# Determining Time-Varying Drivers of Spot Oil Price in a Dynamic Model Averaging Framework

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Models

#### 2.2. Oil Price Drivers

## 3. Data

## 4. Methodology

#### 4.1. Model Specification

^{m}different regression models can be constructed, including the model with constant only. Let t denote the time index, i.e., let t = {0, 1, …}, and let y

_{t}denote the dependent variable. Let x

_{t}

^{(k)}denote independent variables (drivers) in the k-th model.

_{t}

^{(k)}denotes regression parameters of the k-th model. It is assumed that errors are normally distributed, i.e., ε

_{t}

^{(k)}~ N(0,V

_{t}

^{(k)}) and δ

_{t}

^{(k)}~ N(0,W

_{t}

^{(k)}). Starting at t = 0 the initial values have to be assigned to variance matrices V

_{0}

^{(k)}and W

_{0}

^{(k)}. Further, inference of V

_{t}

^{(k)}is estimated by a recursive method of moments estimator [5], and inference of W

_{t}

^{(k)}—by the forgetting procedure. In this paper, W

_{0}

^{(k)}has been estimated following the procedure basing on the whole data sample given by Raftery et al. [5]. It is also necessary to set the initial value for V

_{0}

^{(k)}. This is further discussed more thoroughly in Section 4.3.2. The regression coefficients are updated with the help of the Kalman filter.

_{k}(y

_{t}|Y

^{t}

^{−1}) is the predictive density of the k-th model at y

_{t}, given the data from previous periods, and α is a certain forgetting factor fixed from (0,1). π

_{t|t,k}are called posteriori inclusion probabilities and π

_{t|t}

_{−1,k}are called posteriori predictive probabilities. In order to guarantee non-zero outcomes (which might happen because of numerical approximations) some small constant is added in Equation (4). For example, c = 0.001/K.

#### 4.2. Assumptions and Limitations Involved in the DMA Method

_{t}

^{(k)}can be computed independently for each of K models. Except for in Equation (4) the forgetting factor is also used in the estimation of the state error covariance during the Kalman filtering. The details are in the paper by Raftery et al. [5].

_{t}

^{(k)}and W

_{t}

^{(k)}are time-varying. In the previous applications of DMA to economic and financial problems their Authors have been using stationary data [14,15,16,17,18], but it was rather due to the common habit in economic research, than any real necessity for the applied method.

^{m}. However, it is not desirable to consider an enormous number of potential drivers. Definitely, it should not be taken for granted that more drivers automatically lead to better forecast. Indeed, not only in DMA framework it might happen that quite a high number of variables can worsen the forecast [110]. This effect is theoretically understood when considering model structure estimation [111].

#### 4.3. Model Calibration

#### 4.3.1. Forgetting Factor

^{i}weight. So, α = 1 corresponds to no forgetting at all. In fact [5], if α = 1, then DMA reduces to Bayesian Model Averaging (BMA). If α = 0.99, then in the considered case of monthly data, data from previous quarter is given 97% weight in comparison to the current one. But if α = 0.95, it is given 86% weight. The forgetting factor α should be specified by the researcher. According to the remarks from the Literature review, a few values have been tested, i.e., α = {1, 0.99, 0.95}.

#### 4.3.2. Variance Matrix

_{0}

^{(k)}. This should be done in correspondence to the allowed variability of the used time-series. Therefore, for models with normalized time-series it has been set to the unit matrix. For models with non-normalized ones—to the unit matrix multiplied by 100

^{2}. In other words, for non-normalized models it has been rescaled by an arbitrary big number, as it has been done in previous researches, for example, by Baur et al. [16].

_{t}be scaled. Then, it is transformed into:

_{0}

^{(k}

^{)}, i.e., a few unit matrices multiplied by a few numbers ranging from 0.25 to 4. However, the coefficient of variation for the obtained sample of various MSE was slightly above 1%. Therefore, V

_{0}

^{(k)}= I (i.e., the unit matrix) was used in further estimations. Generally, some pre-simulations for the given data have indicated that higher values on the diagonal of V

_{0}

^{(k)}result in lower MSE in the logarithmic pattern. Indeed, data normalization is highly preferable for DMA, because if time-series are rescaled to fit between 0 and 1, then setting V

_{0}

^{(k)}= I corresponds to reasonably high volatility [112,116,117].

