# Forecasting the Crude Oil Spot Price with Bayesian Symbolic Regression

## Abstract

**:**

## 1. Introduction

## 2. Data Used in the Study

#### 2.1. Response Variables

#### 2.2. Explanatory Variables

#### 2.2.1. Supply and Demand Factors

#### 2.2.2. Financial Markets Indicators

#### 2.2.3. Macroeconomic Indicators

#### 2.2.4. Exchange Rates

#### 2.2.5. Other Commodities Prices

#### 2.2.6. Derivatives Markets

#### 2.3. Data Transformations

## 3. Methodology

#### 3.1. Bayesian Symbolic Regression

_{t}be the forecasted time series, i.e., crude oil price (taken in logarithmic differences). Let x

_{1,t}, …, x

_{n,t}be the explanatory variables time series (also suitably transformed, as explained earlier in the text). Then, it is assumed that y

_{t}= β

_{0}+ β

_{1}* f

_{1}(x

_{1,1,t−1}, …, x

_{1,i,t−1}) + … + β

_{k}* f

_{k}(x

_{k,1,t−1}, …, x

_{k,i,t−1}), with x

_{i,j,t}denoting some of the explanatory variables out of n available ones which are present in the i-th component expression, i.e., f

_{i}, with j = {1, …, n} and i = {1, …, k}. The number of components, k, is fixed and set up at the initial stage, and coefficients β

_{i}are estimated by the ordinary least squares linear regression method. According to [32], higher values of k result in higher forecast accuracy. However, this increase diminishes as larger values of k are taken.

_{i}is built with the help of a symbolic tree consisting of operators such as +, *, 1/x, etc. [123] and [124] observed that the operator lt(x) = a * x + b, with a and b being some real numbers, can noticeably improve the set of potential expressions available to be constructed. The set of operators needs to be specified in the initial stage of symbolic regression (both in the case of BSR and if a standard genetic programming approach is used).

_{1}+ … + f

_{k}and T—the set of nodes, M—the nodes’ features, and ϴ—the parameters. Initially, uniform priors are taken as they represent equal probabilities for selecting possible operators and the nodes’ features. In particular, a node feature represents whether the given node is a terminal one, or extends to one child node, or splits into two child nodes. The probability that a given node is a terminal one is 1—α(1 + d)

^{−β}with α and β being some parameters and d being the depth of the node [32]. Herein, following [32] α = 0.4 and β = −1 were taken. High values of β restrict the depth of the trees and α controls the symmetric shape of the distribution. The priors for a and b for operators lt were Gaussian and centred around the identity function [32].

#### 3.2. Model Averaging Schemes

_{1}, …, y

_{50}be the forecasts obtained from 50 iterations. Then, let w

_{1}, …, w

_{50}be some weights (such that w

_{1}+ … + w

_{50}= 1) ascribed to each of these forecasts (or equivalently: models). The weighted average forecast is then understood as w

_{1}* y

_{1}+ … + w

_{50}* y

_{50}.

_{1}* ϴ

_{1}+ … + w

_{50}* ϴ

_{50}. For example, this is true for dynamic model averaging (DMA) mentioned later in this text. However, analysis of such averages should be understood with caution [133,134,135,136].

#### 3.3. Recursive and Fixed Estimations

#### 3.4. Benchmark Models

#### 3.4.1. Bayesian Model Combination Schemes

#### 3.4.2. LASSO and RIDGE Regressions

#### 3.4.3. Least-Angle Regression

#### 3.4.4. Some Common Models

#### 3.5. Software

#### 3.6. Forecast Evaluation

## 4. Results

## 5. Conclusions

^{2}, are computationally equivalent; however, they are constructed from different operators [161]. How to store and reduce them and avoid computational cost is a subtle task. Moreover, even different functions can be “approximately” very similar for the given data set problem. Similarly, the estimation of regression coefficients and the numerical accuracy of them pose some issues. There are also still some issues with the computational costs of BSR. Although still faster than genetic programming-based symbolic regression, Markov chain Monte Carlo (MCMC) sampling still generated some significant computational time.

