecasting strategies. ES was established as a classical method of analysis for forecasting different econometric and financial real-time data prospects [

The HW approach, which is a variant of ES, is a method commonly used to forecast recurring time series [

The ARIMA approach, also known as the Box?Jenkins method, has been one of the most widely applied linear frameworks in time-series forecasting over the past three decades [

Conversely, studies by [

This study focuses on the major tools of decision making; namely, time-series-based models. We present analyses of the accuracies of the ES, HW and ARIMA approaches in forecasting crude oil prices for the first time and also discuss the significances of the error terms in these forecasting models and the importance of achieving minimal errors. The remainder of the paper is organized as follows: Section 2 describes the data source, methodology and means of quantifying forecasting accuracy. The results from real data sets and a discussion are presented in Section 3. Finally, Section 4 gives our concluding remarks.

West Texas Intermediate (WTI) oil prices from October 2011 and March 2016 served as the central time series used in this study. This time series was chosen because the fluctuating nature of the data endows it with extreme nonlinearity, which means that chaos might pose challenges in forecasting future prices. The datasets consisted of daily data collected from the official websites of the United States Energy Information Administration (EIA). WTI was also chosen because of its geographical location and cost of transport; these are the major factors influencing the oil market at the local and international levels [

The in-sample series contributed approximately 90% of the data, whereas the out-of-sample series contributed approximately 10% (this division was arbitrary). The accuracies of the oil-price forecasts of the three models were determined and interrelated based on six metrics. The next section presents both the respective models both mathematically and theoretically, as well as the mathematical expressions and applications of the six metrics to quantify the forecast accuracies.

ES is a formalization of the familiar learning method, which is a practical basis for forecasting. Typically, the use of ES is significant in cases in which the data pattern is nearly horizontal, particularly when no particular trend or seasonal variation exists in previous data sets. The ES framework for time series

For ES, the h-step-ahead forecast equation for time series

where

The HW model is equally important in its own distinct ways. This model is an extension of the ES framework, but it employs a different set of parameters, unlike those used in rudimentary time series, to smooth the inclination of values. HW-based forecasting can be performed by using three smoothing elements. The HW model is applied to data characterized by seasonality and trend. Its equations are

and

In the HW model, an h-step-ahead forecast of

where

The ARIMA model involves a series transformation to a state of stationary covariance, followed by identification, approximation, diagnosis and prediction. The “I” in ARIMA implies that the dataset undergoes differentiation and that, upon completion of the modelling, the results undergo an integration process to produce final predictions and estimates [

AR model:

MA model:

and

ARMA model:

where

Although one model can occasionally be suitable for any set of data, the correctness of a forecasting model requires that various models be evaluated to identify the one that provides the best results with minimal errors [_{1} and U_{2}. These criteria exhibited particular advantages and disadvantages, as well as specific conditions of applicability.

The first assessment criterion, MSE, is the mean value of the squared error of the overall forecasts. MSE assigns more weight to significant errors. The major limitation of this approach is that it overstates substantial er-

Criteria | Formula | Criteria | Formula |
---|---|---|---|

MSE | RMSE | ||

MAE | MAPE | ||

U_{1} | U_{2} |

Note:

rors. This evaluation criterion also provides limited information regarding whether the framework is an overestimation or underestimation of the actual forecast value. RMSE is the second evaluation criterion. It preserves the units of the estimation variable. This approach is more sensitive and minimizes large errors. Nevertheless, the ability to compare different time series is limited with this criterion. Conversely, MAE, the third fourth criterion, determines the error magnitude for a precise set of forecasts. MAE defines how close forecasts are to the actual outcomes. This metric does not consider the direction of the forecasts. Moreover, these criteria determine the precisions of continuous variables. MAPE, the fourth criterion, enables comparison of distinct time-series data without defining the relation or percent error. This metric is significant in instances in which the measured variables are very large.

The fifth and sixth criteria are U_{1} and U_{2}, respectively. The former enables different predictions to be compared, which implies that actual values are compared with predicted values. U_{1} provides a range of values on a zero-to-one scale. The nearer U_{1} is to zero, the more accurate the prediction is. When faced with alternative predictions, the forecast with the smallest value of U_{1} is regarded as the best and is thus selected. Conversely, U_{2} performs relative comparisons based on random walk models and prediction models (naïve model). The naïve model may be described as the actual predetermined forecast model applied based on an indiscriminate-walk process. When U_{2} levels off at unity, the naïve method is considered to be equally useful for forecasting. U_{2} < 1 indicates that the forecasting model would work better than the naïve approach. Note that U_{1} and U_{2} are more difficult metrics to use than the MAPE metric.

We implemented the ES, HW and ARIMA models in MATLAB (R2014a), which is a powerful mathematical software package used internationally by engineers and scientists. MATLAB contains built-in functions that allow the user to determine model parameters spontaneously; the only requirement in this system is the time series to be analysed. The quality of the parameter set used in this approach is defined by whether it lead to minimal errors between the forecast and actual values. To determine the quality of the parameter set used in this work, we compared the forecast values obtained from the three above-mentioned models with the actual WTI crude oil prices from October 2015 and March 2016. The results of this exercise that were obtained using the ES, HW and ARIMA methods are shown in Figures 1-3, respectively. These three figures show the forecast trends with a confidence interval of 95% and the actual data. The results of the different models are consistent regardless of which of the six different metrics is used to quantify their performances.