_{t}are residuals from modelling the independent variable Y

_{t}, and e

_{t}are residuals from modelling y

_{t}, where Y

_{t}= a·y

_{t}+ b, with a and b being some scaling parameters (which corresponds to normalization), then E

_{t}= a·e

_{t}. So, if Y

_{t}is obtained by normalization of y

_{t}, then residuals can be rescaled from the model including non-normalized time-series (by dividing them by $\underset{i=0,\dots ,t,\dots}{\mathrm{max}}{Y}_{i}-\underset{i=0,\dots ,t,\dots}{\mathrm{min}}{Y}_{i}$) and compared with residuals from the model with normalized time-series. This observation can help to compare, whether time-series normalization really improves the quality of prediction from the DMA model.

#### 4.4. Time-Varying Parameters Preselection

^{st}lag of WTI to the drivers present in the initial Model 1.

^{10}= 1024. Consequently computations would take rather days than minutes.

_{t|t}

_{−1,k}and π

_{t|t,k}with 1/K. In W

_{0}

^{(k)}approximation the forgetting factor 0.99 was used.

#### 4.5. Economic Interpretation

_{t|t,k}can be used for a nice economic interpretation. Indeed, for every period t, we can sum up the a posteriori inclusion probabilities of every model which contain a given driver. In other words, let us compute:

_{t}(X) is the probability that a driver X is useful for forecasting oil price at time t based on weights attached by DMA to models which include this driver. Therefore, this number can be interpreted as the importance of driver X in predicting oil price. As p

_{t}(X) naturally changes with t, it is interesting to observe its time variation.

_{t}(X) includes also joint statistical significance. In other words, a high posteriori inclusion probability π

_{t|t,i}can be indicated by the i-th model containing the given driver and some other driver(s), while a marginally small posteriori inclusion probability π

_{t|t,j}might be indicated by the j-th model containing only the given driver, and the high value of π

_{t|t,i}can be the result of including the other driver(s).

## 5. Results

_{t}(X) described in Section 4.5. In Figure 4 they are presented for the above chosen, the “best” model, i.e., Model 5 with normalized data, in the “reduced” version, and with the forgetting factor α = 0.99. These probabilities express the probability that a driver X is useful for forecasting oil price at time t based on weights attached by DMA to regression models which include this driver.

_{t}(X) start from the same value of 0.5, i.e., p

_{0}(X) = 0.5 for every X. This is just a direct consequence of Equation (3). Afterwards, DMA “learns” from the upcoming new data. Therefore, it is crucial to use DMA for sufficiently long time-series. Of course, this requirement is met in the analysis presented herein. The period of 30 years is covered, with 360 observations. Approximately first 20% of observations play a “learning” role for models. This can be seen in Figure 4 as p

_{t}(X) adapt quickly their values. However, they are not the exact values of p

_{t}(X) which are important to interpret, but their time-paths. In other words, from the economical point of view it is interesting to observe how the probability that a given driver is important in forecasting oil price varies in time.

^{st}and the 2nd lag of VXO marginal values shortly before 2007. Later, the role of this driver suddenly increased. Its role was increasing until around 2012. Recently, its role as an important oil price driver has been decreasing.

_{t}is greater than 0.5. It can be seen that during 1990s the main drivers of oil price were: developed stock markets, Chinese economy and autoregressive components. Later, in the 2000s the importance of these drivers decreased. Especially, the market stress index become less important as an oil price driver. During the oil price surge Chinese economy was an important driver. Later, its role decreased, but recently it has been increasing again. Recently, the role of futures prices has also been increasing. They played an important role in 1990s, but later (in 2000s) their role decreased.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Data Sources

**BRENT**

**WTI, PROD, IMP, INV, CONS and NFP:**

**MSCI:**

**TB3MS and TWEXM:**

**KEI:**

**VXO:**

**CHI (Hang Seng and Shaghai Composite):**

**Glossary**

- BMA—Bayesian Model Averaging
- DMA—Dynamic Model Averaging
- forgetting factor—described in Section 4.1 and 4.3.1
- “full” model—described in Section 4.4
- futures forecast—a forecast is equal to the current price of 1-month futures price
- MSE—mean squared error, i.e., the average of the squares of differences between the real values of a time-series and the forecasted values of this time-series
- naïve forecast—a forecast is equal to the last observed value
- normalization—defined by Equation (7)
- posterior probability—conditional probability assigned after the relevant evidence is taken into account
- posteriori inclusion probability—defined by Equation (5)
- posteriori predictive probability—defined by Equation (4)
- prior probability—probability expressing the belief about it, before some evidence is taken into account
- “reduced” model—described in Section 4.4
- swing producer—supplier of a commodity, controlling its global deposits, able to change the level of supply at minimal cost, and, therefore, able to influence the price and balance the market

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**Figure 4.**Posteriori inclusion probabilities p

_{t}for drivers included in Model 5 (normalized, “reduced”, α = 0.99). Thick line corresponds to model with WTI, dotted—to model with BRENT.