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Results of the Giacomini–Rossi fluctuation test for various parameter μ and loss functions.

Variable Description | Variable Abbreviation | Transformation |
---|---|---|

Cushing, OK WTI Spot Price FOB (Dollars per Barrel) | WTI | 2 |

Europe Brent Spot Price FOB (Dollars per Barrel) | Brent | 2 |

Crude oil, Dubai ($/bbl) | Dubai | 2 |

World production of crude oil including lease condensate (Mb/d) | Prod_glob | 3 |

U.S. production of crude oil including lease condensate (Mb/d) | Prod_US | 3 |

OECD refined petroleum products consumption (Mb/d) | Cons_OECD | 3 |

U.S. Product Supplied of Crude Oil and Petroleum Products (Thousand Barrels) | Cons_US | 3 |

U.S. Ending Stocks excluding SPR of Crude Oil and Petroleum Products (Thousand Barrels) | Stocks | 3 |

U.S. Crude Oil Rotary Rigs in Operation (Count) | Rigs_US | 1 |

Baker Hughes International Rig Count | Rigs_glob | 1 |

S&P 500 | SP | 2 |

MSCI WORLD (DEVELOPED MARKETS, Standard, Large+Mid Cap, USD) | MSCI_World | 2 |

MSCI EM | MSCI_EM | 2 |

Chinese Stock Markets | CHI | 2 |

VXO (Month-end Closing Values) | VXO | 0 |

GPR | GPR | 0 |

U.S 3-Month Treasury Bill: Secondary Market Rate, Percent, Monthly, Not Seasonally Adjusted (TB3MS) | R_short | 0 |

Long-Term Government Bond Yields: 10-year: Main (Including Benchmark) for the United States, Percent, Monthly, Not Seasonally Adjusted (IRLTLT01USM156N) | R_long | 0 |

Term Spread | TS | 0 |

Default Return Spread | Drs | 0 |

Dividend-To-Price Ratio | Dpr | 0 |

Kilian Index | Ec_act | 0 |

U.S. Industrial Production Index, Index 2012 = 100, Monthly, Seasonally Adjusted (INDPRO) | IP | 3 |

U.S. Unemployment Rate, Percent, Monthly, Seasonally Adjusted (UNRATE) | UNE | 0 |

U.S. Real M1 Money Stock, Billions of 1982-84 Dollars, Monthly, Seasonally Adjusted | M1 | 2 |

Consumer Price Index for All Urban Consumers: All Items in U.S. City Average, Index 1982–1984 = 100, Monthly, Seasonally Adjusted (CPIAUCSL) | CPI | 2 |

Real Narrow Effective Exchange Rate for United States, Index 2010 = 100, Monthly, Not Seasonally Adjusted (RNUSBIS) | FX | 2 |

S&P GSCI Commodity Total Return Index | GSCI | 2 |

Gold (ozt/U.S. Dollar, XAUUSD) | Gold | 2 |

Natural gas, US (Henry Hub spot price, Louisiana, $/mmbtu) | Gas | 2 |

Coal, Australian ($/mt) | Coal | 2 |

Cushing, OK Crude Oil Future Contract 1 (Dollars per Barrel) | Fut | 2 |

Dollar Open Interest | OI | 2 |

Working’s T-index | T | 0 |

_{t}→ Y

_{t}; 1–12-month difference: Y

_{t}→ Y

_{t}–Y

_{t−12}; 2–logarithmic difference: Y

_{t}→ log(Y

_{t})–log(Y

_{t−1}); 3–12-month logarithmic difference: Y

_{t}→ log(Y

_{t})–log(Y

_{t−12}); t denoted the time index.