The ES model uses a single parameter

Compared with the ES model forecasts, those of the HW model are relatively accurate. The performance of the HW model relies heavily on the choice of parameters, as shown in

The forecast presented in

and

The ARIMA model consists of four steps: The first step is model selection, wherein the parameters p and q are selected by using the prediction-accuracy method of minimizing the error upon recognition of a variable’s fixed degree. The second step involves estimating the model and its viability. Finally, the chosen and estimated frameworks are replicated forward to obtain the forecast values for the variables in question. We found that ARIMA (2, 1, 2) was the best-fitting model with respect to the WTI data.

Note, however, that ARIMA is a fairly sophisticated model for accurate forecasting.

The ARIMA (2, 1, 2) model equation can be written as

Analysis of Forecast-Model Selection

The preceding section outlined three time-series models that were used to generate forecasts for crude oil prices. Every prediction was compared with the actual value of the respective time series of crude oil prices. The accuracy of each forecast was evaluated by using six metrics, as discussed in the preceding section. Each approach was applied to determine and rank the performances of the models for the given time series.

Variables | Coefficient | p-value |
---|---|---|

Constant | −0.182 | 0.0503 |

AR (1) | −1.505 | <0.0001^{*} |

AR (2) | −0.605 | 0.0126^{*} |

MA (1) | −1.646 | <0.0001^{*} |

MA (2) | −0.723 | 0.0039^{*} |

Note: ^{*}p value < 0.05.

Measures of accuracy | ES | HW | ARIMA |
---|---|---|---|

MSE | 5.985 | 3.122 | 2.337^{*} |

RMSE | 2.446 | 1.767 | 1.529^{*} |

MAE | 1.795 | 1.260 | 1.528^{*} |

MAPE | 2.928 | 2.149 | 1.718^{*} |

U_{1} | 0.726 | 0.528 | 0.343^{*} |

U_{2} | 2.786 | 1.592 | 0.689^{*} |

Model ranking | 3 | 2 | 1 |

Note: ^{*}significance < 1.

summarizes the three models and their forecasting performances. The results demonstrate that the ARIMA model performs better than either of the other models for the given time series. Note, however, that although the HW model exhibits the second best forecast after that of the ARIMA model, the performance of each model relies on the data used. In addition, specific ES models perform inadequately and thus provide poor forecasts, in contrast to the HW and ARIMA models.

Despite all three models being black boxes, the differences between their performances are related to the differences between the methods of determining forecasts in the ES and HW models and in the ARIMA model. The forecasting method in the ES and HW models relies on a weighted average of the past observed values in which the weights decline exponentially, which basically implies that the data for more recent observations contribute significantly more than do previous data. The ARIMA model, however, has three parts: autoregression, integration and moving average, with the future value of a variable being a linear combination of the past values and the associated errors.

Forecasting plays a vital role in the entire process of advising policy makers. Perfect decision making is achieved when changes are viewed from two perspectives: current events and what is likely to occur in the future. However, uncertain a forecast might seem, policy makers are compelled to consider its validity in their decision making. Ideally, policy makers would base their decisions on accurate forecasts to tighten their policies and to achieve ultimate outcomes that differ from those forecasted. In addition, marketing strategists use information acquired from various forecasting models and consider the most accurate models for developing, formulating and implementing marketing strategies. Therefore, the organisations and entities whose main activities rely on the oil industry find such forecasts especially useful for formulating their policies and marketing strategies to adapt to future changes forecasted by the models. With accurate forecasts, entities affiliated with oil production can use the information to make prudent decisions regarding the prices they attach to oil. As a result, with accurate knowledge of the flow of public money and of the regular patterns in oil consumption and demand, they can modify oil prices in such a way as to avoid damaging the financial capability or otherwise affecting their organizations’ objectives.

This paper explored the natures of statistical predictors by presenting time series analyses for oil prices data. Three types of univariate time-series models were investigated: ES, HW and ARIMA. We determined the qualities of their forecasts by comparing the results of the given models with actual data. The ARIMA (2, 1, 2) model provided forecasts that were more accurate than those of the ES or HW models. The six selection criteria used to quantify the qualities of the forecasts yielded their smallest values for the ARIMA forecast, indicating clearly that it is the best of the three methods. As a result, we were able to not only conclude that the ARIMA model provided the most accurate forecasts but also discuss the sophistication and robustness of the competing models in crude oil market. Ultimately, the best decision-making process will be the most effective. Predicting future events based on an appropriate time-series model will help policy makers and marketing strategists make decisions and devise suitable strategic plans in oil industry. To extend this work, future research should examine and compare complex univariate models, such as the autoregressive conditional heteroscedasticity model (ARCH), the generalized ARCH model (GARCH) and the ARIMA/GARCH model.

Gurudeo Anand Tularam,Tareq Saeed,1 1, (2016) Oil-Price Forecasting Based on Various Univariate Time-Series Models. American Journal of Operations Research,06,226-235. doi: 10.4236/ajor.2016.63023