Name | Description | Economic Factor Measured by the Driver (and Units) |
---|---|---|

dependent variable in regression models | ||

WTI | WTI spot price | crude oil spot price (in USD) |

independent variables in regression models (drivers) | ||

MSCI | MSCI World Index | stocks prices (index) |

TB3MS | U.S. 3-month treasury bill secondary market rate | interest rate (in percentages) |

KEI | Kilian’s index of global economy activity [101] | global economic activity (index) |

TWEXM | trade weighted U.S. dollar index | exchange rate (Mar, 1973 = 100) |

PROD | U.S. crude oil production | oil supply (in thousand barrels) |

IMP | daily average of U.S. crude oil import | oil demand (in thousand barrels per day) |

INV | U.S. total ending stocks of commercial crude oil (excluding SPR) | speculative pressures (in thousand barrels) |

VXO | implied volatility of S&P 100 | volatility of stocks market (index) |

CONS | total consumption of petroleum products in OECD | oil demand (in quad BTU) |

CHI | Shanghai Composite Index merged with Hang Seng Index as a representative of Chinese economy | Chinese economy (rescaled index) |

other time-series | ||

NFP | 1-month NYMEX WTI futures prices | alternative forecast of crude oil price (in USD) |

Model | x_{t}^{(k)} (Drivers Considered in the Model) |
---|---|

Model 1 | 1st lag of MSCI, |

1st lag of TB3MS, 1st lag of KEI, 1st lag of TWEXM, 1st lag of PROD, 1st lag of IMP,1st lag of INV, | |

1st lag of VXO, 1st lag of CONS, 1st lag of CHI | |

Model 2 | 1st lag of WTI, |

1st lag of MSCI, | |

1st lag of TB3MS, 1st lag of KEI, 1st lag of TWEXM, 1st lag of PROD, 1st lag of IMP, 1st lag of INV, | |

1st lag of VXO, 1st lag of CONS, 1st lag of CHI | |

Model 3 | 1st lag of WTI, |

1st lag of MSCI, | |

1st lag of TB3MS, 1st lag of KEI, 1st lag of TWEXM, 1st lag of PROD, 1st lag of IMP, 1st lag of INV, | |

1st lag of VXO, 1st lag of CONS, 1st lag of CHI, | |

1st lag of NFP | |

Model 4 | 1st lag of WTI, 2nd lag of WTI, |

1st lag of MSCI, 2nd lag of MSCI, | |

1st lag of TB3MS, 1st lag of KEI, 1st lag of TWEXM, 1st lag of PROD, 1st lag of IMP, 1st lag of INV, | |

1st lag of VXO, 1st lag of CONS, 1st lag of CHI, | |

Model 5 | 1st lag of WTI, 2nd lag of WTI, 1st lag of MSCI, 2nd lag of MSCI, 1st lag of VXO, 2nd lag of VXO, |

1st lag of CHI, 2nd lag of CHI |

Presence of the Driver in the Model with Normalized Data is Indicated by “x”, and in the Model with Non-Normalized Data by “o”. | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |||||||||||

α | 1 | 0.99 | 0.95 | 1 | 0.99 | 0.95 | 1 | 0.99 | 0.95 | 1 | 0.99 | 0.95 | 1 | 0.99 | 0.95 |

1st lag of WTI | x o | x o | x o | x o | x o | x o | x o | x o | x o | x o | x o | x o | |||

2nd lag of WTI | x o | x o | x o | x o | x o | x o | |||||||||

1st lag of MSCI | x o | x o | o | o | x o | x o | o | x o | |||||||

2nd lag of MSCI | x o | o | x o | ||||||||||||

1st lag of TB3MS | x | o | x o | x o | x | x o | x | x o | x o | x | x o | ||||

1st lag of KEI | x | x o | x o | x o | x | x o | |||||||||

1st lag of TWEXM | x o | x o | x o | x | x o | x | x | x o | x o | x | x o | ||||