Variable | Mean | Standard Deviation | Median | Min | Max | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

WTI | 46.71 | 28.70 | 39.34 | 11.31 | 133.90 | 0.77 | −0.49 |

Brent | 48.17 | 32.26 | 39.36 | 9.80 | 133.90 | 0.81 | −0.50 |

Dubai | 45.90 | 31.67 | 34.82 | 10.05 | 131.20 | 0.80 | −0.54 |

Prod_glob | 70,870.00 | 7335.00 | 72,640.00 | 56,970.00 | 84,610.00 | −0.04 | −1.14 |

Prod_US | 7069.00 | 1941.00 | 6482.00 | 3974.00 | 12,860.00 | 1.22 | 0.74 |

Cons_OECD | 46,550.00 | 2805.00 | 47,040.00 | 34,970.00 | 52,710.00 | -0.67 | 0.56 |

Cons_US | 579,400.00 | 42,390.00 | 583,000.00 | 440,700.00 | 671,600.00 | −0.33 | −0.45 |

Stocks | 1,079,000.00 | 123,800.00 | 1,050,000.00 | 861,200.00 | 1,453,000.00 | 1.10 | 0.46 |

Rigs_US | 494.40 | 370.20 | 358.50 | 108.00 | 1596.00 | 1.46 | 1.27 |

Rigs_glob | 2293.00 | 679.00 | 2113.00 | 1016.00 | 3900.00 | 0.59 | −0.68 |

SP | 1329.00 | 800.70 | 1205.00 | 288.90 | 4181.00 | 1.02 | 0.71 |

MSCI_World | 1209.00 | 532.90 | 1169.00 | 423.10 | 2939.00 | 0.60 | −0.18 |

MSCI_EM | 667.60 | 331.20 | 551.50 | 132.70 | 1348.00 | 0.18 | −1.43 |

CHI | 44,910.00 | 26,810.00 | 41,470.00 | 2274.00 | 141,100.00 | 0.40 | -0.12 |

VXO | 19.91 | 8.31 | 18.00 | 7.87 | 61.38 | 1.64 | 3.92 |

GPR | 87.89 | 66.29 | 65.52 | 23.74 | 545.30 | 2.82 | 12.00 |

R_short | 2.77 | 2.39 | 2.36 | 0.01 | 8.82 | 0.44 | −0.92 |

R_long | 4.49 | 2.11 | 4.35 | 0.62 | 9.36 | 0.27 | −0.87 |

TS | 1.72 | 1.11 | 1.66 | −0.53 | 3.76 | 0.03 | −1.09 |

Drs | 3.10 | 1.34 | 3.22 | 0.36 | 5.93 | −0.04 | −1.19 |

Dpr | −1.45 | 0.28 | −1.50 | −2.03 | −0.76 | 0.36 | −0.49 |

Ec_act | 3.41 | 60.12 | -5.03 | −159.80 | 190.60 | 0.77 | 0.86 |

IP | 88.06 | 13.64 | 92.52 | 60.59 | 104.20 | −0.83 | −0.74 |

UNE | 5.90 | 1.71 | 5.50 | 3.50 | 14.80 | 1.28 | 2.25 |

M1 | 1041.00 | 1057.00 | 715.30 | 614.10 | 7100.00 | 4.82 | 23.16 |

CPI | 194.10 | 40.75 | 192.80 | 121.20 | 266.80 | −0.02 | −1.28 |

FX | 108.00 | 10.02 | 104.90 | 92.94 | 133.30 | 0.51 | −1.03 |

GSCI | 3601.00 | 1627.00 | 3114.00 | 1350.00 | 10560.00 | 1.17 | 1.22 |

Gold | 773.50 | 502.00 | 431.80 | 255.80 | 1974.00 | 0.63 | −1.10 |

Gas | 3.60 | 2.15 | 2.87 | 1.19 | 13.52 | 1.72 | 3.53 |

Coal | 57.91 | 30.17 | 49.44 | 22.25 | 180.00 | 1.02 | 0.46 |

Fut | 46.74 | 28.70 | 39.41 | 11.31 | 134.00 | 0.76 | −0.49 |

OI | 89,360,000,000.00 | 92,030,000,000.00 | 38,860,000,000.00 | 3,703,000,000.00 | 29,680,000,0000.00 | 0.64 | −1.12 |

T | 1.08 | 0.04 | 1.08 | 1.01 | 1.24 | 0.61 | 0.11 |

Variable | ADF Stat. | ADF p-Val. | PP Stat. | PP p-Val. | KPSS Stat. | KPSS p-Val. |
---|---|---|---|---|---|---|

WTI | −7.8014 | 0.0100 | −238.5223 | 0.0100 | 0.0481 | 0.1000 |

Brent | −7.7438 | 0.0100 | −244.8415 | 0.0100 | 0.0527 | 0.1000 |

Dubai | −7.9779 | 0.0100 | −222.0210 | 0.0100 | 0.0555 | 0.1000 |

Prod_glob | −2.8855 | 0.2033 | −44.2822 | 0.0100 | 0.5194 | 0.0373 |

Prod_US | −3.0288 | 0.1427 | −38.8618 | 0.0100 | 1.7642 | 0.0100 |

Cons_OECD | −3.8034 | 0.0191 | −154.1004 | 0.0100 | 1.3844 | 0.0100 |

Cons_US | −3.9644 | 0.0110 | −175.7799 | 0.0100 | 0.5408 | 0.0325 |

Stocks | −5.4851 | 0.0100 | −34.7983 | 0.0100 | 0.4158 | 0.0704 |

Rigs_US | −4.8830 | 0.0100 | −20.7352 | 0.0601 | 0.2109 | 0.1000 |

Rigs_glob | −5.8389 | 0.0100 | −28.1249 | 0.0120 | 0.3116 | 0.1000 |

SP | −6.2031 | 0.0100 | −378.5306 | 0.0100 | 0.1225 | 0.1000 |

MSCI_World | −6.4447 | 0.0100 | −362.0396 | 0.0100 | 0.0506 | 0.1000 |

MSCI_EM | −7.0688 | 0.0100 | −330.9499 | 0.0100 | 0.0925 | 0.1000 |

CHI | −6.3836 | 0.0100 | −400.7951 | 0.0100 | 0.1895 | 0.1000 |

VXO | −3.3366 | 0.0649 | −60.1466 | 0.0100 | 0.2561 | 0.1000 |

GPR | −4.2005 | 0.0100 | −101.6276 | 0.0100 | 1.3386 | 0.0100 |

R_short | −3.6701 | 0.0264 | −8.5238 | 0.6332 | 4.4101 | 0.0100 |

R_long | −3.8708 | 0.0157 | −31.7929 | 0.0100 | 6.0267 | 0.0100 |

TS | −3.5345 | 0.0394 | −13.2871 | 0.3667 | 0.2375 | 0.1000 |

Drs | −3.4891 | 0.0438 | −9.8452 | 0.5593 | 0.3312 | 0.1000 |

Dpr | −1.9913 | 0.5808 | −7.0833 | 0.7138 | 1.4468 | 0.0100 |

Ec_act | −2.4409 | 0.3910 | −22.1019 | 0.0446 | 0.7303 | 0.0108 |

IP | −5.1379 | 0.0100 | −27.5228 | 0.0143 | 0.8551 | 0.0100 |

UNE | −2.5066 | 0.3633 | −20.6563 | 0.0612 | 0.4055 | 0.0748 |

M1 | −6.7871 | 0.0100 | −355.2253 | 0.0100 | 0.5388 | 0.0329 |

CPI | −7.1785 | 0.0100 | −201.2820 | 0.0100 | 0.9009 | 0.0100 |