1st lag of PROD | o | o | x | x o | x | x o | x | x | x o | ||||||

1st lag of IMP | x o | o | x | o | x | ||||||||||

1st lag of INV | x o | o | x | o | x | ||||||||||

1st lag of VXO | x | x o | o | x | o | o | o | o | x o | ||||||

2nd lag of VXO | x | x | x | ||||||||||||

1st lag of CONS | o | x | x o | x o | x | x o | x | x o | |||||||

1st lag of CHI | x | x | o | x | x o | ||||||||||

2nd lag of CHI | o | x o | x o | ||||||||||||

1st lag of NFP | x o | o | x o |

Models | α | |||
---|---|---|---|---|

1 | 0.99 | 0.95 | ||

Model 1 (normalized) | reduced | 0.03794 | 0.01949 | 0.00365 |

full | 0.02169 | 0.00817 | 0.00298 | |

Model 1 (non-normalized) | reduced | 0.02176 | 0.01795 | 0.00365 |

full | 0.02235 | 0.00654 | 0.00283 | |

Model 2 (normalized) | reduced | 0.00147 | 0.00145 | 0.00156 |

full | 0.00131 | 0.00135 | 0.00141 | |

Model 2 (non-normalized) | reduced | 0.00215 | 0.00187 | 0.00144 |

full | 0.00159 | 0.00132 | 0.00150 | |

Model 3 (normalized) | reduced | 0.00115 | 0.00138 | 0.00117 |

full | 0.00115 | 0.00113 | 0.00126 | |

Model 3 (non-normalized) | reduced | 0.00120 | 0.00120 | 0.00142 |

full | 0.00115 | 0.00118 | 0.00147 | |

Model 4 (normalized) | reduced | 0.00114 | 0.00112 | 0.00117 |

full | 0.00117 | 0.00112 | 0.00127 | |

Model 4 (non-normalized) | reduced | 0.00119 | 0.00120 | 0.00136 |

full | 0.00135 | 0.00136 | 0.00130 | |

Model 5 (normalized) | reduced | 0.00117 | 0.00113 | 0.00113 |

full | 0.00115 | 0.00113 | 0.00117 | |

Model 5 (non-normalized) | reduced | 0.00122 | 0.00122 | 0.00124 |

full | 0.00121 | 0.00120 | 0.00125 | |

Benchmarks | ||||

1-month futures | 0.00122 | |||

naïve (i.e., the last period’s actuals are used as this period’s forecast) | 0.00124 | |||

Equal-Weighted Averaging | 0.02694 | |||

MSE smaller than those of benchmark forecasts are bolded. |

H_{alt}: Forecast from the Chosen Model is More Accurate than the Forecast from the … | Stat. | p-val. |
---|---|---|

BMA | 0.8676 | 0.1931 |

Equal-Weighted Averaging | 9.5349 | 0.0000 |

1-month futures | 0.7614 | 0.2235 |

naive | 14.6047 | 0.0000 |

Model 1 (normalized) | 8.7911 | 0.0000 |

Model 2 (normalized) | 2.0940 | 0.0185 |

Model 3 (normalized) | 1.8576 | 0.0320 |

Model 4 (normalized) | −0.1078 | 0.5429 |

Strategy | Mean | SD | Sharpe Ratio |
---|---|---|---|

DMA | 0.0069 | 0.0614 | 0.1131 |

“Hold oil” | 0.0066 | 0.0848 | 0.0778 |

Based on futures | −0.0010 | 0.0688 | −0.0141 |

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## Share and Cite

**MDPI and ACS Style**

Drachal, K. Determining Time-Varying Drivers of Spot Oil Price in a Dynamic Model Averaging Framework. *Energies* **2018**, *11*, 1207.
https://doi.org/10.3390/en11051207

**AMA Style**

Drachal K. Determining Time-Varying Drivers of Spot Oil Price in a Dynamic Model Averaging Framework. *Energies*. 2018; 11(5):1207.
https://doi.org/10.3390/en11051207

**Chicago/Turabian Style**

Drachal, Krzysztof. 2018. "Determining Time-Varying Drivers of Spot Oil Price in a Dynamic Model Averaging Framework" *Energies* 11, no. 5: 1207.
https://doi.org/10.3390/en11051207