FX | −7.5795 | 0.0100 | −229.7841 | 0.0100 | 0.0635 | 0.1000 |

GSCI | −6.9948 | 0.0100 | −302.4211 | 0.0100 | 0.2696 | 0.1000 |

Gold | −6.4954 | 0.0100 | −415.0716 | 0.0100 | 0.3922 | 0.0805 |

Gas | −8.7395 | 0.0100 | −349.5609 | 0.0100 | 0.0580 | 0.1000 |

Coal | −7.1172 | 0.0100 | −276.9986 | 0.0100 | 0.0523 | 0.1000 |

Fut | −7.8003 | 0.0100 | −239.2251 | 0.0100 | 0.0533 | 0.1000 |

OI | −7.6357 | 0.0100 | −268.1572 | 0.0100 | 0.0757 | 0.1000 |

T | −3.1799 | 0.0915 | −43.9459 | 0.0100 | 4.4577 | 0.0100 |

**Table 4.**In-sample forecast accuracy for BSR models with the different number of linear components k.

RMSE | MAE | MAPE | MASE | |
---|---|---|---|---|

k = 1 | 2.4903 | 1.3372 | 0.0637 | 1.1415 |

k = 2 | 2.4850 | 1.3436 | 0.0641 | 1.1470 |

k = 3 | 2.2150 | 1.1243 | 0.0559 | 0.9598 |

k = 4 | 2.3694 | 1.2805 | 0.0623 | 1.0931 |

k = 5 | 2.4801 | 1.3126 | 0.0624 | 1.1206 |

k = 6 | 2.4747 | 1.3195 | 0.0629 | 1.1265 |

k = 7 | 2.2079 | 1.1431 | 0.0560 | 0.9758 |

k = 8 | 2.3960 | 1.2927 | 0.0619 | 1.1036 |

k = 9 | 2.2272 | 1.0441 | 0.0507 | 0.8913 |

k = 10 | 2.2456 | 1.0549 | 0.0509 | 0.9006 |

k = 11 | 2.3895 | 1.2730 | 0.0615 | 1.0868 |

k = 12 | 2.3760 | 1.2859 | 0.0621 | 1.0977 |

RMSE | MAE | MAPE | MASE | |
---|---|---|---|---|

BSR rec | 5.0292 | 3.5842 | 0.0702 | 0.9686 |

BSR av MSE rec | 4.8708 | 3.5061 | 0.0697 | 0.9475 |

BSR av EW rec | 4.7540 | 3.4403 | 0.0683 | 0.9297 |

BSR av LL rec | 4.8259 | 3.4482 | 0.0699 | 0.9319 |

GP rec | 4.3060 | 3.0918 | 0.0607 | 0.8355 |

GP av MSE rec | 4.2611 | 3.0553 | 0.0609 | 0.8257 |

GP av EW rec | 4.4706 | 3.1509 | 0.0634 | 0.8515 |

BSR fix | 3,062,769,697.4493 | 180,479,519.6051 | 6,319,413.9686 | 48,773,940.2698 |

BSR av MSE fix | 18.1086 | 4.7788 | 0.0917 | 1.2914 |

BSR av EW fix | 161.7497 | 16.5085 | 0.4792 | 4.4614 |

BSR av LL fix | 17.7764 | 5.7118 | 0.1048 | 1.5436 |

GP fix | 4.5174 | 3.2454 | 0.0631 | 0.8771 |

GP av MSE fix | 4.4448 | 3.1720 | 0.0621 | 0.8572 |

GP av EW fix | 4.6365 | 3.2708 | 0.0654 | 0.8839 |

DMA | 4.3468 | 3.1198 | 0.0595 | 0.8431 |

BMA | 4.2239 | 3.0905 | 0.0604 | 0.8352 |

DMA 1V | 4.2812 | 3.0745 | 0.0606 | 0.8309 |

DMS 1V | 4.2784 | 3.0706 | 0.0605 | 0.8298 |

BMA 1V | 4.2532 | 3.0582 | 0.0605 | 0.8265 |

BMS 1V | 4.2532 | 3.0582 | 0.0605 | 0.8265 |

LASSO | 4.4275 | 3.1413 | 0.0627 | 0.8489 |

RIDGE | 4.5231 | 3.2073 | 0.0634 | 0.8668 |

EN | 4.4654 | 3.1804 | 0.0633 | 0.8595 |

B-LASSO | 4.3256 | 3.1180 | 0.0615 | 0.8426 |

B-RIDGRE | 4.3968 | 3.1407 | 0.0620 | 0.8488 |

LARS | 4.5037 | 3.2205 | 0.0640 | 0.8703 |

TVP | 4.7444 | 3.3103 | 0.0658 | 0.8946 |

TVP f | 4.9461 | 3.3824 | 0.0658 | 0.9141 |

ARIMA | 5.0009 | 3.6229 | 0.0727 | 0.9791 |

HA | 33.2715 | 24.5524 | 0.3751 | 6.6352 |

NAIVE | 5.2934 | 3.7003 | 0.0738 | 1.0000 |

DM Stat. | DM p-Value | DM Stat. | DM p-Value | |
---|---|---|---|---|

BSR rec | 3.1957 | 0.0008 | ||

BSR av MSE rec | 2.8222 | 0.0026 | −0.6487 | 0.2585 |

BSR av EW rec | 2.7703 | 0.0030 | −1.3763 | 0.0849 |

BSR av LL rec | 2.8533 | 0.0023 | −0.9241 | 0.1781 |

GP rec | 0.9791 | 0.1642 | −2.8565 | 0.0023 |

GP av MSE rec | 0.4253 | 0.3355 | −2.9453 | 0.0017 |

GP av EW rec | 1.6761 | 0.0474 | −2.3154 | 0.0106 |

BSR fix | 1.0000 | 0.1591 | 1.0000 | 0.8409 |

BSR av MSE fix | 1.0501 | 0.1473 | 1.0248 | 0.8468 |

BSR av EW fix | 1.1502 | 0.1255 | 1.1499 | 0.8744 |

BSR av LL fix | 1.2209 | 0.1116 | 1.1888 | 0.8822 |

GP fix | 1.8443 | 0.0331 | −1.7019 | 0.0449 |

GP av MSE fix | 1.5280 | 0.0638 | −1.9627 | 0.0253 |

GP av EW fix | 2.3778 | 0.0090 | −1.4804 | 0.0699 |

DMA | 1.0547 | 0.1462 | −2.5925 | 0.0050 |

BMA | −3.1957 | 0.0008 | ||

DMA 1V | 0.5766 | 0.2823 | −2.9581 | 0.0017 |

DMS 1V | 0.5484 | 0.2919 | −2.9668 | 0.0016 |

BMA 1V | 0.3365 | 0.3684 | −3.0684 | 0.0012 |

BMS 1V | 0.3365 | 0.3684 | −3.0684 | 0.0012 |

LASSO | 1.6971 | 0.0454 | −2.3414 | 0.0099 |

RIDGE | 2.2703 | 0.0120 | −1.9027 | 0.0290 |

EN | 1.9670 | 0.0251 | −2.1973 | 0.0144 |

B-LASSO | 1.3271 | 0.0928 | −2.8436 | 0.0024 |

B-RIDGRE | 1.7355 | 0.0419 | −2.4972 | 0.0065 |

LARS | 2.2660 | 0.0121 | −2.1849 | 0.0149 |

TVP | 2.9316 | 0.0018 | −0.9431 | 0.1732 |

TVP f | 3.3510 | 0.0005 | −0.2645 | 0.3958 |

ARIMA | 3.3434 | 0.0005 | −0.1295 | 0.4485 |

HA | 11.2234 | 0.0000 | 11.1954 | 1.0000 |

NAIVE | 3.5068 | 0.0003 | 1.0147 | 0.8444 |

DM Stat. | DM p-Value | |
---|---|---|

BSR | −0.6487 | 0.7415 |

BSR av MSE | −0.5990 | 0.7252 |

BSR av EW | −2.6759 | 0.9961 |

BSR av LL | 1.0000 | 0.1591 |

GP | 1.1502 | 0.1255 |

GP av MSE | 1.9329 | 0.0271 |

GP av EW | 2.0442 | 0.0209 |

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**MDPI and ACS Style**

Drachal, K.
Forecasting the Crude Oil Spot Price with Bayesian Symbolic Regression. *Energies* **2023**, *16*, 4.
https://doi.org/10.3390/en16010004

**AMA Style**

Drachal K.
Forecasting the Crude Oil Spot Price with Bayesian Symbolic Regression. *Energies*. 2023; 16(1):4.
https://doi.org/10.3390/en16010004

**Chicago/Turabian Style**

Drachal, Krzysztof.
2023. "Forecasting the Crude Oil Spot Price with Bayesian Symbolic Regression" *Energies* 16, no. 1: 4.
https://doi.org/10.3390/en16010004